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On the effects of protection zone and directed population flux in prey-predator dynamics

Kousuke Kuto, Kazuhiro Oeda

TL;DR

The work addresses how a prey refuge and directed predator movement shape stationary and time-dependent dynamics in a spatial predator–prey model. It develops a local-in-time well-posedness theory for the time-dependent problem under Neumann boundaries, using a quasilinear evolution framework and a cut-off to preserve ellipticity. For the stationary problem, it establishes bifurcation from semitrivial states, constructs global bifurcation branches, and reveals turning-point structures that arise from the interaction of protection zones and directed movement. In the large-directional-movement limit, the study shows coexistence branches collapse to semitrivial states or converge, after rescaling, to a limiting LP2 system with its own global bifurcation structure, highlighting rich qualitative changes induced by refuges and far-sighted predator motion.

Abstract

We study a spatial predator-prey model in which prey can enter a protection zone (refuge) inaccessible to predators, while predators exhibit directed movement toward prey-rich regions. The directed movement is modeled by a far-sighted population flux motivated by classical movement rules, in contrast to the more commonly analyzed near-sighted chemotaxis-type mechanisms. We first establish local-in-time well-posedness for the corresponding nonstationary problem under Neumann boundary conditions, despite the discontinuity induced by the refuge interface. We then investigate the stationary problem, focusing on how the coexistence states emerge and organize globally in parameter space. In particular, we identify the bifurcation threshold for positive steady states from semitrivial predator-only equilibria, and describe the global continuation of the resulting branches. Our analysis reveals that strong directed movement can induce turning-point structures and multiplicity of coexistence steady states, highlighting a nontrivial interplay between spatial protection and predator movement behavior.

On the effects of protection zone and directed population flux in prey-predator dynamics

TL;DR

The work addresses how a prey refuge and directed predator movement shape stationary and time-dependent dynamics in a spatial predator–prey model. It develops a local-in-time well-posedness theory for the time-dependent problem under Neumann boundaries, using a quasilinear evolution framework and a cut-off to preserve ellipticity. For the stationary problem, it establishes bifurcation from semitrivial states, constructs global bifurcation branches, and reveals turning-point structures that arise from the interaction of protection zones and directed movement. In the large-directional-movement limit, the study shows coexistence branches collapse to semitrivial states or converge, after rescaling, to a limiting LP2 system with its own global bifurcation structure, highlighting rich qualitative changes induced by refuges and far-sighted predator motion.

Abstract

We study a spatial predator-prey model in which prey can enter a protection zone (refuge) inaccessible to predators, while predators exhibit directed movement toward prey-rich regions. The directed movement is modeled by a far-sighted population flux motivated by classical movement rules, in contrast to the more commonly analyzed near-sighted chemotaxis-type mechanisms. We first establish local-in-time well-posedness for the corresponding nonstationary problem under Neumann boundary conditions, despite the discontinuity induced by the refuge interface. We then investigate the stationary problem, focusing on how the coexistence states emerge and organize globally in parameter space. In particular, we identify the bifurcation threshold for positive steady states from semitrivial predator-only equilibria, and describe the global continuation of the resulting branches. Our analysis reveals that strong directed movement can induce turning-point structures and multiplicity of coexistence steady states, highlighting a nontrivial interplay between spatial protection and predator movement behavior.
Paper Structure (8 sections, 18 theorems, 219 equations, 5 figures)

This paper contains 8 sections, 18 theorems, 219 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Local-in-time solvability
  3. Main results on the stationary problem
  4. Proofs of Theorems \ref{['exthm1']} and \ref{['exthm2']}
  5. A priori estimates and the nonexistence region for positive stationary solutions
  6. Bifurcation structure of positive stationary solutions
  7. Proof of Theorem \ref{['abthm']}
  8. Proof of Theorem \ref{['LP2thm']}

Key Result

Theorem 2.1

Assume $N=2$. Under the assumptions (A1)-(A6), (F1) and (F2), there exists a unique local-in-time solution $\boldsymbol{U}$ of evo on some interval $[0,T]$, with and together with Here the existence time $T>0$ depends only on $\|A(\boldsymbol{U}_{0})^{\gamma}\boldsymbol{U}_{0}\|_{\mathbb{L}^{2}}$. Moreover, the solution $\boldsymbol{U}$ satisfies the estimate for some $C>0$ depending on $\|A(\

Figures (5)

  • Figure 1: Protection zone $\Omega_{0}$
  • Figure 2: (Left) Graphs of $\lambda=\sigma_{1}(b\mu\mathrm{1}\space\mathrm{l}_{\Omega_{1}})$ and $\lambda=|\mu|/c$ in the $\lambda\mu$ plane. (Right) Typical bifurcation diagrams: (i) the case $\alpha>\alpha^{*}$ in Theorem \ref{['exthm1']}; (ii) the case $\alpha<\alpha^{*}$ in Theorem \ref{['exthm1']}; (iii) the case in Theorem \ref{['exthm2']}.
  • Figure 3: (Left) Graph of $\lambda = \sigma_{1}(b\mu\mathrm{1}\space\mathrm{l}_{\Omega_{1}})$ in the $\lambda\mu$ plane. (Right) Typical bifurcation diagram expected in Theorem 3.5.
  • Figure 4: A conjectured bifurcation diagram of positive solutions of \ref{['sp']} in the regime of large directional movement strength.
  • Figure 5: Graphs of $\lambda=\widetilde{\ell} (\mu, \alpha)$ and $\mu=m(\lambda, \alpha)$ in the $\lambda\mu$ plane in Propositions \ref{['nonexprop1']} and \ref{['nonexprop2']}.

Theorems & Definitions (31)

  • Theorem 2.1
  • Corollary 2.1
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 21 more