Non-constant Positive Steady States Of A FIV Cross-Diffusion Model

In this paper,we focus on the existence and stability of non-constant positive steady states of a FIV cross-diffusion model.First,stability of nonnegative equilibrium points of the local model are giv-en.By applying linearization method,the local stability of nonnegative equilibrium points in the corresponding sernilinear reaction-diffusion model are discussed.By constructing a Lyapunov function,the global stability of the unique positive equi-librium point is studied.In particular,the effects of cross-diffusion on the stability of the positive equilibrium point are analysed,and sufficient conditions of Turing instability which caused by cross-diffusion are givenThen,by the maximum principle,Harnack inequality and standard elliptic reg-ularity,we derive a priori estimates of positive steady states for the cross-diffusion system.By using the implicit function theorem and the Laray-Schauder degree theory,the non-existence and existence of non-constant positive steady states are obtained,respectively.It is shown that susceptible individuals and infected individ-uals can coexist in certain conditionsFinally,by taking the cross-diffusion coefficient as branch parameter,using asymptotic analysis method,an explicit formula for the non-constant steady states and their stability are studied.