Long time dynamics of a three-species food chain model with Allee effect in the top predator

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• 作者：Rana D. Parshad^a ; ^{rparshad@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Emmanuel Quansah^a ; Kelly Black^a ; Ranjit Kumar Upadhyay^b ; S.K. Tiwari^b ; Nitu Kumari^c
• 关键词：Three species reaction&ndash ; diffusion food chain model ; Global existence ; Global attractor ; Upper semi-continuity ; Allee effect ; Turing instability
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：71
• 期：2
• 页码：503-528
• 全文大小：5916 K

文摘

The Allee effect is an important phenomenon in population biology characterized by positive density dependence, that is a positive correlation between population density and individual fitness. However, the effect is not well studied in multi-level trophic food chains. We consider a ratio dependent spatially explicit three species food chain model, where the top predator is subjected to a strong Allee effect. We show the existence of a global attractor for the model, that is upper semicontinuous in the Allee threshold parameter m$m$. Next, we numerically investigate the decay rate to a target attractor, that is when m=0$m=0$, in terms of m$m$. We find decay estimates that are O(m^γ)$\mathcal{O}\left(m^{\gamma }\right)$, where γ$\gamma$ is found explicitly. Furthermore, we prove various overexploitation theorems for the food chain model, showing that overexploitation has to be driven by the middle predator. In particular overexploitation is not possible without an Allee effect in place. We also uncover a rich class of Turing patterns in the model which depend significantly on the Allee threshold parameter m$m$. Our results have potential applications to trophic cascade control, conservation efforts in food chains, as well as Allee mediated biological control.