Pattern formation of a predator-prey system with Ivlev-type functional response

TL;DR

This paper explores how diffusion influences pattern formation in a predator-prey system with Ivlev-type response, revealing conditions for various bifurcations and the impact of initial conditions on spatial patterns.

Contribution

It provides a detailed bifurcation analysis and numerical simulations of pattern formation in a reaction-diffusion predator-prey model with Ivlev response, highlighting the effects of initial conditions.

Findings

Hopf instability leads to spiral patterns

Turing instability results in chaotic spatial patterns

Initial conditions influence pattern formation in prey-dependent models

Abstract

In this paper, we investigate the emergence of a predator-prey system with Ivlev-type functional response and reaction-diffusion. We study how diffusion affects the stability of predator-prey coexistence equilibrium and derive the conditions for Hopf and Turing bifurcation in the spatial domain. Based on the bifurcation analysis, we give the spatial pattern formation, the evolution process of the system near the coexistence equilibrium point, via numerical simulation. We find that pure Hopf instability leads to the formation of spiral patterns and pure Turing instability destroys the spiral pattern and leads to the formation of chaotic spatial pattern. Furthermore, we perform three categories of initial perturbations which predators are introduced in a small domain to the coexistence equilibrium point to illustrate the emergence of spatiotemporal patterns, we also find that in the beginning of evolution of the spatial pattern, the special initial conditions have an effect on the formation of spatial patterns, though the effect is less and less with the more and more iterations. This indicates that for prey-dependent type predator-prey model, pattern formations do depend on the initial conditions, while for predator-dependent type they do not. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics.