Competition With Cross-diffusion Equations Spike Equilibrium Solution Structure

This paper is concerned with the construction of a class of positive steady states for a quasi-linear cross-diffusion system describing two-species competition.let r =d₁/ρ₁₂, s =1/ρ₁₂,φ= (r + v)u, (?) = v the system(1)can be written asmake the limitρ₁₂→+∞,ρ₁₂/d₁→∞,i.e (s→0⁺,r→0⁺)from the first equation of systems(2),we haveφ_xx→0,if x∈(0, 1),s→0⁺, r→0⁺ sinceφ'(x) = 0,x = 0,1 , we haveφ(x)→τ,if x∈(0,1),s→0⁺,r→0⁺ make the limit of system(2),we have the shadow system:Chapterâ…¡mainly based on the methods of the calculation of scores and implicit function theorem to find the structure of the shadow system This chapter of the major findings:Theorem: suppose that a₁/a₂≥1/4 b₁/b₂+3/4 c₁/c₂and b₁/b₂1/c₂ set up , and d₂ issmall enough ,the systems (2) have a spike solution :Chapterâ…¢mainly takes use of perturbation theory return to the non-shadow system equations to find a solutionTheorem: suppose that a₁/a₂≥1/4 b₁/b₂+3/4 c₁/c₂ and b₁/b₂1/c₂ set up , there exists smalld₀ >0,for each Fixed 0 < d₂ < d₀ , there exists sufficiently largeα,such that ifα(?)ρ₁₂/d₁ >αandρ₁₂ >α, Equations (1) has a non-constant spike steady state (u_α,ρ12(x),u_α,ρ12(x)), ifα→∞andρ₁₂→∞,(u_α,ρ12(x),v_α,ρ12(x))→(Ï„_ε/(?)_ε(x),(?)_ε(x))...