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The Influence of Exclusion Zones on the Coexistence of Predator and Prey with an Allee Effect

Henri Berestycki, William F. Fagan, Alex Safsten

TL;DR

A reaction--diffusion model of predator--prey interaction in which the predators occupy only a subset of the prey's territory, leaving a predator-free exclusion zone is proposed, showing that as the predator domain becomes large, it may exhibit thresholding behavior, highlighting the need for careful analysis in the management of predator--prey systems.

Abstract

We propose a reaction--diffusion model of predator--prey interaction in which the predators occupy only a subset of the prey's territory, leaving a predator-free exclusion zone. Ecological examples include marine protected areas where it is illegal to fish, or buffer zones left between the territories of rival predators. The prey are subject to a strong Allee effect, so excessive predation may lead to the extinction of both species. The exclusion zone mitigates this problem by providing the prey with a refuge in which to proliferate without predation. Thus, paradoxically, a smaller predator territory may be able to support a more substantial population than a larger one. Using a topological degree argument, we show in any dimensions that, provided the exclusion zone is large enough, the system possesses spatially heterogeneous coexistence equilibria with positive populations of both species. This result is global in the sense that it does not rely on local bifurcations from semi-trivial stationary states. We also show that as the predator domain becomes asymptotically small, the total predator population does not vanish, and in some cases may actually be maximized in this limit of shrinking predation area. Conversely, we show that as the predator domain becomes large, it may exhibit thresholding behavior, passing suddenly from a regime with coexistence solutions to one in which extinction becomes unavoidable, highlighting the need for careful analysis in the management of predator--prey systems.

The Influence of Exclusion Zones on the Coexistence of Predator and Prey with an Allee Effect

TL;DR

A reaction--diffusion model of predator--prey interaction in which the predators occupy only a subset of the prey's territory, leaving a predator-free exclusion zone is proposed, showing that as the predator domain becomes large, it may exhibit thresholding behavior, highlighting the need for careful analysis in the management of predator--prey systems.

Abstract

We propose a reaction--diffusion model of predator--prey interaction in which the predators occupy only a subset of the prey's territory, leaving a predator-free exclusion zone. Ecological examples include marine protected areas where it is illegal to fish, or buffer zones left between the territories of rival predators. The prey are subject to a strong Allee effect, so excessive predation may lead to the extinction of both species. The exclusion zone mitigates this problem by providing the prey with a refuge in which to proliferate without predation. Thus, paradoxically, a smaller predator territory may be able to support a more substantial population than a larger one. Using a topological degree argument, we show in any dimensions that, provided the exclusion zone is large enough, the system possesses spatially heterogeneous coexistence equilibria with positive populations of both species. This result is global in the sense that it does not rely on local bifurcations from semi-trivial stationary states. We also show that as the predator domain becomes asymptotically small, the total predator population does not vanish, and in some cases may actually be maximized in this limit of shrinking predation area. Conversely, we show that as the predator domain becomes large, it may exhibit thresholding behavior, passing suddenly from a regime with coexistence solutions to one in which extinction becomes unavoidable, highlighting the need for careful analysis in the management of predator--prey systems.
Paper Structure (14 sections, 25 theorems, 124 equations, 6 figures, 1 table)

This paper contains 14 sections, 25 theorems, 124 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. The role of exclusion zones in predator-prey systems
  3. The Predator-Prey model with exclusion zones
  4. Coexistence solutions in the 1D problem
  5. Coexistence solutions in higher dimensions
  6. Related literature
  7. Existence of non-trivial equilibria
  8. Coexistence equilibrium in higher dimensions
  9. Persistence owing to exclusion zones
  10. Thin-limit analysis in 1D
  11. Further Properties and Open Problems
  12. Computations of equilibria 1D
  13. Discussion
  14. Future work

Key Result

Proposition 1.1

If $\alpha<\gamma$, then all solutions to eq:higher_dimensional_model with $0\leq u\leq 1$ and $v\geq 0$ satisfy $\lim_{t\to\infty}\Vert v\Vert_{L^1}$.

Figures (6)

  • Figure 1: A diagram of $A\subset\Omega$ illustrating the assumptions of Theorem \ref{['thm:existenceND']}.
  • Figure 2: In some parameter regimes, solutions to \ref{['eq:higher_dimensional_model']} approach a coexistence equilibrium (left). In others, solutions approach a limit cycle (right).
  • Figure 3: These contour plots show the population densities of prey (top) and predators (bottom) depending on time and space in as solutions of \ref{['eq:higher_dimensional_model']} approach a limit cycle.
  • Figure 4: A plot of the limiting profiles exhibiting an interior maximum, Hopf bifurcation, and extinction.
  • Figure 5: A plot of the limiting profiles exhibiting a local maximum in the thin-limit, a Hopf bifurcation and a global maximum at extinction.
  • ...and 1 more figures

Theorems & Definitions (44)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 34 more