Hyperbolic predators vs parabolic preys

TL;DR

This paper introduces a nonlinear predator-prey model combining a nonlocal conservation law for predators with a parabolic equation for preys, proving well-posedness and illustrating solution behaviors through numerical simulations.

Contribution

It develops a novel predator-prey system with nonlocal predator movement and provides rigorous mathematical analysis of existence, uniqueness, and stability of solutions.

Findings

Numerical simulations reveal qualitative solution features.

The model ensures well-posedness in any space dimension.

Nonlocal predator movement influences prey distribution dynamics.

Abstract

We present a nonlinear predator-prey system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for preys. The drift term in the predators' equation is a nonlocal function of the prey density, so that the movement of predators can be directed towards region with high prey density. Moreover, Lotka-Volterra type right hand sides describe the feeding. A theorem ensuring existence, uniqueness, continuous dependence of weak solutions and various stability estimates is proved, in any space dimension. Numerical integrations show a few qualitative features of the solutions.