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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 $$u_{t}=Du_{xx}+f(u) \text{ for }x\in {\Bbb R},t\geq0 $$ with $u(x,0)=u_{0}(x)$ for $x\in {\Bbb R}$, where $u \in{\Bbb R^n}$, $D=\operatorname{diag}(d_{1},d_{2},\ldots,d_{N})$, $d_{i}>0$, and $u_{0}(x)$ is a bounded uniformly continuous function on ${\Bbb R}$. 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.

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
Full Text: DOI arXiv
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