The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system

Xin Wei,Junjie Wei

doi:10.15388/na.2018.5.7

Abstract

This paper deals with an arbitrary-order autocatalysis model with delayed feedback subject to Neumann boundary conditions. We perform a detailed analysis about the effect of the delayed feedback on the stability of the positive equilibrium of the system. By analyzing the distribution of eigenvalues, the existence of Hopf bifurcation is obtained. Then we derive an algorithm for determining the direction and stability of the bifurcation by computing the normal form on the center manifold. Moreover, some numerical simulations are given to illustrate the analytical results. Our studies show that the delayed feedback not only breaks the stability of the positive equilibrium of the system and results in the occurrence of Hopf bifurcation, but also breaks the stability of the spatial inhomogeneous periodic solutions. In addition, the delayed feedback also makes the unstable equilibrium become stable under certain conditions.

Highlights

Autocatalysis is the process whereby a chemical is involved in its own production
We study its effect on the stability of the positive equilibrium E∗ and prove the existence of Hopf bifurcation
We have studied the effect of delayed feedback on the dynamics of an arbitrary-order autocatalysis model

Summary

Introduction

Autocatalysis is the process whereby a chemical is involved in its own production. In recent years, the diffusive autocatalysis reaction models, which attracts much attention, have been extensively used in the studies of Turing instability or Turing pattern. The authors discussed the effect of diffusion coefficients on the existence of Hopf bifurcation. Motivated by the idea of Ott and Grebogi, many investigators have studied the effect of time delay in ecological and chemical models (see [1, 2, 7, 9, 13, 16, 24, 28, 30]). We study its effect on the stability of the positive equilibrium E∗ and prove the existence of Hopf bifurcation. The simulations show that the delayed feedback breaks the stability of E∗ and results in the occurrence of the Hopf bifurcation, and effects the stability of the spatial inhomogeneous periodic solutions

Stability analysis

Stability and direction of Hopf bifurcation

Numerical simulations

Conclusion