Document Zbl 1472.92190

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Biosci. Eng.</a> <a href="/?q=in%3A468460" title="Articles in this Issue">17, No. 3, 2616-2635 (2020)</a>. </div> <div class="abstract">The paper treats a particular case of a predator-prey 4-dimensional ODE model that has been treated in [<span class="zbmathjax-textit">L. Castellino</span> et al., in: AIP Conference Proceedings 1776, 020006 (2016), <a href="https://doi.org/10.1063/1.4965312">https://doi.org/10.1063/1.4965312</a>]. The model in the 2016 paper is of the Lotka-Volterra type (but with carrying capacities on the prey) with both predator and prey having two genotypes. The particular case treated here consists in assuming that the hunting ability of each predator genotype does not depend on the prey genotype and allows this paper to go deeper than the previous paper into the qualitative behaviour of the (simpler) model. The study of the model is performed on an a dimensionalized version.<br class="zbmathjax-paragraph">The paper starts by studying the stability of the model in the absence of predators and then reminds the known stability results of the model with just one genotype for both preys and predators. <br class="zbmathjax-paragraph">The full model is then studied. There are two boundary equilibria. One corresponds to the extinction of both species and is unstable. The other corresponds to the extinction of just the predators and, depending on the value of a parameter $h$ compared to some critical value $h_c$, is unstable for $h<h_c$, globally stable for $h>h_c$ or has a local centre manifold characterized in the paper for $h=h_c$. A necessary and sufficient condition for the existence of a unique positive equilibrium of the full model is that $h<h_c$ and this equilibrium is stable. Thus, the model has a transcritical bifurcation. The paper concludes with numerical examples and corresponding figures illustrating its different possible qualitative behaviours.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A28">Carlos A. dos Santos Braumann (Évora)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7387908">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D25" title="MSC2020">92D25</a> </td> <td class="space"> Population dynamics (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D10" title="MSC2020">92D10</a> </td> <td class="space"> Genetics and epigenetics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34D20" title="MSC2020">34D20</a> </td> <td class="space"> Stability of solutions to ordinary differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Apredator-prey+model">predator-prey model</a>; <a href="/?q=ut%3Agenotypes">genotypes</a>; <a href="/?q=ut%3Aequilibrium">equilibrium</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3Acenter+manifold">center manifold</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 1472.92190" data-ciurl="/ci/07387908" data-biburl="/bibtex/07387908.bib" data-amsurl="/amsrefs/07387908.bib" data-xmlurl="/xml/07387908.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07387908.pdf" title="Zbl 1472.92190 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/10.3934/mbe.2020143" aria-label="DOI for “A predator-prey model with genetic differentiation both in the predator and prey”" title="10.3934/mbe.2020143">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">A. J. Lotka, Contribution to the theory of periodic reaction, J. Phys. Chem., 14 (1910), 271-274.</td> </tr><tr> <td>[2]</td> <td class="space">V. Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Menor.Accad. Lincei., 6 (1926), 31-113. · <a href="/52.0450.06" class="nowrap">JFM 52.0450.06</a></td> </tr><tr> <td>[3]</td> <td class="space">C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320.</td> </tr><tr> <td>[4]</td> <td class="space">M. L. Rosenzweig, R. H. MacArthur, Graphical representation and stability conditions of predatorprey interactions, Am. Nat., 97 (1963), 209-223.</td> </tr><tr> <td>[5]</td> <td class="space">C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, The Memoirs of the Entomological Society of Canada, 97 (1965), 5-60.</td> </tr><tr> <td>[6]</td> <td class="space">J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology 56 (1975), 855-867.</td> </tr><tr> <td>[7]</td> <td class="space">R. M. Anderson, R. M. May, Regulation and stability of host-parasite population interactions I. Regulation processes, J. Anim. Ecol., 47 (1978), 219-247.</td> </tr><tr> <td>[8]</td> <td class="space">R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 2001. · <a href="/1044.92047" class="nowrap">Zbl 1044.92047</a></td> </tr><tr> <td>[9]</td> <td class="space">M. Mobilia, I. T. Georgiev, U. C. Täuber, Phase transitions and spatio-temporal fluctuations in stochastic lattice Lotka-Volterra models, J. Stat. Phys., 128 (2007), 447-483. · <a href="/1117.82029" class="nowrap">Zbl 1117.82029</a></td> </tr><tr> <td>[10]</td> <td class="space">U. Dobramysl, M. Mobilia, M. Pleimling, U. C. Täuber, Stochastic population dynamics in spatially extended predator-prey systems, J. Phys. A Math. Theor., 51 (2018), 063001. · <a href="/1385.92041" class="nowrap">Zbl 1385.92041</a></td> </tr><tr> <td>[11]</td> <td class="space">J. A. Langa, A. Rodríguez-Bernal, A. Suárez, On the long time behavior of non-autonomous Lotka-Volterra models with diffusion via the sub-supertrajectory method, J. Differ. Equations, 249 (2010), 414-445. · <a href="/1195.35178" class="nowrap">Zbl 1195.35178</a></td> </tr><tr> <td>[12]</td> <td class="space">F. M. De Oca, M. Vivas, Extinction in a two dimensional Lotka-Volterra system with infinite delay, Nonlinear Anal. Real World Appl., 7 (2006), 1042-1047. · <a href="/1122.34058" class="nowrap">Zbl 1122.34058</a></td> </tr><tr> <td>[13]</td> <td class="space">R. Bhattacharyya, B. Mukhopadhyay, On an eco-epidemiological model with prey harvesting and predator · <a href="/1205.34048" class="nowrap">Zbl 1205.34048</a></td> </tr><tr> <td>[14]</td> <td class="space">B. W. Kooi, E. Venturino, Ecoepidemic predator-prey model with feeding satiation, prey herd behavior and abandoned infected prey, Math. Biosci., 274 (2016), 58-72. · <a href="/1336.37029" class="nowrap">Zbl 1336.37029</a></td> </tr><tr> <td>[15]</td> <td class="space">Y. Xiao, L. Chen, Analysis of a three species eco-epidemiological model, J. Math. Anal. Appl., 258 (2001), 733-754. · <a href="/0967.92017" class="nowrap">Zbl 0967.92017</a></td> </tr><tr> <td>[16]</td> <td class="space">J. Zhao, J. Jiang, Permanence in nonautonomous Lotka-Volterra system with predator-prey, Appl.Math. Comput., 152 (2004), 99-109. · <a href="/1047.92050" class="nowrap">Zbl 1047.92050</a></td> </tr><tr> <td>[17]</td> <td class="space">J. Chattopadhyay, S. Chatterjee, E. Venturino, Patchy agglomeration as a transition from monospecies to recurrent plankton blooms, J. Theor. Biol., 253 (2008), 289-295. · <a href="/1398.92270" class="nowrap">Zbl 1398.92270</a></td> </tr><tr> <td>[18]</td> <td class="space">V. Ajraldi, M. Pittavino, E. Venturino, Modeling herd behavior in population systems, NonlinearAnal. Real World Appl., 12 (2011), 2319-2338. · <a href="/1225.49037" class="nowrap">Zbl 1225.49037</a></td> </tr><tr> <td>[19]</td> <td class="space">P. A. Braza, Predator-prey dynamics with square root functional responses, Nonlinear Anal. RealWorld Appl., 13 (2012), 1837-1843. · <a href="/1254.92072" class="nowrap">Zbl 1254.92072</a></td> </tr><tr> <td>[20]</td> <td class="space">T. Zhang, Y. Xing, H. Zang, M. Han, Spatio-temporal dynamics of a reaction-diffusion system for a predator-prey model with hyperbolic mortality, Nonlinear Dynam., 78 (2014), 265-277.</td> </tr><tr> <td>[21]</td> <td class="space">C. Xu, S. Yuan, T. Zhang, Global dynamics of a predator-prey model with defence mechanism for prey, Appl. Math. Lett., 62 (2016), 42-48. · <a href="/1351.34052" class="nowrap">Zbl 1351.34052</a></td> </tr><tr> <td>[22]</td> <td class="space">E. Venturino, An ecogenetic model, Appl. Math. Lett., 25 (2012), 1230-1233. · <a href="/1250.92043" class="nowrap">Zbl 1250.92043</a></td> </tr><tr> <td>[23]</td> <td class="space">M. D. Rahman, Modelling of an eco-genetic</td> </tr><tr> <td>[24]</td> <td class="space">C. Viberti, E. Venturino, A predator-prey model with genetically distinguishable predators, in Recent Advances in Environmental Sciences, Proceedings of the 9th International Conference onEnergy, Environment, Ecosystems and Sustainable Development (EEESD13), Lemesos, Cyprus,March 21st-23rd 2013 (eds. A. Kanarachos, N. E. Mastorakis), WSEAS Press, (2013), 87-92.</td> </tr><tr> <td>[25]</td> <td class="space">C. Viberti, E. Venturino, An ecosystem with HTII response and predators’ genetic variability, Math. Model. Anal., 19 (2014), 371-394. · <a href="/1488.92088" class="nowrap">Zbl 1488.92088</a></td> </tr><tr> <td>[26]</td> <td class="space">L. Castellino, S. Peretti, S. Rivoira, E. Venturino, A mathematical ecogenetic predator-prey model where both populations are genetically distinguishable, AIP Conference Proceedings, 1776 (2016), 020006.</td> </tr><tr> <td>[27]</td> <td class="space">I. Azzali, G. Marcaccio, R. Turrisi, E. Venturino, A genetically distinguishable competition model, in Modern Mathematical Methods and High Performance Computing in Science and Technology,Springer Proceedings in Mathematics and Statistics (eds. V. Singh, H. Srivastava, E. Venturino, M. Resch, V. Gupta), Springer, Singapore, 171 (2016), 129-140.</td> </tr><tr> <td>[28]</td> <td class="space">R. M. Anderson, R. May, The invasion persistence and spread of infectious diseases within animal and plant communities, Philos. T. R. Soc. B, 314 (1986), 533-570.</td> </tr><tr> <td>[29]</td> <td class="space">K. P. Hadeler, H. I. Freedman, Predator-prey populations with parasitic infection, J. Math. Biol., 27 (1989), 609-631. · <a href="/0716.92021" class="nowrap">Zbl 0716.92021</a></td> </tr><tr> <td>[30]</td> <td class="space">E. Venturino, The influence of diseases on Lotka-Volterra systems, Rocky Mt. J. Math., 24 (1994), 381-402. · <a href="/0799.92017" class="nowrap">Zbl 0799.92017</a></td> </tr><tr> <td>[31]</td> <td class="space">J. Chattopadhyay, O. Arino, A predator-prey model with disease in the prey, Nonlinear Anal., 36 (1999), 747-766. · <a href="/0922.34036" class="nowrap">Zbl 0922.34036</a></td> </tr><tr> <td>[32]</td> <td class="space">Y. Xiao, L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math.Biosci., 171 (2001), 59-82. · <a href="/0978.92031" class="nowrap">Zbl 0978.92031</a></td> </tr><tr> <td>[33]</td> <td class="space">L. Han, Z. Ma, H. W. Hethcote, Four predator prey models with infectious diseases, Math. Comput.Model., 34 (2001), 849-858. · <a href="/0999.92032" class="nowrap">Zbl 0999.92032</a></td> </tr><tr> <td>[34]</td> <td class="space">E. Venturino, Epidemics in predator-prey</td> </tr><tr> <td>[35]</td> <td class="space">P. van den Driessche, M. L. Zeeman, Disease induced oscillations between two computing species, SIAM J. Appl. Dyn. Syst., 3 (2004), 601-619. · <a href="/1055.92058" class="nowrap">Zbl 1055.92058</a></td> </tr><tr> <td>[36]</td> <td class="space">H. W. Hethcote, W. Wang, L. Han, Z. Ma, A predator-prey model with infected prey, Theor. Popul.Biol., 66 (2004), 259-268.</td> </tr><tr> <td>[37]</td> <td class="space">M. Haque, J. Zhen, E. Venturino, An ecoepidemiological predator-prey model with standard disease incidence, Math. Method. Appl. Sci., 32 (2009), 875-898. · <a href="/1158.92035" class="nowrap">Zbl 1158.92035</a></td> </tr><tr> <td>[38]</td> <td class="space">X. Zhou, J. Cui, X. Shi, X. Song, A modified Leslie-Grower predator-prey model with prey infection, J. Appl. Math. Comput., 33 (2010), 471-487. · <a href="/1205.34061" class="nowrap">Zbl 1205.34061</a></td> </tr><tr> <td>[39]</td> <td class="space">S. Belvisi, E. Venturino, An ecoepidemic model with diseased predators and prey group defense, Simul. Model. Pract. Th., 34 (2013), 144-155.</td> </tr><tr> <td>[40]</td> <td class="space">S. Sharma, G. P. Samanta, A ratio-dependent predator-prey model with Allee effect and disease in prey, J. Appl. Math. Comput., 47 (2015), 345-364. · <a href="/1320.92074" class="nowrap">Zbl 1320.92074</a></td> </tr><tr> <td>[41]</td> <td class="space">C. Chicone, Ordinary Differential Equations with Applications, 2^nd edition, Texts in Applied Mathematics 34, Springer, New York, 2006. · <a href="/1120.34001" class="nowrap">Zbl 1120.34001</a></td> </tr><tr> <td>[42]</td> <td class="space">J. Hofbauer, K. Sigmund, The Theory of Evolution and Dynamical</td> </tr><tr> <td>[43]</td> <td class="space">J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981. · <a href="/0464.58001" class="nowrap">Zbl 0464.58001</a></td> </tr><tr> <td>[44]</td> <td class="space">T. C. Sideris, Ordinary Differential Equations and Dynamical Systems, Atlantis Press, Paris France, 2013. · <a href="/1293.34001" class="nowrap">Zbl 1293.34001</a></td> </tr><tr> <td>[45]</td> <td class="space">J. Zhao, X. Guo, Z. Han, Z. Chen, Average conditions for competitive exclusion in a nonautonomous two dimensional Lotka-Volterra system, Math. Comput. Model., 57 (2013), 1131-1138.</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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