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Existence of travelling wave solutions of predator-Prey systems via the connection index. (English) Zbl 0541.35044
The author studies the existence of travelling wave solutions of the system (1) $\partial u\sb 1/\partial t=d\sb 1\partial\sp 2u\sb 1/\partial x\sp 2+u\sb 1f\sb 1(u),\quad \partial u\sb 2/\partial t=d\sb 2\partial\sp 2u\sb 2/\partial x\sp 2+u\sb 2f\sb 2(u)$ in ${\bbfR}\times {\bbfR}\sp+$, $u=(u\sb 1,u\sb 2)$. It is assumed that $\partial f\sb 1/\partial u\sb 2<0$ and $\partial f\sb 2/\partial u\sb 1>0$. One is interested in solutions of (1) which depend on the single variable $\xi =x-ct$, $c=cons\tan t$ (wave velocity). A possible biological interpretation is a predator-prey model for diffusing species. The author then proves the existence of travelling wave solutions under appropriate hypotheses. The proofs use a variant of the Conley index (the ”connection index”). These topological means are introduced in the paper for non-specialists, making the clearly written paper accessible for a broad audience.
Reviewer: R.Sperb

35K55Nonlinear parabolic equations
35B40Asymptotic behavior of solutions of PDE
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