Predator-prey models with prey-dependent diffusion on predators in spatially heterogeneous habitat

Abstract

This article considers a predator–prey system with a certain type of prey-dependent diffusion for predators where the source of prey population depends on location in a habitat with spatial heterogeneity distributed within a bounded domain. In particular, it is assumed that the spread rate of predators can change depending on the satisfaction of predators according to the amount of available prey in the vicinity of predators in the habitat. First, how prey-dependent dispersal sensitively affects the migration mechanism of predators is examined. More precisely, it is shown that predators via such prey-dependent diffusion can invade a habitat region by investigating stability analysis of the semitrivial solution of the system where the predator is absent. Additionally, the existence and uniqueness of a positive steady state are studied using the fixed point index theory in a positive cone in a Banach space. The coexistence state is found to be unique if the diffusion rate of the prey is above a certain threshold and the average of the resource function of the prey is within a specific range represented through the equilibrium value of the prey.