Competition With Cross-diffusion Equations Is The Number Of The Existence Of Equilibrium Solution

This paper is concerned with the existence of a class of positive steady states for a quasi-linear cross-diffusion system describing two-species competition.This is a biological model derived from the classical competitive model. For studying the steady solution of (1), we just need to study the following elliptic problemThis paper contains the following two parts:In chapter 2, we first study the stability of negative constant solutions of (1), we prove that ,if u_* satisfies some conditions andγis big enough, then the positive constant solution (u_*,v_*) turns to be instable. We try to prove that there exists non-constant positive solutions of (2) , which bifurcate from (u_*,v_*) . Let <φ> = (d₁ +γv)u,ψ=d₂v, the systems (2) can be rewritten asBy applying bifucation theory to (3), we obtain the existence of bifurcation solution, at last we discuss the stability of the bifurcation solution. The main result is:Theoreml If d₁ > 0 fixed,(?)< d₂ < c₁, u_* > max{(?)}, thenthere exists nontrivial solution curve of (3)whereδis a small positive constant ,φ₀ = K(γ₁u_*f_u^*- (d₁ +γ₁v_*)f_v^*,d₂f_u^*-d₂Ï€²(d₁+γ₁v_*)) cosÏ€x, K satisfies ||φ₀|| = 1,(?)₁(s), (?)₂(s),γ(s) are smooth functions with respect to s and satisfy ((?)₁(0),(?)₂(0),γ(0)) = (0,0,γ₁). And all solutions of (3) near (φ_*,ψ_*,γ₁) are either on the above nontravial solution curve or on the travial solution curve {(φ_*,ψ_*,γ)}. In chapter 3, we apply shadow system methods to constuct spiky solution of (2). Firstly, we obtain the shadow system of (2) by taking limits .And then we apply Implicit Function Theorem to shadow system and obtain its solution. At last, by applying Implicit Function Theorem and perturbation theory, we obtain the solution of non-shadow system.The main result is thatTheorem2 If (?),there exists a small d, such that for any d₂∈(0,(?)],there exists a big enough (?), whenα≥(?),γ≥(?), (3) has a non-constant positive solution (u_α,γ,v_α,γ), and whenαâ†'∞,γâ†'∞, then (u_α,γ,v_α,γ)â†'((?),w), where (λ,w) is the positive solution of shadow system.