| 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 haveφxx'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:ChapterⅡmainly 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 :ChapterⅢmainly 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 (uα,ρ12(x),uα,ρ12(x)), ifα'∞andρ12'∞,(uα,ρ12(x),vα,ρ12(x))'(τε/(?)ε(x),(?)ε(x))... |
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