A Class Of Competition Equations With Cross-diffusion Spike Equilibrium Solution Instability

This paper is concerned with the instability of a class of positive spiky steady states for a quasi-linear cross-diffusion system describing two-species competition.For a course of transition,the system (1.1)can be written asWhen a is large enough,the system (2) is a perturbation of the below systemBased on the theory of perturbation,the instability of the non-constant steady states of the system (2)is equal to the system (3)'s. Hence,we only need consider the instability of the non-constant steady states of the system (3).When d₁ is large enough,we can get the shadow system. 1.In the second section,we use the shadow system analysis method to consider the instbility of the non-constant steady states with spikey layer. First,we proved that the non-constant steady states with spikey layer of shadow system in [2] is a function with exponential dacay. Then we get the linearized system of(4).(5)The corresponding eigenvalue problem of (5) is...