Unified neutral theory of biodiversity

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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Theory of evolutionary biology</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox vcard"><caption class="infobox-title" style="font-size:125%; font-style:italic; padding-bottom:0.2em;">The Unified Neutral Theory of Biodiversity and Biogeography <span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Unified+Neutral+Theory+of+Biodiversity+and+Biogeography&rft.author=%5B%5BStephen+P.+Hubbell%5D%5D&rft.date=2001&rft.pub=%5B%5BPrinceton+University+Press%5D%5D&rft.place=%5B%5BUnited+States%5D%5D&rft.pages=375&rft.series=Monographs+in+Population+Biology"></span></caption><tbody><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Hubbell_Unified_Neutral_Theory_Cover.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/2/2a/Hubbell_Unified_Neutral_Theory_Cover.jpg/150px-Hubbell_Unified_Neutral_Theory_Cover.jpg" decoding="async" width="150" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/2/2a/Hubbell_Unified_Neutral_Theory_Cover.jpg/225px-Hubbell_Unified_Neutral_Theory_Cover.jpg 1.5x, //upload.wikimedia.org/wikipedia/en/2/2a/Hubbell_Unified_Neutral_Theory_Cover.jpg 2x" data-file-width="259" data-file-height="400" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Author</th><td class="infobox-data"><a href="/wiki/Stephen_P._Hubbell" title="Stephen P. Hubbell">Stephen P. Hubbell</a></td></tr><tr><th scope="row" class="infobox-label">Language</th><td class="infobox-data">English</td></tr><tr><th scope="row" class="infobox-label">Series</th><td class="infobox-data">Monographs in Population Biology</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Release number</div></th><td class="infobox-data">32</td></tr><tr><th scope="row" class="infobox-label">Publisher</th><td class="infobox-data"><a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Publication date</div></th><td class="infobox-data">2001</td></tr><tr><th scope="row" class="infobox-label">Publication place</th><td class="infobox-data"><a href="/wiki/United_States" title="United States">United States</a></td></tr><tr><th scope="row" class="infobox-label">Pages</th><td class="infobox-data">375</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a></th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/Special:BookSources/0-691-02129-5" title="Special:BookSources/0-691-02129-5">0-691-02129-5</a></td></tr></tbody></table> <p>The <b>unified neutral theory of biodiversity and biogeography</b> (here <b>"Unified Theory"</b> or <b>"UNTB"</b>) is a theory and the title of a monograph by <a href="/wiki/Ecology" title="Ecology">ecologist</a> <a href="/wiki/Stephen_P._Hubbell" title="Stephen P. Hubbell">Stephen P. Hubbell</a>.<sup id="cite_ref-Hubbell2001_1-0" class="reference"><a href="#cite_note-Hubbell2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It aims to explain the diversity and relative abundance of species in ecological communities. Like other neutral theories of ecology, Hubbell assumes that the differences between members of an ecological community of <a href="/wiki/Trophic_level" title="Trophic level">trophically</a> similar species are "neutral", or irrelevant to their success. This implies that <a href="/wiki/Ecological_niche" title="Ecological niche">niche</a> differences do not influence abundance and the abundance of each species follows a <a href="/wiki/Random_walk" title="Random walk">random walk</a>.<sup id="cite_ref-McGill2003_2-0" class="reference"><a href="#cite_note-McGill2003-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The theory has sparked controversy,<sup id="cite_ref-Pocheville2015_3-0" class="reference"><a href="#cite_note-Pocheville2015-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Science_Daily_4-0" class="reference"><a href="#cite_note-Science_Daily-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Connolly2014_5-0" class="reference"><a href="#cite_note-Connolly2014-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and some authors consider it a more complex version of other null models that fit the data better.<sup id="cite_ref-Nee2003_6-0" class="reference"><a href="#cite_note-Nee2003-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>"Neutrality" means that at a given <a href="/wiki/Trophic_level" title="Trophic level">trophic level</a> in a <a href="/wiki/Food_web" title="Food web">food web</a>, species are equivalent in birth rates, death rates, dispersal rates and speciation rates, when measured on a per-capita basis.<sup id="cite_ref-Hubbell2005_7-0" class="reference"><a href="#cite_note-Hubbell2005-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This can be considered a <a href="/wiki/Null_hypothesis" title="Null hypothesis">null hypothesis</a> to <a href="/wiki/Ecological_niche" title="Ecological niche">niche theory</a>. Hubbell built on earlier neutral models, including <a href="/wiki/Robert_MacArthur" class="mw-redirect" title="Robert MacArthur">Robert MacArthur</a> and <a href="/wiki/E.O._Wilson" class="mw-redirect" title="E.O. Wilson">E.O. Wilson</a>'s theory of <a href="/wiki/Island_biogeography" class="mw-redirect" title="Island biogeography">island biogeography</a><sup id="cite_ref-Hubbell2001_1-1" class="reference"><a href="#cite_note-Hubbell2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Stephen_Jay_Gould" title="Stephen Jay Gould">Stephen Jay Gould</a>'s concepts of symmetry and null models.<sup id="cite_ref-Hubbell2005_7-1" class="reference"><a href="#cite_note-Hubbell2005-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>An "ecological community" is a group of trophically similar, <a href="/wiki/Sympatric" class="mw-redirect" title="Sympatric">sympatric</a> <a href="/wiki/Species" title="Species">species</a> that actually or potentially compete in a local area for the same or similar resources.<sup id="cite_ref-Hubbell2001_1-2" class="reference"><a href="#cite_note-Hubbell2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Under the Unified Theory, complex <a href="/wiki/Ecological" class="mw-redirect" title="Ecological">ecological</a> interactions are permitted among individuals of an ecological community (such as competition and cooperation), provided that all individuals obey the same rules. Asymmetric phenomena such as <a href="/wiki/Parasite" class="mw-redirect" title="Parasite">parasitism</a> and <a href="/wiki/Predator" class="mw-redirect" title="Predator">predation</a> are ruled out by the terms of reference; but cooperative strategies such as swarming, and negative interaction such as competing for limited food or light are allowed (so long as all individuals behave alike). </p><p>The theory predicts the existence of a fundamental biodiversity constant, conventionally written <i>θ</i>, that appears to govern species richness on a wide variety of spatial and temporal scales. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Saturation">Saturation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unified_neutral_theory_of_biodiversity&action=edit&section=1" title="Edit section: Saturation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although not strictly necessary for a neutral theory, many <a href="/wiki/Stochastic" title="Stochastic">stochastic</a> models of biodiversity assume a fixed, finite community size (total number of individual organisms). There are unavoidable physical constraints on the total number of individuals that can be packed into a given space (although space <i>per se</i> isn't necessarily a resource, it is often a useful surrogate variable for a limiting resource that is distributed over the landscape; examples would include <a href="/wiki/Sunlight" title="Sunlight">sunlight</a> or hosts, in the case of parasites). </p><p>If a wide range of species are considered (say, <a href="/wiki/Giant_sequoia" class="mw-redirect" title="Giant sequoia">giant sequoia</a> trees and <a href="/wiki/Duckweed" class="mw-redirect" title="Duckweed">duckweed</a>, two species that have very different saturation densities), then the assumption of constant community size might not be very good, because density would be higher if the smaller species were monodominant. Because the Unified Theory refers only to communities of trophically similar, competing species, it is unlikely that population density will vary too widely from one place to another. </p><p>Hubbell considers the fact that community sizes are constant and interprets it as a general principle: <i>large landscapes are always biotically saturated with individuals</i>. Hubbell thus treats communities as being of a fixed number of individuals, usually denoted by <i>J</i>. </p><p>Exceptions to the saturation principle include disturbed ecosystems such as the <a href="/wiki/Serengeti" title="Serengeti">Serengeti</a>, where saplings are trampled by <a href="/wiki/Elephant" title="Elephant">elephants</a> and <a href="/wiki/Blue_wildebeest" title="Blue wildebeest">Blue wildebeests</a>; or <a href="/wiki/Garden" title="Garden">gardens</a>, where certain species are systematically removed. </p> <div class="mw-heading mw-heading3"><h3 id="Species_abundances">Species abundances</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unified_neutral_theory_of_biodiversity&action=edit&section=2" title="Edit section: Species abundances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When abundance data on natural populations are collected, two observations are almost universal: </p> <ul><li>The most common species accounts for a substantial fraction of the individuals sampled;</li> <li>A substantial fraction of the species sampled are very rare. Indeed, a substantial fraction of the species sampled are singletons, that is, species which are sufficiently rare for only a single individual to have been sampled.</li></ul> <p>Such observations typically generate a large number of questions. Why are the rare species rare? Why is the most abundant species so much more abundant than the median species abundance? </p><p>A non neutral explanation for the rarity of rare species might suggest that rarity is a result of poor adaptation to local conditions. The UNTB suggests that it is not necessary to invoke adaptation or niche differences because neutral dynamics alone can generate such patterns. </p><p><a href="/wiki/Species_composition" class="mw-redirect" title="Species composition">Species composition</a> in any community will change randomly with time. Any particular abundance structure will have an associated probability. The UNTB predicts that the probability of a community of <i>J</i> individuals composed of <i>S</i> distinct species with abundances <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee784b70e772f55ede5e6e0bdc929994bff63413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{1}}"></span> for species 1, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840e456e3058bc0be28e5cf653b170cdbfcc3be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{2}}"></span> for species 2, and so on up to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0b01901c8174ac451dfc68b1bc2052ecbf5ca8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.687ex; height:2.009ex;" alt="{\displaystyle n_{S}}"></span> for species <i>S</i> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(n_{1},n_{2},\ldots ,n_{S}|\theta ,J)={\frac {J!\theta ^{S}}{1^{\phi _{1}}2^{\phi _{2}}\cdots J^{\phi _{J}}\phi _{1}!\phi _{2}!\cdots \phi _{J}!\Pi _{k=1}^{J}(\theta +k-1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo>,</mo> <mi>J</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>J</mi> <mo>!</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> </mrow> <mrow> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>⋯</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> </mrow> </msup> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>!</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>!</mo> <mo>⋯</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> <mo>!</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(n_{1},n_{2},\ldots ,n_{S}|\theta ,J)={\frac {J!\theta ^{S}}{1^{\phi _{1}}2^{\phi _{2}}\cdots J^{\phi _{J}}\phi _{1}!\phi _{2}!\cdots \phi _{J}!\Pi _{k=1}^{J}(\theta +k-1)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1222d5cfc5daadcfe3723b8c1502edc731569652" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:68.796ex; height:7.009ex;" alt="{\displaystyle \Pr(n_{1},n_{2},\ldots ,n_{S}|\theta ,J)={\frac {J!\theta ^{S}}{1^{\phi _{1}}2^{\phi _{2}}\cdots J^{\phi _{J}}\phi _{1}!\phi _{2}!\cdots \phi _{J}!\Pi _{k=1}^{J}(\theta +k-1)}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =2J\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mi>J</mi> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =2J\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07acbb770a6c07d76a3ce0a2e9558b974296c1f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.055ex; height:2.176ex;" alt="{\displaystyle \theta =2J\nu }"></span> is the fundamental biodiversity number (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> is the speciation rate), and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0182dbf29b54844c92fd9b0311778a02a38398ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.185ex; height:2.509ex;" alt="{\displaystyle \phi _{i}}"></span> is the number of species that have <i>i</i> individuals in the sample. </p><p>This equation shows that the UNTB implies a nontrivial dominance-diversity equilibrium between speciation and extinction. </p><p>As an example, consider a community with 10 individuals and three species "a", "b", and "c" with abundances 3, 6 and 1 respectively. Then the formula above would allow us to assess the <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood</a> of different values of <i>θ</i>. There are thus <i>S</i> = 3 species and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{1}=\phi _{3}=\phi _{6}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{1}=\phi _{3}=\phi _{6}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd207d6622205623a367045dbeb7dba5d99f25f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.777ex; height:2.509ex;" alt="{\displaystyle \phi _{1}=\phi _{3}=\phi _{6}=1}"></span>, all other <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>'s being zero. The formula would give </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(3,6,1|\theta ,10)={\frac {10!\theta ^{3}}{1^{1}\cdot 3^{1}\cdot 6^{1}\cdot 1!1!1!\cdot \theta (\theta +1)(\theta +2)\cdots (\theta +9)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo>,</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>10</mn> <mo>!</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mn>1</mn> <mo>!</mo> <mn>1</mn> <mo>!</mo> <mn>1</mn> <mo>!</mo> <mo>⋅</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mn>9</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(3,6,1|\theta ,10)={\frac {10!\theta ^{3}}{1^{1}\cdot 3^{1}\cdot 6^{1}\cdot 1!1!1!\cdot \theta (\theta +1)(\theta +2)\cdots (\theta +9)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae148ac4e0d7d9ba99075bd311a169e9503b257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:62.98ex; height:6.676ex;" alt="{\displaystyle \Pr(3,6,1|\theta ,10)={\frac {10!\theta ^{3}}{1^{1}\cdot 3^{1}\cdot 6^{1}\cdot 1!1!1!\cdot \theta (\theta +1)(\theta +2)\cdots (\theta +9)}}}"></span></dd></dl> <p>which could be maximized to yield an estimate for <i>θ</i> (in practice, <a href="/wiki/Numerical_methods" class="mw-redirect" title="Numerical methods">numerical methods</a> are used). The <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimate for <i>θ</i> is about 1.1478. </p><p>We could have labelled the species another way and counted the abundances being 1,3,6 instead (or 3,1,6, etc. etc.). Logic tells us that the probability of observing a pattern of abundances will be the same observing any <a href="/wiki/Permutation" title="Permutation">permutation</a> of those abundances. Here we would have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(3;3,6,1)=\Pr(3;1,3,6)=\Pr(3;3,1,6)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>;</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>;</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(3;3,6,1)=\Pr(3;1,3,6)=\Pr(3;3,1,6)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0baef9e08dfeb740df859ea85648c9e12b67ef7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.363ex; height:2.843ex;" alt="{\displaystyle \Pr(3;3,6,1)=\Pr(3;1,3,6)=\Pr(3;3,1,6)}"></span></dd></dl> <p>and so on. </p><p>To account for this, it is helpful to consider only ranked abundances (that is, to sort the abundances before inserting into the formula). A ranked dominance-diversity configuration is usually written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(S;r_{1},r_{2},\ldots ,r_{s},0,\ldots ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>;</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(S;r_{1},r_{2},\ldots ,r_{s},0,\ldots ,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1372888224b2350c49251f24afafb0c7b5005a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.844ex; height:2.843ex;" alt="{\displaystyle \Pr(S;r_{1},r_{2},\ldots ,r_{s},0,\ldots ,0)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"></span> is the abundance of the <i>i</i>th most abundant species: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}"></span> is the abundance of the most abundant, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}"></span> the abundance of the second most abundant species, and so on. For convenience, the expression is usually "padded" with enough zeros to ensure that there are <i>J</i> species (the zeros indicating that the extra species have zero abundance). </p><p>It is now possible to determine the <a href="/wiki/Expected_value" title="Expected value">expected</a> abundance of the <i>i</i>th most abundant species: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(r_{i})=\sum _{k=1}^{C}r_{i}(k)\cdot \Pr(S;r_{1},r_{2},\ldots ,r_{s},0,\ldots ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>;</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(r_{i})=\sum _{k=1}^{C}r_{i}(k)\cdot \Pr(S;r_{1},r_{2},\ldots ,r_{s},0,\ldots ,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/554ad29c9cf700941661480b347b53b82f32cf6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.666ex; height:7.343ex;" alt="{\displaystyle E(r_{i})=\sum _{k=1}^{C}r_{i}(k)\cdot \Pr(S;r_{1},r_{2},\ldots ,r_{s},0,\ldots ,0)}"></span></dd></dl> <p>where <i>C</i> is the total number of configurations, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e014c51733e12d571f8fe8a7cd7b35bbd3ccce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.869ex; height:2.843ex;" alt="{\displaystyle r_{i}(k)}"></span> is the abundance of the <i>i</i>th ranked species in the <i>k</i>th configuration, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Pr(\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Pr(\ldots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41525fa397177652c5b1d5716412f1b78ff1cc18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.327ex; height:2.843ex;" alt="{\displaystyle Pr(\ldots )}"></span> is the dominance-diversity probability. This formula is difficult to manipulate mathematically, but relatively simple to simulate computationally. </p><p>The model discussed so far is a model of a regional community, which Hubbell calls the <a href="/wiki/Metacommunity" title="Metacommunity">metacommunity</a>. Hubbell also acknowledged that on a local scale, dispersal plays an important role. For example, seeds are more likely to come from nearby parents than from distant parents. Hubbell introduced the parameter m, which denotes the probability of immigration in the local community from the metacommunity. If m = 1, dispersal is unlimited; the local community is just a random sample from the metacommunity and the formulas above apply. If m < 1, dispersal is limited and the local community is a dispersal-limited sample from the metacommunity for which different formulas apply. </p><p>It has been shown<sup id="cite_ref-Volkov2003_8-0" class="reference"><a href="#cite_note-Volkov2003-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi _{n}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi _{n}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c31919df2ef7dc19285e43ce1dd05812ab02c11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.413ex; height:2.843ex;" alt="{\displaystyle \langle \phi _{n}\rangle }"></span>, the expected number of species with abundance n, may be calculated by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta {\frac {J!}{n!(J-n)!}}{\frac {\Gamma (\gamma )}{\Gamma (J+\gamma )}}\int _{y=0}^{\gamma }{\frac {\Gamma (n+y)}{\Gamma (1+y)}}{\frac {\Gamma (J-n+\gamma -y)}{\Gamma (\gamma -y)}}\exp(-y\theta /\gamma )\,dy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>J</mi> <mo>!</mo> </mrow> <mrow> <mi>n</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>J</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>J</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>J</mi> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mi>γ</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>y</mi> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>γ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta {\frac {J!}{n!(J-n)!}}{\frac {\Gamma (\gamma )}{\Gamma (J+\gamma )}}\int _{y=0}^{\gamma }{\frac {\Gamma (n+y)}{\Gamma (1+y)}}{\frac {\Gamma (J-n+\gamma -y)}{\Gamma (\gamma -y)}}\exp(-y\theta /\gamma )\,dy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7649e45b71cc6c99b0ba23536b62144b003ed1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:69.398ex; height:6.509ex;" alt="{\displaystyle \theta {\frac {J!}{n!(J-n)!}}{\frac {\Gamma (\gamma )}{\Gamma (J+\gamma )}}\int _{y=0}^{\gamma }{\frac {\Gamma (n+y)}{\Gamma (1+y)}}{\frac {\Gamma (J-n+\gamma -y)}{\Gamma (\gamma -y)}}\exp(-y\theta /\gamma )\,dy}"></span></dd></dl> <p>where <i>θ</i> is the fundamental biodiversity number, <i>J</i> the community size, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> is the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =(J-1)m/(1-m)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>J</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =(J-1)m/(1-m)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4cd30e89b327b76aae94d05787e717c2c7eeea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.7ex; height:2.843ex;" alt="{\displaystyle \gamma =(J-1)m/(1-m)}"></span>. This formula is an approximation. The correct formula is derived in a series of papers, reviewed and synthesized by Etienne and Alonso in 2005:<sup id="cite_ref-Etienne2005_9-0" class="reference"><a href="#cite_note-Etienne2005-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\theta }{(I)_{J}}}{J \choose n}\int _{0}^{1}(Ix)_{n}(I(1-x))_{J-n}{\frac {(1-x)^{\theta -1}}{x}}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mrow> <mo stretchy="false">(</mo> <mi>I</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>J</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>I</mi> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> <mo>−</mo> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\theta }{(I)_{J}}}{J \choose n}\int _{0}^{1}(Ix)_{n}(I(1-x))_{J-n}{\frac {(1-x)^{\theta -1}}{x}}\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a4bcbb5a3de0a73b73189d6c79123e48fd7b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.092ex; height:6.676ex;" alt="{\displaystyle {\frac {\theta }{(I)_{J}}}{J \choose n}\int _{0}^{1}(Ix)_{n}(I(1-x))_{J-n}{\frac {(1-x)^{\theta -1}}{x}}\,dx}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=(J-1)*m/(1-m)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>J</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=(J-1)*m/(1-m)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35670b5ca9579baab0057f048e1ad70c6ddbba0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.804ex; height:2.843ex;" alt="{\displaystyle I=(J-1)*m/(1-m)}"></span> is a parameter that measures dispersal limitation. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi _{n}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi _{n}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c31919df2ef7dc19285e43ce1dd05812ab02c11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.413ex; height:2.843ex;" alt="{\displaystyle \langle \phi _{n}\rangle }"></span> is zero for <i>n</i> > <i>J</i>, as there cannot be more species than individuals. </p><p>This formula is important because it allows a quick evaluation of the Unified Theory. It is not suitable for testing the theory. For this purpose, the appropriate likelihood function should be used. For the metacommunity this was given above. For the local community with dispersal limitation it is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(n_{1},n_{2},\ldots ,n_{S}|\theta ,m,J)={\frac {J!}{\prod _{i=1}^{S}n_{i}\prod _{j=1}^{J}\Phi _{j}!}}{\frac {\theta ^{S}}{(I)_{J}}}\sum _{A=S}^{J}K({\overrightarrow {D}},A){\frac {I^{A}}{(\theta )_{A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>J</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>J</mi> <mo>!</mo> </mrow> <mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </munderover> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>I</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>=</mo> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo>→</mo> </mover> </mrow> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(n_{1},n_{2},\ldots ,n_{S}|\theta ,m,J)={\frac {J!}{\prod _{i=1}^{S}n_{i}\prod _{j=1}^{J}\Phi _{j}!}}{\frac {\theta ^{S}}{(I)_{J}}}\sum _{A=S}^{J}K({\overrightarrow {D}},A){\frac {I^{A}}{(\theta )_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf58d7d84a9d6d33636086dbe148d985daf8767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:69.982ex; height:7.843ex;" alt="{\displaystyle \Pr(n_{1},n_{2},\ldots ,n_{S}|\theta ,m,J)={\frac {J!}{\prod _{i=1}^{S}n_{i}\prod _{j=1}^{J}\Phi _{j}!}}{\frac {\theta ^{S}}{(I)_{J}}}\sum _{A=S}^{J}K({\overrightarrow {D}},A){\frac {I^{A}}{(\theta )_{A}}}}"></span></dd></dl> <p>Here, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K({\overrightarrow {D}},A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo>→</mo> </mover> </mrow> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K({\overrightarrow {D}},A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c38c1a3116d2aa91b4852d82294ac409c2ae803d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.105ex; height:4.176ex;" alt="{\displaystyle K({\overrightarrow {D}},A)}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=S,...,J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>S</mi> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=S,...,J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c17cdcebec38a005ae78bcae42faa5c8570e05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.982ex; height:2.509ex;" alt="{\displaystyle A=S,...,J}"></span> are coefficients fully determined by the data, being defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K({\overrightarrow {D}},A):=\sum _{\{a_{1},...,a_{S}|\sum _{i=1}^{S}a_{i}=A\}}\prod _{i=1}^{S}{\frac {{\overline {s}}\left(n_{i},a_{i}\right){\overline {s}}\left(a_{i},1\right)}{{\overline {s}}\left(n_{i},1\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo>→</mo> </mover> </mrow> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K({\overrightarrow {D}},A):=\sum _{\{a_{1},...,a_{S}|\sum _{i=1}^{S}a_{i}=A\}}\prod _{i=1}^{S}{\frac {{\overline {s}}\left(n_{i},a_{i}\right){\overline {s}}\left(a_{i},1\right)}{{\overline {s}}\left(n_{i},1\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb9f48e19061db7a0fb0bbd31e8c03255804f86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:50.934ex; height:8.509ex;" alt="{\displaystyle K({\overrightarrow {D}},A):=\sum _{\{a_{1},...,a_{S}|\sum _{i=1}^{S}a_{i}=A\}}\prod _{i=1}^{S}{\frac {{\overline {s}}\left(n_{i},a_{i}\right){\overline {s}}\left(a_{i},1\right)}{{\overline {s}}\left(n_{i},1\right)}}}"></span></dd></dl> <p>This seemingly complicated formula involves <a href="/wiki/Stirling_number" title="Stirling number">Stirling numbers</a> and <a href="/wiki/Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbols</a>, but can be very easily calculated.<sup id="cite_ref-Etienne2005_9-1" class="reference"><a href="#cite_note-Etienne2005-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>An example of a species abundance curve can be found in Scientific American.<sup id="cite_ref-Scientific_American_10-0" class="reference"><a href="#cite_note-Scientific_American-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Stochastic_modelling_of_species_abundances">Stochastic modelling of species abundances</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unified_neutral_theory_of_biodiversity&action=edit&section=3" title="Edit section: Stochastic modelling of species abundances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>UNTB distinguishes between a dispersal-limited local community of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> and a so-called metacommunity from which species can (re)immigrate and which acts as a <a href="/wiki/Heat_bath" class="mw-redirect" title="Heat bath">heat bath</a> to the local community. The distribution of species in the metacommunity is given by a dynamic equilibrium of speciation and extinction. Both community dynamics are modelled by appropriate <a href="/wiki/Urn_problem" title="Urn problem">urn processes</a>, where each individual is represented by a ball with a color corresponding to its species. With a certain rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> randomly chosen individuals reproduce, i.e. add another ball of their own color to the urn. Since one basic assumption is saturation, this reproduction has to happen at the cost of another random individual from the urn which is removed. At a different rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> single individuals in the metacommunity are replaced by mutants of an entirely new species. Hubbell calls this simplified model for <a href="/wiki/Speciation" title="Speciation">speciation</a> a <a href="/wiki/Point_mutation" title="Point mutation">point mutation</a>, using the terminology of the <a href="/wiki/Neutral_theory_of_molecular_evolution" title="Neutral theory of molecular evolution">Neutral theory of molecular evolution</a>. The urn scheme for the metacommunity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7efad4293e290678ca0c5a02a7b467dc719e62d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.249ex; height:2.509ex;" alt="{\displaystyle J_{M}}"></span> individuals is the following. </p><p>At each time step take one of the two possible actions : </p> <ol><li>With probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\nu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaeed353c9065c7839c8880682b79c8290759753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.044ex; height:2.843ex;" alt="{\displaystyle (1-\nu )}"></span> draw an individual at random and replace another random individual from the urn with a copy of the first one.</li> <li>With probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> draw an individual and replace it with an individual of a new species.</li></ol> <p>The size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7efad4293e290678ca0c5a02a7b467dc719e62d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.249ex; height:2.509ex;" alt="{\displaystyle J_{M}}"></span> of the metacommunity does not change. This is a <a href="/wiki/Point_process" title="Point process">point process</a> in time. The length of the time steps is distributed exponentially. For simplicity one can assume that each time step is as long as the mean time between two changes which can be derived from the reproduction and mutation rates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span>. The probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> is given as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =\mu /(r+\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =\mu /(r+\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696fe99a5e7f1ed8e740060fb0315589e9bacf53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.995ex; height:2.843ex;" alt="{\displaystyle \nu =\mu /(r+\mu )}"></span>. </p><p>The species abundance distribution for this urn process is given by <a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens's sampling formula</a> which was originally derived in 1972 for the distribution of <a href="/wiki/Allele" title="Allele">alleles</a> under neutral mutations. The expected number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{M}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{M}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92adc62f60e2cbff87022fce9e25ab7747cdaf50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.588ex; height:2.843ex;" alt="{\displaystyle S_{M}(n)}"></span> of species in the metacommunity having exactly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> individuals is:<sup id="cite_ref-Vallade2003_11-0" class="reference"><a href="#cite_note-Vallade2003-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{M}(n)={\frac {\theta }{n}}{\frac {\Gamma (J_{M}+1)\Gamma (J_{M}+\theta -n)}{\Gamma (J_{M}+1-n)\Gamma (J_{M}+\theta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mi>n</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>+</mo> <mi>θ</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{M}(n)={\frac {\theta }{n}}{\frac {\Gamma (J_{M}+1)\Gamma (J_{M}+\theta -n)}{\Gamma (J_{M}+1-n)\Gamma (J_{M}+\theta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f76bbfa7e924fb7a8a9db4cc25862aac9ae94777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.945ex; height:6.509ex;" alt="{\displaystyle S_{M}(n)={\frac {\theta }{n}}{\frac {\Gamma (J_{M}+1)\Gamma (J_{M}+\theta -n)}{\Gamma (J_{M}+1-n)\Gamma (J_{M}+\theta )}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =(J_{M}-1)\nu /(1-\nu )\approx J_{M}\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =(J_{M}-1)\nu /(1-\nu )\approx J_{M}\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae0496afd37afeaf8ac912cbd2a481c1fef1d89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.269ex; height:2.843ex;" alt="{\displaystyle \theta =(J_{M}-1)\nu /(1-\nu )\approx J_{M}\nu }"></span> is called the fundamental biodiversity number. For large metacommunities and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\ll J_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≪</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\ll J_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4a587b96ddd85d3690d3e05d9d1c25444b9da0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.258ex; height:2.509ex;" alt="{\displaystyle n\ll J_{M}}"></span> one recovers the <a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">Fisher Log-Series</a> as species distribution. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{M}(n)\approx {\frac {\theta }{n}}\left({\frac {J_{M}}{J_{M}+\theta }}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mi>n</mi> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>+</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{M}(n)\approx {\frac {\theta }{n}}\left({\frac {J_{M}}{J_{M}+\theta }}\right)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6406cd1affb5c15c3debb75d9381411972b69b4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.573ex; height:6.176ex;" alt="{\displaystyle S_{M}(n)\approx {\frac {\theta }{n}}\left({\frac {J_{M}}{J_{M}+\theta }}\right)^{n}}"></span></dd></dl> <p>The urn scheme for the local community of fixed size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> is very similar to the one for the metacommunity. </p><p>At each time step take one of the two actions : </p> <ol><li>With probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-m)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-m)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1859b9045c66dae8c76bbf040539de7f8f573c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.853ex; height:2.843ex;" alt="{\displaystyle (1-m)}"></span> draw an individual at random and replace another random individual from the urn with a copy of the first one.</li> <li>With probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> replace a random individual with an immigrant drawn from the metacommunity.</li></ol> <p>The metacommunity is changing on a much larger timescale and is assumed to be fixed during the evolution of the local community. The resulting distribution of species in the local community and expected values depend on four parameters, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7efad4293e290678ca0c5a02a7b467dc719e62d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.249ex; height:2.509ex;" alt="{\displaystyle J_{M}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> (or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>) and are derived by Etienne and Alonso (2005),<sup id="cite_ref-Etienne2005_9-2" class="reference"><a href="#cite_note-Etienne2005-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> including several simplifying limit cases like the one presented in the previous section (there called <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi _{n}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi _{n}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c31919df2ef7dc19285e43ce1dd05812ab02c11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.413ex; height:2.843ex;" alt="{\displaystyle \langle \phi _{n}\rangle }"></span>). The parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is a dispersal parameter. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6100c5ebd48c6fd848709f2be624465203eb173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=1}"></span> then the local community is just a sample from the metacommunity. For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57f21007575fd03e3be0da20af34d25829cc9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}"></span> the local community is completely isolated from the metacommunity and all species will go extinct except one. This case has been analyzed by Hubbell himself.<sup id="cite_ref-Hubbell2001_1-3" class="reference"><a href="#cite_note-Hubbell2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<m<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>m</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<m<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b1f1a9de0e4a6ccce84c1b64c2b818992d91e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.562ex; height:2.176ex;" alt="{\displaystyle 0<m<1}"></span> is characterized by a <a href="/wiki/Unimodal" class="mw-redirect" title="Unimodal">unimodal</a> species distribution in a <a href="/w/index.php?title=Preston_Diagram&action=edit&redlink=1" class="new" title="Preston Diagram (page does not exist)">Preston Diagram</a> and often fitted by a <a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal distribution</a>. This is understood as an intermediate state between domination of the most common species and a sampling from the metacommunity, where singleton species are most abundant. UNTB thus predicts that in dispersal limited communities rare species become even rarer. The log-normal distribution describes the maximum and the abundance of common species very well but underestimates the number of very rare species considerably which becomes only apparent for very large sample sizes.<sup id="cite_ref-Hubbell2001_1-4" class="reference"><a href="#cite_note-Hubbell2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Species-area_relationships">Species-area relationships</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unified_neutral_theory_of_biodiversity&action=edit&section=4" title="Edit section: Species-area relationships"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Unified Theory unifies <i>biodiversity</i>, as measured by species-abundance curves, with <i>biogeography</i>, as measured by species-area curves. Species-area relationships show the rate at which species diversity increases with area. The topic is of great interest to conservation biologists in the design of reserves, as it is often desired to harbour as many species as possible. </p><p>The most commonly encountered relationship is the power law given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=cA^{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>c</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=cA^{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dfa07a4a2af18740676615e17f981565172064a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.349ex; height:2.343ex;" alt="{\displaystyle S=cA^{z}}"></span></dd></dl> <p>where <i>S</i> is the number of species found, <i>A</i> is the area sampled, and <i>c</i> and <i>z</i> are constants. This relationship, with different constants, has been found to fit a wide range of empirical data. </p><p>From the perspective of Unified Theory, it is convenient to consider <i>S</i> as a function of total community size <i>J</i>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=kJ^{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>k</mi> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=kJ^{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0258b7f751b289703e55cbfcb3d337a3285e7633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.336ex; height:2.343ex;" alt="{\displaystyle S=kJ^{z}}"></span> for some constant <i>k</i>, and if this relationship were exactly true, the species area line would be straight on log scales. It is typically found that the curve is not straight, but the slope changes from being steep at small areas, shallower at intermediate areas, and steep at the largest areas. </p><p>The formula for <a href="/wiki/Species_composition" class="mw-redirect" title="Species composition">species composition</a> may be used to calculate the expected number of species present in a community under the assumptions of the Unified Theory. In symbols </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left\{S|\theta ,J\right\}={\frac {\theta }{\theta }}+{\frac {\theta }{\theta +1}}+{\frac {\theta }{\theta +2}}+\cdots +{\frac {\theta }{\theta +J-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>{</mo> <mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo>,</mo> <mi>J</mi> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mi>θ</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mrow> <mi>θ</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mrow> <mi>θ</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mrow> <mi>θ</mi> <mo>+</mo> <mi>J</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left\{S|\theta ,J\right\}={\frac {\theta }{\theta }}+{\frac {\theta }{\theta +1}}+{\frac {\theta }{\theta +2}}+\cdots +{\frac {\theta }{\theta +J-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6006998fdd1523c48ff9ce2a002bc64250a1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:51.439ex; height:5.676ex;" alt="{\displaystyle E\left\{S|\theta ,J\right\}={\frac {\theta }{\theta }}+{\frac {\theta }{\theta +1}}+{\frac {\theta }{\theta +2}}+\cdots +{\frac {\theta }{\theta +J-1}}}"></span></dd></dl> <p>where θ is the fundamental biodiversity number. This formula specifies the expected number of species sampled in a community of size <i>J</i>. The last term, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta /(\theta +J-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>J</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta /(\theta +J-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d031b5ae296653b5560099f31547d45b20456c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.467ex; height:2.843ex;" alt="{\displaystyle \theta /(\theta +J-1)}"></span>, is the expected number of <i>new</i> species encountered when adding one new individual to the community. This is an increasing function of θ and a decreasing function of <i>J</i>, as expected. </p><p>By making the substitution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=\rho A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mi>ρ</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=\rho A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a40e9a9a920b7daf46823ea702acc063d39c51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.515ex; height:2.676ex;" alt="{\displaystyle J=\rho A}"></span> (see section on saturation above), then the expected number of species becomes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma \theta /(\theta +\rho A-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>ρ</mi> <mi>A</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma \theta /(\theta +\rho A-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a2ac7ff81472c241764d86a5d136235314d8a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.619ex; height:2.843ex;" alt="{\displaystyle \Sigma \theta /(\theta +\rho A-1)}"></span>. </p><p>The formula above may be approximated to an <a href="/wiki/Integral" title="Integral">integral</a> giving </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\theta )=1+\theta \ln \left(1+{\frac {J-1}{\theta }}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>θ</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>J</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>θ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\theta )=1+\theta \ln \left(1+{\frac {J-1}{\theta }}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c9c20dd9ed5b8436cff9c0f85bf60521bd6e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.298ex; height:6.176ex;" alt="{\displaystyle S(\theta )=1+\theta \ln \left(1+{\frac {J-1}{\theta }}\right).}"></span></dd></dl> <p>This formulation is predicated on a random placement of individuals. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unified_neutral_theory_of_biodiversity&action=edit&section=5" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the following (synthetic) dataset of 27 individuals: </p><p>a,a,a,a,a,a,a,a,a,a,b,b,b,b,c,c,c,c,d,d,d,d,e,f,g,h,i </p><p>There are thus 27 individuals of 9 species ("a" to "i") in the sample. Tabulating this would give: </p> <pre> a b c d e f g h i 10 4 4 4 1 1 1 1 1 </pre> <p>indicating that species "a" is the most abundant with 10 individuals and species "e" to "i" are singletons. Tabulating the table gives: </p> <pre>species abundance 1 2 3 4 5 6 7 8 9 10 number of species 5 0 0 3 0 0 0 0 0 1 </pre> <p>On the second row, the 5 in the first column means that five species, species "e" through "i", have abundance one. The following two zeros in columns 2 and 3 mean that zero species have abundance 2 or 3. The 3 in column 4 means that three species, species "b", "c", and "d", have abundance four. The final 1 in column 10 means that one species, species "a", has abundance 10. </p><p>This type of dataset is typical in biodiversity studies. Observe how more than half the biodiversity (as measured by species count) is due to singletons. </p><p>For real datasets, the species abundances are binned into logarithmic categories, usually using base 2, which gives bins of abundance 0–1, abundance 1–2, abundance 2–4, abundance 4–8, etc. Such abundance classes are called <i>octaves</i>; early developers of this concept included <a href="/wiki/Frank_W._Preston" title="Frank W. Preston">F. W. Preston</a> and histograms showing number of species as a function of abundance octave are known as <a href="/w/index.php?title=Preston_diagram&action=edit&redlink=1" class="new" title="Preston diagram (page does not exist)">Preston diagrams</a>. </p><p>These bins are not mutually exclusive: a species with abundance 4, for example, could be considered as lying in the 2-4 abundance class or the 4-8 abundance class. Species with an abundance of an exact power of 2 (i.e. 2,4,8,16, etc.) are conventionally considered as having 50% membership in the lower abundance class 50% membership in the upper class. Such species are thus considered to be evenly split between the two adjacent classes (apart from singletons which are classified into the rarest category). Thus in the example above, the Preston abundances would be </p> <pre>abundance class 1 1-2 2-4 4-8 8-16 species 5 0 1.5 1.5 1 </pre> <p>The three species of abundance four thus appear, 1.5 in abundance class 2–4, and 1.5 in 4–8. </p><p>The above method of analysis cannot account for species that are unsampled: that is, species sufficiently rare to have been recorded zero times. Preston diagrams are thus truncated at zero abundance. <a href="/wiki/Frank_W._Preston" title="Frank W. Preston">Preston</a> called this the <i>veil line</i> and noted that the cutoff point would move as more individuals are sampled. </p> <div class="mw-heading mw-heading2"><h2 id="Dynamics">Dynamics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unified_neutral_theory_of_biodiversity&action=edit&section=6" title="Edit section: Dynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All biodiversity patterns previously described are related to time-independent quantities. For biodiversity evolution and species preservation, it is crucial to compare the dynamics of ecosystems with models (Leigh, 2007). An easily accessible index of the underlying evolution is the so-called species turnover distribution (STD), defined as the probability P(r,t) that the population of any species has varied by a fraction r after a given time t. </p><p>A neutral model that can analytically predict both the relative species abundance (RSA) at steady-state and the STD at time t has been presented in Azaele et al. (2006).<sup id="cite_ref-Azaele2006_12-0" class="reference"><a href="#cite_note-Azaele2006-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Within this framework the population of any species is represented by a continuous (random) variable x, whose evolution is governed by the following Langevin equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {x}}=b-x/\tau +{\sqrt {Dx}}\xi (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>b</mi> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>D</mi> <mi>x</mi> </msqrt> </mrow> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}}=b-x/\tau +{\sqrt {Dx}}\xi (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c3bafddf15e39d2926d050380dd98c5f165339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.669ex; height:3.176ex;" alt="{\displaystyle {\dot {x}}=b-x/\tau +{\sqrt {Dx}}\xi (t)}"></span></dd></dl> <p>where b is the immigration rate from a large regional community, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x/\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x/\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91133e55508aa84425bea8eec175c77617b23c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.502ex; height:2.843ex;" alt="{\displaystyle -x/\tau }"></span> represents competition for finite resources and D is related to demographic stochasticity; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d14b0b0146f7196ff6234dd7fc36608035da5b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.679ex; height:2.843ex;" alt="{\displaystyle \xi (t)}"></span> is a Gaussian white noise. The model can also be derived as a continuous approximation of a master equation, where birth and death rates are independent of species, and predicts that at steady-state the RSA is simply a gamma distribution. </p><p>From the exact time-dependent solution of the previous equation, one can exactly calculate the STD at time t under stationary conditions: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(r,t)=A{\frac {\lambda +1}{\lambda }}{\frac {(e^{t/\tau })^{b/2D}}{1-e^{-t/\tau }}}\left({\frac {\sinh({\frac {t}{2\tau }})}{\lambda }}\right)^{{\frac {b}{D}}+1}\left({\frac {4\lambda ^{2}}{(\lambda +1)^{2}e^{t/\tau }-4\lambda }}\right)^{{\frac {b}{D}}+{\frac {1}{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>λ</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>λ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>D</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mrow> <mn>2</mn> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>D</mi> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>D</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(r,t)=A{\frac {\lambda +1}{\lambda }}{\frac {(e^{t/\tau })^{b/2D}}{1-e^{-t/\tau }}}\left({\frac {\sinh({\frac {t}{2\tau }})}{\lambda }}\right)^{{\frac {b}{D}}+1}\left({\frac {4\lambda ^{2}}{(\lambda +1)^{2}e^{t/\tau }-4\lambda }}\right)^{{\frac {b}{D}}+{\frac {1}{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d43353fe0fdd2f58b4bbc34a05a636880de6a843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:72.219ex; height:8.843ex;" alt="{\displaystyle P(r,t)=A{\frac {\lambda +1}{\lambda }}{\frac {(e^{t/\tau })^{b/2D}}{1-e^{-t/\tau }}}\left({\frac {\sinh({\frac {t}{2\tau }})}{\lambda }}\right)^{{\frac {b}{D}}+1}\left({\frac {4\lambda ^{2}}{(\lambda +1)^{2}e^{t/\tau }-4\lambda }}\right)^{{\frac {b}{D}}+{\frac {1}{2}}}.}"></span></dd></dl> <p>This formula provides good fits of data collected in the Barro Colorado tropical forest from 1990 to 2000. From the best fit one can estimate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> ~ 3500 years with a broad uncertainty due to the relative short time interval of the sample. This parameter can be interpreted as the relaxation time of the system, i.e. the time the system needs to recover from a perturbation of species distribution. In the same framework, the estimated mean species lifetime is very close to the fitted temporal scale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span>. This suggests that the neutral assumption could correspond to a scenario in which species originate and become extinct on the same timescales of fluctuations of the whole ecosystem. </p> <div class="mw-heading mw-heading2"><h2 id="Testing">Testing</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unified_neutral_theory_of_biodiversity&action=edit&section=7" title="Edit section: Testing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The theory has provoked much controversy as it "abandons" the role of ecology when modelling ecosystems.<sup id="cite_ref-Ricklefs2006_13-0" class="reference"><a href="#cite_note-Ricklefs2006-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The theory has been criticized as it requires an equilibrium, yet climatic and geographical conditions are thought to change too frequently for this to be attained.<sup id="cite_ref-Ricklefs2006_13-1" class="reference"><a href="#cite_note-Ricklefs2006-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Tests on bird and tree abundance data demonstrate that the theory is usually a poorer match to the data than alternative null hypotheses that use fewer parameters (a log-normal model with two tunable parameters, compared to the neutral theory's three<sup id="cite_ref-Nee2003_6-1" class="reference"><a href="#cite_note-Nee2003-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>), and are thus more parsimonious.<sup id="cite_ref-McGill2003_2-1" class="reference"><a href="#cite_note-McGill2003-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The theory also fails to describe coral reef communities, studied by <a href="/wiki/Maria_Dornelas" title="Maria Dornelas">Dornelas</a> et al.,<sup id="cite_ref-Dornelas2006_14-0" class="reference"><a href="#cite_note-Dornelas2006-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> and is a poor fit to data in intertidal communities.<sup id="cite_ref-Wootton2005_15-0" class="reference"><a href="#cite_note-Wootton2005-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> It also fails to explain why families of tropical trees have statistically highly correlated numbers of species in phylogenetically unrelated and geographically distant forest plots in Central and South America, Africa, and South East Asia.<sup id="cite_ref-Ricklefs2012_16-0" class="reference"><a href="#cite_note-Ricklefs2012-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>While the theory has been heralded as a valuable tool for palaeontologists,<sup id="cite_ref-Hubbell2005_7-2" class="reference"><a href="#cite_note-Hubbell2005-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> little work has so far been done to test the theory against the fossil record.<sup id="cite_ref-Bonuso2007_17-0" class="reference"><a href="#cite_note-Bonuso2007-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unified_neutral_theory_of_biodiversity&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Biodiversity_Action_Plan" class="mw-redirect" title="Biodiversity Action Plan">Biodiversity Action Plan</a></li> <li><a href="/wiki/Functional_equivalence_(ecology)" title="Functional equivalence (ecology)">Functional equivalence (ecology)</a></li> <li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens's sampling formula</a></li> <li><a href="/w/index.php?title=Metabolic_Scaling_Theory_(Metabolic_theory_of_ecology)&action=edit&redlink=1" class="new" title="Metabolic Scaling Theory (Metabolic theory of ecology) (page does not exist)">Metabolic Scaling Theory (Metabolic theory of ecology)</a></li> <li><a href="/wiki/Neutral_theory_of_molecular_evolution" title="Neutral theory of molecular evolution">Neutral theory of molecular evolution</a></li> <li><a href="/wiki/Warren_Ewens" title="Warren Ewens">Warren Ewens</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unified_neutral_theory_of_biodiversity&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Hubbell2001-1"><span 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"Global correlations in tropical tree species richness and abundance reject neutrality". <i>Science</i>. <b>335</b> (6067): 464–467. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012Sci...335..464R">2012Sci...335..464R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1215182">10.1126/science.1215182</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22282811">22282811</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15347595">15347595</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Global+correlations+in+tropical+tree+species+richness+and+abundance+reject+neutrality&rft.volume=335&rft.issue=6067&rft.pages=464-467&rft.date=2012&rft_id=info%3Adoi%2F10.1126%2Fscience.1215182&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15347595%23id-name%3DS2CID&rft_id=info%3Apmid%2F22282811&rft_id=info%3Abibcode%2F2012Sci...335..464R&rft.au=Ricklefs%2C+R.+E.&rft.au=S.+S.+Renner&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnified+neutral+theory+of+biodiversity" class="Z3988"></span></span> </li> <li id="cite_note-Bonuso2007-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bonuso2007_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBonuso2007" class="citation journal cs1">Bonuso, N. (2007). "Shortening the Gap Between Modern Community Ecology and Evolutionary Paleoecology". <i>PALAIOS</i>. <b>22</b> (5): 455–456. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007Palai..22..455B">2007Palai..22..455B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2110%2Fpalo.2007.S05">10.2110/palo.2007.S05</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121753503">121753503</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PALAIOS&rft.atitle=Shortening+the+Gap+Between+Modern+Community+Ecology+and+Evolutionary+Paleoecology&rft.volume=22&rft.issue=5&rft.pages=455-456&rft.date=2007&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121753503%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.2110%2Fpalo.2007.S05&rft_id=info%3Abibcode%2F2007Palai..22..455B&rft.aulast=Bonuso&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnified+neutral+theory+of+biodiversity" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unified_neutral_theory_of_biodiversity&action=edit&section=10" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGilbertLechowicz_MJ2004" class="citation journal cs1">Gilbert, B; Lechowicz MJ (2004). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC419661">"Neutrality, niches, and dispersal in a temperate forest understory"</a>. <i>PNAS</i>. <b>101</b> (20): 7651–7656. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004PNAS..101.7651G">2004PNAS..101.7651G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.0400814101">10.1073/pnas.0400814101</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC419661">419661</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/15128948">15128948</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PNAS&rft.atitle=Neutrality%2C+niches%2C+and+dispersal+in+a+temperate+forest+understory&rft.volume=101&rft.issue=20&rft.pages=7651-7656&rft.date=2004&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC419661%23id-name%3DPMC&rft_id=info%3Apmid%2F15128948&rft_id=info%3Adoi%2F10.1073%2Fpnas.0400814101&rft_id=info%3Abibcode%2F2004PNAS..101.7651G&rft.aulast=Gilbert&rft.aufirst=B&rft.au=Lechowicz+MJ&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC419661&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnified+neutral+theory+of+biodiversity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeigh_E.G._(Jr)2007" class="citation journal cs1">Leigh E.G. (Jr) (2007). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1420-9101.2007.01410.x">"Neutral theory: a historical perspective"</a>. <i><a href="/wiki/Journal_of_Evolutionary_Biology" title="Journal of Evolutionary Biology">Journal of Evolutionary Biology</a></i>. <b>20</b> (6): 2075–2091. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1420-9101.2007.01410.x">10.1111/j.1420-9101.2007.01410.x</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17956380">17956380</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Evolutionary+Biology&rft.atitle=Neutral+theory%3A+a+historical+perspective&rft.volume=20&rft.issue=6&rft.pages=2075-2091&rft.date=2007&rft_id=info%3Adoi%2F10.1111%2Fj.1420-9101.2007.01410.x&rft_id=info%3Apmid%2F17956380&rft.au=Leigh+E.G.+%28Jr%29&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252Fj.1420-9101.2007.01410.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnified+neutral+theory+of+biodiversity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPreston1962" class="citation journal cs1">Preston, F. W. (1962). "The Canonical Distribution of Commonness and Rarity: Part I". <i>Ecology</i>. <b>43</b> (2). Ecology, Vol. 43, No. 2: 185–215. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1931976">10.2307/1931976</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1931976">1931976</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ecology&rft.atitle=The+Canonical+Distribution+of+Commonness+and+Rarity%3A+Part+I&rft.volume=43&rft.issue=2&rft.pages=185-215&rft.date=1962&rft_id=info%3Adoi%2F10.2307%2F1931976&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1931976%23id-name%3DJSTOR&rft.aulast=Preston&rft.aufirst=F.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnified+neutral+theory+of+biodiversity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPueyoHe,_F.Zillio,_T.2007" class="citation journal cs1">Pueyo, S.; He, F.; Zillio, T. (2007). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2121135">"The maximum entropy formalism and the idiosyncratic theory of biodiversity"</a>. <i>Ecology Letters</i>. <b>10</b> (11): 1017–1028. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1461-0248.2007.01096.x">10.1111/j.1461-0248.2007.01096.x</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2121135">2121135</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17692099">17692099</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ecology+Letters&rft.atitle=The+maximum+entropy+formalism+and+the+idiosyncratic+theory+of+biodiversity&rft.volume=10&rft.issue=11&rft.pages=1017-1028&rft.date=2007&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2121135%23id-name%3DPMC&rft_id=info%3Apmid%2F17692099&rft_id=info%3Adoi%2F10.1111%2Fj.1461-0248.2007.01096.x&rft.aulast=Pueyo&rft.aufirst=S.&rft.au=He%2C+F.&rft.au=Zillio%2C+T.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2121135&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnified+neutral+theory+of+biodiversity" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unified_neutral_theory_of_biodiversity&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.sciam.com/article.cfm?articleID=000D401A-8D9D-1CDA-B4A8809EC588EEDF">Scientific American Interview with Steve Hubbell</a></li> <li><a rel="nofollow" class="external text" href="http://finzi.psych.upenn.edu/R/library/untb/html/untb.package.html">R package for implementing UNTB</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190918204703/http://finzi.psych.upenn.edu/R/library/untb/html/untb.package.html">Archived</a> September 18, 2019, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130806083430/http://news.cell.com/discussions/trends-in-ecology-and-evolution/ecological-neutral-theory-useful-model-or-statement-of-ignorance">"Ecological neutral theory: useful model or statement of 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href="/wiki/Ecosystem_model" title="Ecosystem model">Modelling ecosystems</a>: <a href="/wiki/Trophic_level" title="Trophic level">Trophic</a> components</div></th></tr><tr><th scope="row" class="navbox-group" style="width:7.5em">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abiotic_component" title="Abiotic component">Abiotic component</a></li> <li><a href="/wiki/Abiotic_stress" title="Abiotic stress">Abiotic stress</a></li> <li><a href="/wiki/Behavioral_ecology" title="Behavioral ecology">Behaviour</a></li> <li><a href="/wiki/Biogeochemical_cycle" title="Biogeochemical cycle">Biogeochemical cycle</a></li> <li><a href="/wiki/Biomass_(ecology)" title="Biomass (ecology)">Biomass</a></li> <li><a href="/wiki/Biotic_component" class="mw-redirect" title="Biotic component">Biotic component</a></li> <li><a href="/wiki/Biotic_stress" title="Biotic stress">Biotic stress</a></li> <li><a href="/wiki/Carrying_capacity" title="Carrying capacity">Carrying capacity</a></li> <li><a href="/wiki/Competition_(biology)" title="Competition (biology)">Competition</a></li> <li><a href="/wiki/Ecosystem" title="Ecosystem">Ecosystem</a></li> <li><a href="/wiki/Ecosystem_ecology" title="Ecosystem ecology">Ecosystem ecology</a></li> <li><a href="/wiki/Ecosystem_model" title="Ecosystem model">Ecosystem model</a></li> <li><a href="/wiki/Green_world_hypothesis" title="Green world hypothesis">Green world hypothesis</a></li> <li><a href="/wiki/Keystone_species" title="Keystone species">Keystone species</a></li> <li><a href="/wiki/List_of_feeding_behaviours" title="List of feeding behaviours">List of feeding behaviours</a></li> <li><a href="/wiki/Metabolic_theory_of_ecology" title="Metabolic theory of ecology">Metabolic theory of ecology</a></li> <li><a href="/wiki/Productivity_(ecology)" title="Productivity (ecology)">Productivity</a></li> <li><a href="/wiki/Resource_(biology)" title="Resource (biology)">Resource</a></li> <li><a href="/wiki/Restoration_ecology" class="mw-redirect" title="Restoration ecology">Restoration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em"><a href="/wiki/Autotroph" title="Autotroph">Producers</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autotroph" title="Autotroph">Autotrophs</a></li> <li><a href="/wiki/Chemosynthesis" title="Chemosynthesis">Chemosynthesis</a></li> <li><a href="/wiki/Chemotroph" title="Chemotroph">Chemotrophs</a></li> <li><a href="/wiki/Foundation_species" title="Foundation species">Foundation species</a></li> <li><a href="/wiki/Kinetotroph" title="Kinetotroph">Kinetotrophs</a></li> <li><a href="/wiki/Mixotroph" title="Mixotroph">Mixotrophs</a></li> <li><a href="/wiki/Myco-heterotrophy" title="Myco-heterotrophy">Myco-heterotrophy</a></li> <li><a href="/wiki/Mycotroph" title="Mycotroph">Mycotroph</a></li> <li><a href="/wiki/Organotroph" title="Organotroph">Organotrophs</a></li> <li><a href="/wiki/Photoheterotroph" title="Photoheterotroph">Photoheterotrophs</a></li> <li><a href="/wiki/Photosynthesis" title="Photosynthesis">Photosynthesis</a></li> <li><a href="/wiki/Photosynthetic_efficiency" title="Photosynthetic efficiency">Photosynthetic efficiency</a></li> <li><a href="/wiki/Phototroph" title="Phototroph">Phototrophs</a></li> <li><a href="/wiki/Primary_nutritional_groups" title="Primary nutritional groups">Primary nutritional groups</a></li> <li><a href="/wiki/Primary_production" title="Primary production">Primary production</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em"><a href="/wiki/Consumer_(food_chain)" title="Consumer (food chain)">Consumers</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Apex_predator" title="Apex predator">Apex predator</a></li> <li><a href="/wiki/Bacterivore" title="Bacterivore">Bacterivore</a></li> <li><a href="/wiki/Carnivore" title="Carnivore">Carnivores</a></li> <li><a href="/wiki/Chemoorganotroph" class="mw-redirect" title="Chemoorganotroph">Chemoorganotroph</a></li> <li><a href="/wiki/Foraging" title="Foraging">Foraging</a></li> <li><a href="/wiki/Generalist_and_specialist_species" title="Generalist and specialist species">Generalist and specialist species</a></li> <li><a href="/wiki/Intraguild_predation" title="Intraguild predation">Intraguild predation</a></li> <li><a href="/wiki/Herbivore" title="Herbivore">Herbivores</a></li> <li><a href="/wiki/Heterotroph" title="Heterotroph">Heterotroph</a></li> <li><a href="/wiki/Heterotrophic_nutrition" title="Heterotrophic nutrition">Heterotrophic nutrition</a></li> <li><a href="/wiki/Insectivore" title="Insectivore">Insectivore</a></li> <li><a href="/wiki/Mesopredator" title="Mesopredator">Mesopredators</a></li> <li><a href="/wiki/Mesopredator_release_hypothesis" title="Mesopredator release hypothesis">Mesopredator release hypothesis</a></li> <li><a href="/wiki/Omnivore" title="Omnivore">Omnivores</a></li> <li><a href="/wiki/Optimal_foraging_theory" title="Optimal foraging theory">Optimal foraging theory</a></li> <li><a href="/wiki/Planktivore" title="Planktivore">Planktivore</a></li> <li><a href="/wiki/Predation" title="Predation">Predation</a></li> <li><a href="/wiki/Prey_switching" title="Prey switching">Prey switching</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em"><a href="/wiki/Decomposer" title="Decomposer">Decomposers</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chemoorganoheterotrophy" class="mw-redirect" title="Chemoorganoheterotrophy">Chemoorganoheterotrophy</a></li> <li><a href="/wiki/Decomposition" title="Decomposition">Decomposition</a></li> <li><a href="/wiki/Detritivore" title="Detritivore">Detritivores</a></li> <li><a href="/wiki/Detritus" title="Detritus">Detritus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em"><a href="/wiki/Microorganism#Habitats_and_ecology" title="Microorganism">Microorganisms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Archaea" title="Archaea">Archaea</a></li> <li><a href="/wiki/Bacteriophage" title="Bacteriophage">Bacteriophage</a></li> <li><a href="/wiki/Lithoautotroph" title="Lithoautotroph">Lithoautotroph</a></li> <li><a href="/wiki/Lithotroph" title="Lithotroph">Lithotrophy</a></li> <li><a href="/wiki/Marine_microorganisms" title="Marine microorganisms">Marine</a></li> <li><a href="/wiki/Microbial_cooperation" title="Microbial cooperation">Microbial cooperation</a></li> <li><a href="/wiki/Microbial_ecology" title="Microbial ecology">Microbial ecology</a></li> <li><a href="/wiki/Microbial_food_web" title="Microbial food web">Microbial food web</a></li> <li><a href="/wiki/Microbial_intelligence" title="Microbial intelligence">Microbial intelligence</a></li> <li><a href="/wiki/Microbial_loop" title="Microbial loop">Microbial loop</a></li> <li><a href="/wiki/Microbial_mat" title="Microbial mat">Microbial mat</a></li> <li><a href="/wiki/Microbial_metabolism" title="Microbial metabolism">Microbial metabolism</a></li> <li><a href="/wiki/Phage_ecology" title="Phage ecology">Phage ecology</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em"><a href="/wiki/Food_web" title="Food web">Food webs</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Biomagnification" title="Biomagnification">Biomagnification</a></li> <li><a href="/wiki/Ecological_efficiency" title="Ecological efficiency">Ecological efficiency</a></li> <li><a href="/wiki/Ecological_pyramid" title="Ecological pyramid">Ecological pyramid</a></li> <li><a href="/wiki/Energy_flow_(ecology)" title="Energy flow (ecology)">Energy flow</a></li> <li><a href="/wiki/Food_chain" title="Food chain">Food chain</a></li> <li><a href="/wiki/Trophic_level" title="Trophic level">Trophic level</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em">Example webs</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lake_ecosystem#Trophic_relationships" title="Lake ecosystem">Lakes</a></li> <li><a href="/wiki/River_ecosystem#Trophic_relationships" title="River ecosystem">Rivers</a></li> <li><a href="/wiki/Soil_food_web" title="Soil food web">Soil</a></li> <li><a href="/wiki/Tritrophic_interactions_in_plant_defense" title="Tritrophic interactions in plant defense">Tritrophic interactions in plant defense</a></li> <li><a href="/wiki/Marine_food_web" title="Marine food web">Marine food webs</a> <ul><li><a href="/wiki/Cold_seep" title="Cold seep">cold seeps</a></li> <li><a href="/wiki/Hydrothermal_vent#Biological_communities" title="Hydrothermal vent">hydrothermal vents</a></li> <li><a href="/wiki/Intertidal_ecology" title="Intertidal ecology">intertidal</a></li> <li><a href="/wiki/Kelp_forest#Trophic_ecology" title="Kelp forest">kelp forests</a></li> <li><a href="/wiki/North_Pacific_Gyre" title="North Pacific Gyre">North Pacific Gyre</a></li> <li><a href="/wiki/Ecology_of_the_San_Francisco_Estuary#Food_web" title="Ecology of the San Francisco Estuary">San Francisco Estuary</a></li> <li><a href="/wiki/Tide_pool" title="Tide pool">tide pool</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em">Processes</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ascendency" title="Ascendency">Ascendency</a></li> <li><a href="/wiki/Bioaccumulation" title="Bioaccumulation">Bioaccumulation</a></li> <li><a href="/wiki/Cascade_effect_(ecology)" title="Cascade effect (ecology)">Cascade effect</a></li> <li><a href="/wiki/Climax_community" title="Climax community">Climax community</a></li> <li><a href="/wiki/Competitive_exclusion_principle" title="Competitive exclusion principle">Competitive exclusion principle</a></li> <li><a href="/wiki/Consumer%E2%80%93resource_interactions" title="Consumer–resource interactions">Consumer–resource interactions</a></li> <li><a href="/wiki/Copiotroph" title="Copiotroph">Copiotrophs</a></li> <li><a href="/wiki/Dominance_(ecology)" title="Dominance (ecology)">Dominance</a></li> <li><a href="/wiki/Ecological_network" title="Ecological network">Ecological network</a></li> <li><a href="/wiki/Ecological_succession" title="Ecological succession">Ecological succession</a></li> <li><a href="/wiki/Energy_quality" title="Energy quality">Energy quality</a></li> <li><a href="/wiki/Energy_systems_language" title="Energy systems language">Energy systems language</a></li> <li><a href="/wiki/F-ratio_(oceanography)" title="F-ratio (oceanography)">f-ratio</a></li> <li><a href="/wiki/Feed_conversion_ratio" title="Feed conversion ratio">Feed conversion ratio</a></li> <li><a href="/wiki/Feeding_frenzy" title="Feeding frenzy">Feeding frenzy</a></li> <li><a href="/wiki/Mesotrophic_soil" title="Mesotrophic soil">Mesotrophic soil</a></li> <li><a href="/wiki/Nutrient_cycle" title="Nutrient cycle">Nutrient cycle</a></li> <li><a href="/wiki/Oligotroph" title="Oligotroph">Oligotroph</a></li> <li><a href="/wiki/Paradox_of_the_plankton" title="Paradox of the plankton">Paradox of the plankton</a></li> <li><a href="/wiki/Trophic_cascade" title="Trophic cascade">Trophic cascade</a></li> <li><a href="/wiki/Trophic_mutualism" title="Trophic mutualism">Trophic mutualism</a></li> <li><a href="/wiki/Trophic_state_index" title="Trophic state index">Trophic state index</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em">Defense,<br />counter</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Animal_coloration" title="Animal coloration">Animal coloration</a></li> <li><a href="/wiki/Anti-predator_adaptation" title="Anti-predator adaptation">Anti-predator adaptations</a></li> <li><a href="/wiki/Camouflage" title="Camouflage">Camouflage</a></li> <li><a href="/wiki/Deimatic_behaviour" title="Deimatic behaviour">Deimatic behaviour</a></li> <li><a href="/wiki/Herbivore_adaptations_to_plant_defense" title="Herbivore adaptations to plant defense">Herbivore adaptations to plant defense</a></li> <li><a href="/wiki/Mimicry" title="Mimicry">Mimicry</a></li> <li><a href="/wiki/Plant_defense_against_herbivory" title="Plant defense against herbivory">Plant defense against herbivory</a></li> <li><a href="/wiki/Shoaling_and_schooling#Predator_avoidance" title="Shoaling and schooling">Predator avoidance in schooling fish</a></li></ul> </div></td></tr></tbody></table></div><div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Ecology:_Modelling_ecosystems:_Other_components" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Modelling_ecosystems" title="Template:Modelling ecosystems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Modelling_ecosystems" title="Template talk:Modelling ecosystems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Modelling_ecosystems" title="Special:EditPage/Template:Modelling ecosystems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Ecology:_Modelling_ecosystems:_Other_components" style="font-size:114%;margin:0 4em"><a href="/wiki/Ecology" title="Ecology">Ecology</a>: <a href="/wiki/Ecosystem_model" title="Ecosystem model">Modelling ecosystems</a>: Other components</div></th></tr><tr><th scope="row" class="navbox-group" style="width:7.5em"><a href="/wiki/Population_ecology" title="Population ecology">Population<br />ecology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundance_(ecology)" title="Abundance (ecology)">Abundance</a></li> <li><a href="/wiki/Allee_effect" title="Allee effect">Allee effect</a></li> <li><a href="/wiki/Consumer-resource_model" title="Consumer-resource model">Consumer-resource model</a></li> <li><a href="/wiki/Depensation" title="Depensation">Depensation</a></li> <li><a href="/wiki/Ecological_yield" title="Ecological yield">Ecological yield</a></li> <li><a href="/wiki/Effective_population_size" title="Effective population size">Effective population size</a></li> <li><a href="/wiki/Intraspecific_competition" title="Intraspecific competition">Intraspecific competition</a></li> <li><a href="/wiki/Logistic_function" title="Logistic function">Logistic function</a></li> <li><a href="/wiki/Malthusian_growth_model" title="Malthusian growth model">Malthusian growth model</a></li> <li><a href="/wiki/Maximum_sustainable_yield" title="Maximum sustainable yield">Maximum sustainable yield</a></li> <li><a href="/wiki/Overpopulation" title="Overpopulation">Overpopulation</a></li> <li><a href="/wiki/Overexploitation" title="Overexploitation">Overexploitation</a></li> <li><a href="/wiki/Population_cycle" title="Population cycle">Population cycle</a></li> <li><a href="/wiki/Population_dynamics" title="Population dynamics">Population dynamics</a></li> <li><a href="/wiki/Population_model" title="Population model">Population modeling</a></li> <li><a href="/wiki/Population_size" title="Population size">Population size</a></li> <li><a href="/wiki/Lotka%E2%80%93Volterra_equations" title="Lotka–Volterra equations">Predator–prey (Lotka–Volterra) equations</a></li> <li><a href="/wiki/Recruitment_(biology)" title="Recruitment (biology)">Recruitment</a></li> <li><a href="/wiki/Small_population_size" title="Small population size">Small population size</a></li> <li><a href="/wiki/Ecological_stability" title="Ecological stability">Stability</a> <ul><li><a href="/wiki/Ecological_resilience" title="Ecological resilience">Resilience</a></li> <li><a href="/wiki/Resistance_(ecology)" title="Resistance (ecology)">Resistance</a></li></ul></li> <li><a href="/wiki/Random_generalized_Lotka%E2%80%93Volterra_model" title="Random generalized Lotka–Volterra model">Random generalized Lotka–Volterra model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em">Species</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Biodiversity" title="Biodiversity">Biodiversity</a></li> <li><a href="/wiki/Density_dependence" title="Density dependence">Density-dependent inhibition</a></li> <li><a href="/wiki/Ecological_effects_of_biodiversity" title="Ecological effects of biodiversity">Ecological effects of biodiversity</a></li> <li><a href="/wiki/Ecological_extinction" title="Ecological extinction">Ecological extinction</a></li> <li><a href="/wiki/Endemism" title="Endemism">Endemic species</a></li> <li><a href="/wiki/Flagship_species" title="Flagship species">Flagship species</a></li> <li><a href="/wiki/Ordination_(statistics)" title="Ordination (statistics)">Gradient analysis</a></li> <li><a href="/wiki/Bioindicator" title="Bioindicator">Indicator species</a></li> <li><a href="/wiki/Introduced_species" title="Introduced species">Introduced species</a></li> <li><a href="/wiki/Invasive_species" title="Invasive species">Invasive species</a> / <a href="/wiki/Native_species" title="Native species">Native species</a></li> <li><a href="/wiki/Latitudinal_gradients_in_species_diversity" title="Latitudinal gradients in species diversity">Latitudinal gradients in species diversity</a></li> <li><a href="/wiki/Minimum_viable_population" title="Minimum viable population">Minimum viable population</a></li> <li><a class="mw-selflink selflink">Neutral theory</a></li> <li><a href="/wiki/Occupancy%E2%80%93abundance_relationship" title="Occupancy–abundance relationship">Occupancy–abundance relationship</a></li> <li><a href="/wiki/Population_viability_analysis" title="Population viability analysis">Population viability analysis</a></li> <li><a href="/wiki/Priority_effect" title="Priority effect">Priority effect</a></li> <li><a href="/wiki/Rapoport%27s_rule" title="Rapoport's rule">Rapoport's rule</a></li> <li><a href="/wiki/Relative_abundance_distribution" title="Relative abundance distribution">Relative abundance distribution</a></li> <li><a href="/wiki/Relative_species_abundance" title="Relative species abundance">Relative species abundance</a></li> <li><a href="/wiki/Species_diversity" title="Species diversity">Species diversity</a></li> <li><a href="/wiki/Species_homogeneity" title="Species homogeneity">Species homogeneity</a></li> <li><a href="/wiki/Species_richness" title="Species richness">Species richness</a></li> <li><a href="/wiki/Species_distribution" title="Species distribution">Species distribution</a></li> <li><a href="/wiki/Species%E2%80%93area_relationship" title="Species–area relationship">Species–area curve</a></li> <li><a href="/wiki/Umbrella_species" title="Umbrella species">Umbrella species</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em">Species<br />interaction</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antibiosis" title="Antibiosis">Antibiosis</a></li> <li><a href="/wiki/Biological_interaction" title="Biological interaction">Biological interaction</a></li> <li><a href="/wiki/Commensalism" title="Commensalism">Commensalism</a></li> <li><a href="/wiki/Community_(ecology)" title="Community (ecology)">Community ecology</a></li> <li><a href="/wiki/Ecological_facilitation" title="Ecological facilitation">Ecological facilitation</a></li> <li><a href="/wiki/Interspecific_competition" title="Interspecific competition">Interspecific competition</a></li> <li><a href="/wiki/Mutualism_(biology)" title="Mutualism (biology)">Mutualism</a></li> <li><a href="/wiki/Parasitism" title="Parasitism">Parasitism</a></li> <li><a href="/wiki/Storage_effect" title="Storage effect">Storage effect</a></li> <li><a href="/wiki/Symbiosis" title="Symbiosis">Symbiosis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em"><a href="/wiki/Spatial_ecology" title="Spatial ecology">Spatial<br />ecology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Biogeography" title="Biogeography">Biogeography</a></li> <li><a href="/wiki/Cross-boundary_subsidy" title="Cross-boundary subsidy">Cross-boundary subsidy</a></li> <li><a href="/wiki/Cline_(biology)" title="Cline (biology)">Ecocline</a></li> <li><a href="/wiki/Ecotone" title="Ecotone">Ecotone</a></li> <li><a href="/wiki/Ecotype" title="Ecotype">Ecotype</a></li> <li><a href="/wiki/Disturbance_(ecology)" title="Disturbance (ecology)">Disturbance</a></li> <li><a href="/wiki/Edge_effects" title="Edge effects">Edge effects</a></li> <li><a href="/wiki/Foster%27s_rule" title="Foster's rule">Foster's rule</a></li> <li><a href="/wiki/Habitat_fragmentation" title="Habitat fragmentation">Habitat fragmentation</a></li> <li><a href="/wiki/Ideal_free_distribution" title="Ideal free distribution">Ideal free distribution</a></li> <li><a href="/wiki/Intermediate_disturbance_hypothesis" title="Intermediate disturbance hypothesis">Intermediate disturbance hypothesis</a></li> <li><a href="/wiki/Insular_biogeography" title="Insular biogeography">Insular biogeography</a></li> <li><a href="/wiki/Land_change_modeling" title="Land change modeling">Land change modeling</a></li> <li><a href="/wiki/Landscape_ecology" title="Landscape ecology">Landscape ecology</a></li> <li><a href="/wiki/Landscape_epidemiology" title="Landscape epidemiology">Landscape epidemiology</a></li> <li><a href="/wiki/Landscape_limnology" title="Landscape limnology">Landscape limnology</a></li> <li><a href="/wiki/Metapopulation" title="Metapopulation">Metapopulation</a></li> <li><a href="/wiki/Patch_dynamics" title="Patch dynamics">Patch dynamics</a></li> <li><a href="/wiki/R/K_selection_theory" title="R/K selection theory"><i>r</i>/<i>K</i> selection theory</a></li> <li><a href="/wiki/Resource_selection_function" title="Resource selection function">Resource selection function</a></li> <li><a href="/wiki/Source%E2%80%93sink_dynamics" title="Source–sink dynamics">Source–sink dynamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em"><a href="/wiki/Ecological_niche" title="Ecological niche">Niche</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ecological_niche" title="Ecological niche">Ecological niche</a></li> <li><a href="/wiki/Ecological_trap" title="Ecological trap">Ecological trap</a></li> <li><a href="/wiki/Ecosystem_engineer" title="Ecosystem engineer">Ecosystem engineer</a></li> <li><a href="/wiki/Species_distribution_modelling" title="Species distribution modelling">Environmental niche modelling</a></li> <li><a href="/wiki/Guild_(ecology)" title="Guild (ecology)">Guild</a></li> <li><a href="/wiki/Habitat" title="Habitat">Habitat</a> <ul><li><a href="/wiki/Marine_habitat" title="Marine habitat">marine habitats</a></li></ul></li> <li><a href="/wiki/Limiting_similarity" title="Limiting similarity">Limiting similarity</a></li> <li><a href="/wiki/Niche_apportionment_models" title="Niche apportionment models">Niche apportionment models</a></li> <li><a href="/wiki/Niche_construction" title="Niche construction">Niche construction</a></li> <li><a href="/wiki/Niche_differentiation" class="mw-redirect" title="Niche differentiation">Niche differentiation</a></li> <li><a href="/wiki/Ontogenetic_niche_shift" title="Ontogenetic niche shift">Ontogenetic niche shift</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em"><a href="/wiki/Non-trophic_networks" title="Non-trophic networks">Other<br />networks</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Assembly_rules" title="Assembly rules">Assembly rules</a></li> <li><a href="/wiki/Bateman%27s_principle" title="Bateman's principle">Bateman's principle</a></li> <li><a href="/wiki/Bioluminescence" title="Bioluminescence">Bioluminescence</a></li> <li><a href="/wiki/Ecological_collapse" class="mw-redirect" title="Ecological collapse">Ecological collapse</a></li> <li><a href="/wiki/Ecological_debt" title="Ecological debt">Ecological debt</a></li> <li><a href="/wiki/Ecological_debt" title="Ecological debt">Ecological deficit</a></li> <li><a href="/wiki/Energy_flow_(ecology)" title="Energy flow (ecology)">Ecological energetics</a></li> <li><a href="/wiki/Ecological_indicator" title="Ecological indicator">Ecological indicator</a></li> <li><a href="/wiki/Ecological_threshold" title="Ecological threshold">Ecological threshold</a></li> <li><a href="/wiki/Ecosystem_diversity" title="Ecosystem diversity">Ecosystem diversity</a></li> <li><a href="/wiki/Emergence" title="Emergence">Emergence</a></li> <li><a href="/wiki/Extinction_debt" title="Extinction debt">Extinction debt</a></li> <li><a href="/wiki/Kleiber%27s_law" title="Kleiber's law">Kleiber's law</a></li> <li><a href="/wiki/Liebig%27s_law_of_the_minimum" title="Liebig's law of the minimum">Liebig's law of the minimum</a></li> <li><a href="/wiki/Marginal_value_theorem" title="Marginal value theorem">Marginal value theorem</a></li> <li><a href="/wiki/Thorson%27s_rule" title="Thorson's rule">Thorson's rule</a></li> <li><a href="/wiki/Xerosere" title="Xerosere">Xerosere</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.5em">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Allometry" title="Allometry">Allometry</a></li> <li><a href="/wiki/Alternative_stable_state" title="Alternative stable state">Alternative stable state</a></li> <li><a href="/wiki/Balance_of_nature" title="Balance of nature">Balance of nature</a></li> <li><a href="/wiki/Biological_data_visualization" title="Biological data visualization">Biological data visualization</a></li> <li><a href="/wiki/Ecological_economics" title="Ecological economics">Ecological economics</a></li> <li><a href="/wiki/Ecological_footprint" title="Ecological footprint">Ecological footprint</a></li> <li><a href="/wiki/Ecological_forecasting" title="Ecological forecasting">Ecological forecasting</a></li> <li><a href="/wiki/Environmental_humanities" title="Environmental humanities">Ecological humanities</a></li> <li><a href="/wiki/Ecological_stoichiometry" title="Ecological stoichiometry">Ecological stoichiometry</a></li> <li><a href="/wiki/Ecopath" title="Ecopath">Ecopath</a></li> <li><a href="/wiki/Ecosystem_based_fisheries" class="mw-redirect" title="Ecosystem based fisheries">Ecosystem based fisheries</a></li> <li><a href="/wiki/Endolith" title="Endolith">Endolith</a></li> <li><a href="/wiki/Evolutionary_ecology" title="Evolutionary ecology">Evolutionary ecology</a></li> <li><a href="/wiki/Functional_ecology" title="Functional ecology">Functional ecology</a></li> <li><a href="/wiki/Industrial_ecology" title="Industrial ecology">Industrial ecology</a></li> <li><a href="/wiki/Macroecology" title="Macroecology">Macroecology</a></li> <li><a href="/wiki/Microecosystem" title="Microecosystem">Microecosystem</a></li> <li><a href="/wiki/Natural_environment" title="Natural environment">Natural environment</a></li> <li><a href="/wiki/Regime_shift" title="Regime shift">Regime shift</a></li> <li><a href="/wiki/Sexecology" title="Sexecology">Sexecology</a></li> 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Unified neutral theory of biodiversity - Wikipedia