Turing Instability And Spatial Patterns Of A Predator-prey System With Nonlinear Cross-diffusion

In this thesis,we mainly investigate the Turing instability and stationary problem of the following predator-prey system with nonlinear cross-diffusion and BeddingtonDe Angelis functional response???where ????R^N is a bounded domain with smooth boundary ????,n is the outward unit normal vector on ????,the parameters d_i,?_i,?_i?i=1,2?,K,a,b,d,e,m are all positive constants.The initial values u₀ and v₀ are non-negative smooth functions which are not identically zero.The functions u?x,t? and v?x,t? stand for the population densities of prey and predator at time and the location ,respectively.In Chapter 2,we first study the stability of the unique positive constant solution for this system and the corresponding reaction system and then obtain sufficient conditions of the appearance of Turing patterns by using the linearization method.Our results imply that sufficiently large cross-diffusion effect ?₁may lead to the occurrence of Turing instability.Subsequently in Chapter 3,based on the Maximum principle and Harnack inequality,we give a priori positive lower and upper bounds for the steady-state problem of?1.5?in order to calculate the topological degree.In Chapter 4,the non-existence of non-constant positive steady states with vanished cross-diffusions is discussed.Furthermore,we establish the existence of non-constant positive steady states for the cross-diffusion system?1.5?and then show that large cross-diffusion effect ?₁ or ?₂ can create non-constant positive steady states.Our results exhibit that large cross-diffusion effect of the prey will lead to the appearance of Turing instability as well as non-constant positive steady state.