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

TL;DR

This paper analyzes a spatial predator-prey model with prey refuges and directed predator movement, revealing complex bifurcation structures and multiple coexistence states influenced by spatial protection and movement behavior.

Contribution

It introduces a novel predator movement model based on far-sighted flux and studies its impact on coexistence states and bifurcations in spatial predator-prey systems.

Findings

Directed movement induces bifurcation thresholds.

Strong directed movement causes multiple coexistence states.

Protection zones influence predator-prey dynamics significantly.

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.