Hopf bifurcations in general systems of Brusselator type

Abstract

This paper is concerned with general models of Brusselator type subject to the homogeneous Neumann boundary condition. The existence of Hopf bifurcation for the ODE and PDE models is obtained. By the center manifold theory and the normal form method, the bifurcation direction and stability of bifurcating periodic solutions are established. Moreover, some numerical simulations are shown to support the analytical results.

Introduction

Many physical, chemical, biological, environmental and even sociological processes are driven by reaction–diffusion systems. Turing  [1] showed that a system of coupled reaction–diffusion equations can be used to describe patterns and forms in biological systems. Turing’s theory shows that diffusion could destabilize an otherwise stable equilibrium of the reaction–diffusion system and lead to nonuniform spatial patterns. This kind of instability is usually called Turing instability or diffusion-driven instability.

Turing’s analysis stimulated considerable theoretical research on mathematical models of pattern formation, and a great deal of research has been devoted to the study of Turing instability in chemical and biology contexts. The Brusselator model was introduced in 1968 by Prigogine and Lefever  [2] as a model for a chemical oscillating reaction of self catalysis. It consists of the following four reaction steps:

$$A\to X\text{,}B+X\to Y+D\text{,}2X+Y\to 3X\text{,}X\to E\text{.}$$

The over-all reaction is $A+B\to D+E$. By some scaling and change of variable, the mathematical model corresponding to the Brusselator system is

$$\{\begin{array}{cc}\frac{\partial u}{\partial t}-d_1\Delta u=a-(b+1)u+u^{2}v\text{,}\hfill & x\in \Omega ,t>0\text{,}\hfill \\ \frac{\partial v}{\partial t}-d_2\Delta v=bu-u^{2}v\text{,}\hfill & x\in \Omega ,t>0\hfill \end{array}$$

subject to the homogeneous Neumann boundary conditions. Here $\Omega \subset R^N$ is a smooth and bounded domain, the unknowns $u,v$ represent the concentration of the two reactants having the diffusion rates $d_1,d_2>0$, and $a,b>0$ are the fixed concentrations. Many scholars have investigated the Brusselator model and other models of Brusselator type. In paper  [3], the authors considered the following general Brusselator model

$$\{\begin{array}{cc}\frac{\partial u}{\partial t}-d_1\Delta u=a-(b+1)u+u^{p}v\text{,}\hfill & x\in \Omega ,t>0\text{,}\hfill \\ \frac{\partial v}{\partial t}-d_2\Delta v=bu-u^{p}v\text{,}\hfill & x\in \Omega ,t>0\hfill \end{array}$$

subject to the homogeneous Neumann boundary conditions, where $p>0$. Existence and non-existence of positive steady states are obtained. Pattern formation in system of Brusselator type is also investigated in  [4], [5]. In paper  [5], a more general Brusselator model

$$\{\begin{array}{cc}\frac{\partial u}{\partial t}-d_1\Delta u=a-(b+1)u+f\left(u\right)v\text{,}\hfill & x\in \Omega ,t>0\text{,}\hfill \\ \frac{\partial v}{\partial t}-d_2\Delta v=bu-f\left(u\right)v\text{,}\hfill & x\in \Omega ,t>0\text{,}\hfill \\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0\text{,}\hfill & x\in \partial \Omega ,t>0\text{,}\hfill \\ u(x,0)=u_0\left(x\right)\ge 0\text{,}v(x,0)=v_0\left(x\right)\ge 0\text{,}\hfill & x\in \Omega \text{,}\hfill \end{array}$$

is investigated. Here function $f\in C^1(0,\infty )\cap C[0,\infty )$ is a non-decreasing function. The authors revealed the fact that the dynamics of the evolution system (1.1) and its associated steady-state is strongly related to the behavior of the nonlinearity $f$. The authors assumed that $f$ is either sublinear or superlinear, and discussed Turning pattern. They pointed out the crucial role played by the nonlinearity $f$ in the existence of Turing patterns. More precisely, they showed that if $f$ has a sublinear growth, then no Turing patterns occur, while if $f$ has a superlinear growth, then existence of such patterns is strongly related to the inter-dependence between the parameters $a,b$ and the diffusion coefficients $d_1,d_2$.

In this paper, we shall consider the Hopf analysis in the problem (1.1). Concerning with the Hopf bifurcation analysis, in paper  [6], the authors discussed the following Brusselator model

$$\{\begin{array}{cc}\frac{\partial u}{\partial t}-d_1\Delta u=1-(b+1)u+b{u}^{2}v\text{,}\hfill & x\in \Omega ,t>0\text{,}\hfill \\ \frac{\partial v}{\partial t}-d_2\Delta v=a^2(u-u^{2}v)\text{,}\hfill & x\in \Omega ,t>0\text{.}\hfill \end{array}$$

Treating $a$ as a bifurcation parameter, the authors derived the Hopf bifurcation phenomenon. The systems of Brusselator type have been extensively investigated in the last decades from both analytical and numerical point of view (see  [7], [8], [9], [10], [11], [12], [13], [14]).

In this paper, we focus on the existence, stability and direction of Hopf bifurcation for problem (1.1) and its corresponding ordinary differential model. Employing the center manifold theory and normal method, an algorithm for determining the properties of the Hopf bifurcation is derived. Some numerical simulations for illustrating the analysis results are carried out. We refer to  [15], [16] for the detailed method used in this paper. We shall choose some parameter concerning with the function $f$ as a Hopf bifurcation parameter, and aim to reveal the role of $f$ in the existence of Hopf bifurcation. Our analysis will show that if $f$ satisfies that $a{f}^{\prime }\left(a\right)\le f\left(a\right)$, then no Hopf bifurcation occurs, and while $a{f}^{\prime }\left(a\right)>f\left(a\right)$ holds, Hopf bifurcation emerges in ODE and PDE models. Compared with  [5], we discuss the same problem, but the content, assumption and conclusions are different. The ideal of the present paper is from  [17], and the authors discovered the general Glycolysis reaction–diffusion system with $f$.

The paper is organized as follows. In Section  2, Hopf bifurcation analysis of ODE system, including the existence and the stability, is obtained. In Section  3, for reaction–diffusion problem, we discuss the local stability of positive constant equilibrium, and derive the existence of Hopf bifurcation. Stability and direction of spatial Hopf bifurcation are given in Section  4. Numerical simulations are shown in Section  5.