Turing Instability And Patterns Formation Of A Class Of Prey-Predator System

As one of the important tools to describe the interaction between the species of ecological system, the prey-predator model can generate rich dynamics and its dynamics behavior depends mainly on the properties of its steady states. In this thesis, we are mainly concerned about the Turing instability and the existence and non-existence of non-constant positive steady states, that is patterns, of a class of ratio-dependent Holling-Tanner prey-predator system with diffusion and cross-diffusion.In Chapter 2, we change the prey-predator system with diffusion and cross-diffusion into the other equivalent system by the method of transformation variable and discuss the stabilities of the two equilibria by the Linearization method; the corresponding ordinary system is also considered; finally, we analyze the effects of the cross-diffusion and self-diffusion to the stabilities of the equilibria.In Chapter 3, a priori estimates for the positive solutions of the corresponding elliptic problem are firstly obtained by the maximum principle and Harnack inequality; then we give some sufficient conditions for the non-existence of non-constant positive steady states by energy integral; finally, we obtain the existence of non-constant positive steady states by Leray-Schauder theory.Our result shows that the cross-diffusion and self-diffusion of the prey are beneficial to the constant spreads of the species; when the birth rate of the prey is very large, the cross-diffusion and self-diffusion of the predator are also beneficial to the constant spreads of the species; however, when the birth rate of the prey is in certain ranges, the cross-diffusion and self-diffusion of the predator may lead to Turing instability. Moreover, our result shows that even though the unique constant positive steady state is globally asymptotically stable for the system without cross-diffusion, non-constant positive steady state can still exist due to the emergence of the cross-diffusion for the system, which suggests that Turing pattern occurs.