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Nonlinear Sci. Numer. Simul.</a> <a href="/?q=in%3A506569" title="Articles in this Issue">128, Article ID 107596, 31 p. (2024)</a>. </div> <div class="abstract">Summary: Predator-prey interactions are among the most complicated interactions between biological species, in which there may be both delay digestion and stochastic fluctuation. In the literature, due to the computational complexity and the lack of available methods, few papers have directly considered the predator-prey model for the combination of general infinite time delays with stochasticity. In this sense, this work is devoted to studying the long-term qualitative behavior of a stochastic predator-prey system with general infinite distributed delay. First, we establish sufficient and necessary conditions for exponential extinction and persistence of species. Notably, not only do our results cover the predator-prey model without stochastic noises and time delays, but also reveal the impact of environmental noises on population dynamics. Moreover, there are some challenges to analyze the stationary distribution of the numbers of prey and predator under species persistence. Based on the recent work [<span class="zbmathjax-textit">B. Zhou</span> et al., Chaos Solitons Fractals 137, Article ID 109865, 10 p. (2020; <a href="/1489.92192">Zbl 1489.92192</a>)] where the density function of a three-dimensional avian influenza model with non-degenerate diffusion is concerned, we further generalize the relevant theories to a five-dimensional population system with degenerate diffusion. Via solving the corresponding Fokker-Planck equations, several approximate expressions of the density function of the stochastic predator-prey system with special distribution delay kernels are obtained. Finally, we provide some numerical simulations to supplement the analytical results and study the impact of stochastic noises.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A7784253">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A7784253">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34K60" title="MSC2020">34K60</a> </td> <td class="space"> Qualitative investigation and simulation of models involving functional-differential equations </td> </tr><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%3A34K50" title="MSC2020">34K50</a> </td> <td class="space"> Stochastic functional-differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34K25" title="MSC2020">34K25</a> </td> <td class="space"> Asymptotic theory of functional-differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Astochastic+predator-prey+system">stochastic predator-prey system</a>; <a href="/?q=ut%3AInfinite+distributed+delay">Infinite distributed delay</a>; <a href="/?q=ut%3Aextinction+exponentially">extinction exponentially</a>; <a href="/?q=ut%3Apersistence">persistence</a>; <a href="/?q=ut%3AFokker-Planck+equation">Fokker-Planck equation</a>; <a href="/?q=ut%3Adensity+function">density function</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1489.92192">Zbl 1489.92192</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <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;"> <!-- alert for connection errors etc --> </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"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1531.34075" data-ciurl="/ci/07784253" data-biburl="/bibtex/07784253.bib" data-amsurl="/amsrefs/07784253.bib" data-xmlurl="/xml/07784253.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07784253.pdf" title="Zbl 1531.34075 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/10.1016/j.cnsns.2023.107596" aria-label="DOI for “Threshold dynamics and probability density functions of a stochastic predator-prey model with general distributed delay”" title="10.1016/j.cnsns.2023.107596">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Zhang, S.; Meng, X.; Feng, T.; Zhang, T., Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects. 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