具有交错扩散Brusselator模型的Turing不稳定

Turing instability of Brusselator model with cross-diffusion

为了研究交错扩散对Turing不稳定现象的影响,通过在原有的自扩散Brusselator模型中加入交错扩散项,建立了一类Neumann边值条件下具有交错扩散的Brusselator模型。首先,给出常微分系统正平衡点的渐近稳定性。其次,运用线性化方法,得到线性系统对应的特征方程,给出交错扩散系统在正平衡解处发生Turing不稳定性的条件。进一步,在一维情况下,得到交错扩散系统发生Turing不稳定性更为具体的充分条件,在实际应用中更容易验证。最后,利用Matlab进行数值模拟,分析了交错扩散对于Turing不稳定性现象的影响。结果表明:在一定条件下,不管自扩散系统是稳定的还是不稳定的,加入交错扩散后都可以改变原来自扩散系统正平衡解的稳定性。因此,交错扩散可以造成系统产生Turing不稳定性,也可以使Turing不稳定性消失。

In order to study the effect of the cross-diffusion on Turing instability,on the basis of the original self-diffusion Brusselator model,the cross-diffusion term was incorporated,and Brusselator model with the cross-diffusion under Neumann boundary value conditions was established.Firstly,the asymptotic stability of the positive equilibrium point for the ordinary differential system was given.Secondly,the corresponding characteristic equation of the linear system was obtained by using linearization method,and the conditions for the occurrence of Turing instability at a positive equilibrium point in the cross-diffusion system were given.Furthermore,in a one-dimensional case,a more specific sufficient condition for the occurrence of Turing instability in the cross-diffusion system was obtained,which was easier to verify in practical applications.Finally,numerical simulation was presented using Matlab to analyze the impact of cross-diffusion diffusion on Turing instability.The results show that under certain conditions,no matter whether the self-diffusion system is stable or unstable,the stability of the positive equilibrium solution of the original self-diffusion system can be changed by adding cross-diffusion.Therefore,the cross-diffusion can cause the system to produce Turing instability,and can also make Turing instability vanish.