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Permanence and Extinction of a Diffusive Predator–Prey Model with Robin Boundary Conditions

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Abstract

The main concern of this paper is to study the dynamic of a predator–prey system with diffusion. It incorporates the Holling-type-II and a modified Leslie–Gower functional responses under Robin boundary conditions. More concretely, we study the dissipativeness of the system by using the comparison principle, and we derive a criteria for permanence and for predator extinction.

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Notes

  1. We must thank the referee who drew our attention to this very recent works (Bassett et al. 2017; Kurowski et al. 2017) and which helped us to better present this work, especially with regard to the use of Robin’s boundary conditions.

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Acknowledgements

This work was partially supported by the ERDF (XTerm Project) and Normandie Region, France. And also by The LMAH-FR-CNRS-3335.

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Correspondence to M. A. Aziz-Alaoui.

Appendix: Permanence

Appendix: Permanence

For the convenience of the reader, we will summarize some facts contained in Hale and Waltman, see Hale and Waltman (1989), about the permanence for abstract dynamical systems. Suppose that \(\varOmega\) is a complete metric space with \(\varOmega =\varOmega _0\cup \partial \varOmega _0\) for an open set \(\varOmega _0\), where \(\partial \varOmega _0\) is the boundary of the set \(\varOmega _0\). We will typically choose \(\varOmega _0\) to be the positive cone in an ordered Banach space. A flow or semiflow on \(\varOmega\) under which \(\varOmega _0\) and \(\partial \varOmega _0\) are forward invariant is said to be permanent if it is dissipative and if there is a number \(\eta >0\) such that any trajectory starting in \(\varOmega _0\) will be at least a distance \(\eta\) from \(\partial \varOmega _0\) for all sufficiently large t. To state a theorem implying permanence we need a few definitions. An invariant set M for the flow or semiflow is said to be isolated if it has a neighborhood U such that M is the maximal invariant subset of U. Let \(\omega (\partial \varOmega _0) \subset \partial \varOmega _0\) denote the union of the sets \(\omega (u)\) over \(u\in \varOmega _0\) (This differs from the standard definition of the \(\omega\)-limit set of a set but it is more convenient for our purposes; see Hutson and Schmitt (1992) for a discussion). The set \(\omega (\varOmega _0)\) is said to be isolated if it has a covering \(M=\cup _{k=1}^{N}M_{k}\) of pairwise disjoint, both sets \(M_k\) which are isolated and invariant with respect to the flow or semiflow both on \(\partial \varOmega _0\) and on \(\varOmega =\varOmega _0\cup \partial \varOmega _0\). The covering M is then called an isolated covering. Suppose \(N_1\) and \(N_2\) are isolated invariant sets (not necessarily distinct). The set \(N_1\) is said to be chained to \(N_2\) (denoted \(N_{1}\rightarrow N_{2}\)) if there exists \(u\in N_{1}\cup N_{2}\) with \(u\in {W}^{u} (N_{1})\cap {W}^{s}(N_{2}).\) (As usual, \(W^{u}\) and \(W^{s}\) denote the unstable and stable manifolds, respectively). A finite sequence \(N_{1}, N_{2},\cdots , N_{k}\) of isolated invariant sets is a chain if \(N_{1}\rightarrow N_{2}\rightarrow N_{3}\rightarrow \cdots \rightarrow N_{k}\). (This is possible for \(k = 1\) if \(N_{1}\rightarrow N_{1}.)\) The chain is called a cycle if \(N_{k}= N_{1}.\)The set \(\omega (\partial \varOmega _{0})\) is said to be acyclic if there exists an isolated covering \(\cup _{k=1}^{N}M_{k}\) such that no subset of \(\left\{ M_{k}\right\}\) is a cycle. We now state a theorem that can be used to establish permanence.

Theorem 3

(Hale and Waltman 1989) Suppose that \(\varOmega\) is a complete metric space with \(\varOmega =\varOmega _0\cup \partial \varOmega _0\) where \(\varOmega _0\) is open. Suppose that a semiflow on \(\varOmega\) leaves both \(\varOmega _{0}\) and \(\partial \varOmega _{0}\) forward invariant, maps bounded sets in \(\varOmega\) to precompact set for \(t>0\), and is dissipative. If in addition:

  1. (i)

    \(\omega (\partial \varOmega _{0})\) is isolated and acyclic,

  2. (ii)

    \(W^{s}(M_{k})\cap \varOmega _{0}=\emptyset\) for all k, where \(\cup _{k=1}^{N}M_{k}\) is the isolated covering used in the definition of acyclicity of \(\partial \varOmega _{0}\), then the semiflow is permanent; i.e., there exist \(\eta >0\) such that any trajectory with initial data in \(\varOmega _{0}\) will be bounded away from \(\partial \varOmega _{0}\) by a distance greater than \(\eta\) for t sufficiently large.

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Aziz-Alaoui, M.A., Daher Okiye, M. & Moussaoui, A. Permanence and Extinction of a Diffusive Predator–Prey Model with Robin Boundary Conditions. Acta Biotheor 66, 367–378 (2018). https://doi.org/10.1007/s10441-018-9332-0

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