Document Zbl 1410.60094

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Appl. Probab.</a> <a href="/?q=in%3A410703" title="Articles in this Issue">28, No. 3, 1893-1942 (2018)</a>. </div> <div class="abstract">Summary: In recent years there has been a growing interest in the study of the dynamics of stochastic populations. A key question in population biology is to understand the conditions under which populations coexist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of $n$ populations that live in a stochastic environment and which can interact nonlinearly (through competition for resources, predator-prey behavior, etc.). Our models are described by $n$-dimensional Kolmogorov systems with white noise (stochastic differential equations-SDE). We give sharp conditions under which the populations converge exponentially fast to their unique stationary distribution as well as conditions under which some populations go extinct exponentially fast.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6919741">Cited in <strong>3</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A6919741">Cited in <strong>116</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A60K35" title="MSC2020">60K35</a> </td> <td class="space"> Interacting random processes; statistical mechanics type models; percolation theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37H15" title="MSC2020">37H15</a> </td> <td class="space"> Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60H10" title="MSC2020">60H10</a> </td> <td class="space"> Stochastic ordinary differential equations (aspects of stochastic analysis) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60J05" title="MSC2020">60J05</a> </td> <td class="space"> Discrete-time Markov processes on general state spaces </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AKolmogorov+system">Kolmogorov system</a>; <a href="/?q=ut%3Aergodicity">ergodicity</a>; <a href="/?q=ut%3ALotka-Volterra">Lotka-Volterra</a>; <a href="/?q=ut%3ALyapunov+exponent">Lyapunov exponent</a>; <a href="/?q=ut%3Astochastic+environment">stochastic environment</a>; <a href="/?q=ut%3Apredator-prey">predator-prey</a>; <a href="/?q=ut%3Apopulation+dynamics">population dynamics</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1410.60094" data-ciurl="/ci/06919741" data-biburl="/bibtex/06919741.bib" data-amsurl="/amsrefs/06919741.bib" data-xmlurl="/xml/06919741.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06919741.pdf" title="Zbl 1410.60094 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1214/17-AAP1347" aria-label="DOI for “Coexistence and extinction for stochastic Kolmogorov systems”" title="10.1214/17-AAP1347">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1704.06984"title="Note: arXiv document may differ from published version">arXiv</a> <a class="btn btn-default btn-xs" type="button" href="https://projecteuclid.org/euclid.aoap/1527840035" title="Full Text at Project Euclid">Euclid</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Benaïm, M. and Schreiber, S. 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