Coexistence of heterogenous predator-prey systems with density-dependent dispersal

TL;DR

This study investigates the conditions for coexistence, non-existence, and uniqueness of positive steady states in a predator-prey system with density-dependent dispersal, revealing its beneficial role in species survival.

Contribution

It introduces a novel analytical approach using variable transformation and index theory to classify solutions in a complex predator-prey model with cross-diffusion.

Findings

Density-dependent dispersal promotes species coexistence.

Unique positive solutions are characterized for various diffusion rates.

Dispersal strategies enhance predator survival chances.

Abstract

This paper is concerned with existence, non-existence and uniqueness of positive (coexistence) steady states to a predator-prey system with density-dependent dispersal. To overcome the analytical obstacle caused by the cross-diffusion structure embedded in the density-dependent dispersal, we use a variable transformation to convert the problem into an elliptic system without cross-diffusion structure. The transformed system and pre-transformed system are equivalent in terms of the existence or non-existence of positive solutions. Then we employ the index theory alongside the method of the principle eigenvalue to give a nearly complete classification for the existence and non-existence of positive solutions. Furthermore we show the uniqueness of positive solutions and characterize the asymptotic profile of solutions for small or large diffusion rates of species. Our results pinpoint the positive role of density-dependent dispersal on the population dynamics for the first time by showing that the density-dependent dispersal is a beneficial strategy promoting the coexistence of species in the predator-prey system by increasing the chance of predator's survival.