Bifurcation, Chaos, and Pattern Formation for the Discrete Predator-Prey Reaction-Diffusion Model

Lili Meng,Zhiyi Lu,Yutao Han,Guang Zhang

doi:10.1155/2019/9592878

Abstract

In this paper, a discrete predator-prey system with the periodic boundary conditions will be considered. First, we get the conditions for producing Turing instability of the discrete predator-prey system according to the linear stability analysis. Then, we show that the discrete model has the flip bifurcation and Turing bifurcation under the critical parameter values. Finally, a series of numerical simulations are carried out in the Turing instability region of the discrete predator-prey model; some new Turing patterns such as striped, bar, and horizontal bar are observed.

Highlights

Interaction between species and their natural environment is the main characteristic of ecological systems ([1])
Predator-prey models follow two principles: one is that population dynamics can be decomposed into birth and death processes; the other is the conservation of mass principle, stating that predators can grow only as a function of what they have eaten ([6])
Patterns are ever-present in the chemical and biological worlds; pattern formation is a fundamental problem in the study of far-from-equilibrium phenomena in spatially extended systems

Summary

Introduction

Interaction between species and their natural environment is the main characteristic of ecological systems ([1]). A reaction-diffusion system exhibits diffusion-driven instability or Turing instability if the homogeneous steady state is stable to small perturbations in the absence of diffusion, but it is unstable to small spatial perturbations when diffusion is present This approach allows us to understand and predict a variety of different phenomena, including the formation of structures that are similar to the patterns we observe in the living world ([8,9,10]). Gu = a21V, gV = −a22V, at the fixed point E(u∗, V∗), fu + gV < 0, fugV − fVgu > 0, but fu + dgV < 0; we can conclude that such a simple continuous predator-prey system can not generate Turing instability.

Turing Instability Analysis for the Discrete L-V Predator-Prey Systems

Bifurcations and Center Manifolds

Numerical Simulation

Conclusions