Competition Equations With Small Parameter Traveling Wave Solutions Exist

This paper is concerned with the existence of travelling waves for the following quasilinear reaction-diffusion systems with cross-diffusion and self-diffusion terms, which describes two species competition dynamics under inter- and inro- species pressure,whereHere u and v are the population densities, α_i,β_i,γ_i(i=1,2) are nonnegative constants, β_i is the self-diffusion papameter, γ_i is the cross-diffusion parameters,a_i, b_i,c_i> 0, (i = 1,2). Applying geometric singular perturbation methods to (1) and (2) and by detailed analysis on the fast and slow manifolds , We obtained the existence of the travelling waves for different cases.1. In Chapter 2, we consider the following non-cross diffusion systems:Assume (A.1)-(A.3) and A > max(B,C)(or C < A < B) hold, there exists a maximal wave speed c^* = -2(α₁-c₁β^*)^1/2 for each fixed c(< c^*),there exist small ∈₁ > 0,such that for each fixed 0 < ∈< ∈₁, there exists travelling wave solution U^∈(x -∈ct) of (2.0.1) connecting (0, a₂/c₂) and (a₁/b₁, 0)(or (0, a₂/c₂) and ((a₂c₁-a₁c₂)/(c₁b₂-c₂b₁),(a₁b₂-a₂b₁)/(c₁b₂-c₂b₁)),where U^∈(x - ∈ct) are travelling waves with transition layers.Furthermore,there exist travelling waves connecting two adjacent equilibrium points along the fast manifold.2. In Chapter 3, We consider the following systems with cross-diffusion:...