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Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems. (English) Zbl 1242.35147

The authors consider systems of reaction-diffusion equations of the type

u t =Du xx +f(u)forx,t0

with u(x,0)=u 0 (x) for x, where u n , D=diag(d 1 ,d 2 ,...,d N ), d i >0, and u 0 (x) is a bounded uniformly continuous function on . The authors establish the spreading speed for a large class of non-cooperative systems based on those for cooperative systems. Further, the asymptotic behavior of the traveling wave solutions in terms of eigenvalues and eigenvectors for both cooperative and non-cooperative systems is obtained. The results are applied to a partially cooperative system describing interactions between ungulates and grass.

MSC:
35K57Reaction-diffusion equations
35C07Traveling wave solutions of PDE
35K45Systems of second-order parabolic equations, initial value problems
35Q92PDEs in connection with biology and other natural sciences
35B40Asymptotic behavior of solutions of PDE
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