一类三种群食物链模型中交错扩散引起的Turing不稳定

Turing instability induced by cross-diffusion in a three-species food chain model

研究了一类三种群食物链模型的强耦合交错扩散系统。首先通过构造Lyapunov函数证明唯一的正平衡点在ODE系统下是全局渐近稳定的,当交错扩散系数均为零时,唯一的正平衡点仍是全局渐近稳定的。当引入交错扩散时,正平衡点则变得不稳定。利用Routh-Hurwitz准则和Descartes符号法则证明了大的交错扩散系数(k21或k32足够大时)可以导致平衡点由原来的稳定变得不稳定。最后利用数学软件Matlab对我们的结果进行数值模拟,得到了不同类型的Turing斑图,包括六边形、条状以及二者共存的斑图。

This paper considers a strong coupled cross-diffusion system about a three-species food chain model. We first prove that the unique positive equilibrium solution is globally linearly stable for the ODE system and remains globally linearly stable when the reaction-diffusion system without cross-diffusion by constructing Lyapunov functions. Then we use the Routh-Hurwitz criterion and Descartes＇ rule to illustrate that the unique positive equilibrium solution becomes linearly unstable only when the cross-diffusion plays a role in this reaction-diffusion system. Finally, numerical simulations are performed to test our theoretical results by means of Matlab. We can obtain different types of patterns including spotted, striped and mixture patterns.