一类扩散Holling-Tanner模型行波解的存在性

THE EXISTENCE OF TRAVELLING WAVE SOLUTIONS OF THE DIFFUSION HOLLING-TANNER MODEL

在生态学中,可以用非线性反应扩散方程来描述种群在时间上的变化和在空间中的分布及扩散情况.对于扩散的生物种群模型,通过研究模型中方程的渐近性态,可以知道该种群是持续生存还是趋向灭绝.在非线性反应扩散方程的研究中,行波解由于其形式简单,研究比较方便,为研究偏微分方程的动力学行为提供了一些途径.文章对一类添加扩散项的扩散Holling-Tanner系统进行了定性分析,得到了系统平衡点局部渐近稳定的充分条件.再通过构造Liapunov函数的方法,得到扩散Holling-Tanner系统平衡点全局渐近稳定的条件,以及该系统行波解存在的充分条件,并进行了数值模拟.

In ecology, the variation of population＇s density as time evolves and space changes can be described by nonlinear reaction diffusion equations. By considering the asymp- totic behavior of the differential equation models, the survival or extinction of the species can be obtained. In the study of nonlinear reaction diffusion equations, it is convenient to study travelling wave solution for its simple form. And it provides some ways for the study of dynamic behavior of the partial differential equations. In this paper, we study qualitative analysis on the diffusion Holling-Tanner system, find the sufficient condition of system local asymptotic stability. By constructing a Liapunov function, we obtain the condition of local asymptotic stability for the diffusion Holling-Tanner system, and then get the sufficient condition for the existence of traveling wave solutions. At last, we give some examples of numerical simulation as well.