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) \(\partial u_ 1/\partial t=d_ 1\partial^ 2u_ 1/\partial x^ 2+u_ 1f_ 1(u),\quad \partial u_ 2/\partial t=d_ 2\partial^ 2u_ 2/\partial x^ 2+u_ 2f_ 2(u)\) in \({\mathbb{R}}\times {\mathbb{R}}^+\), \(u=(u_ 1,u_ 2)\). It is assumed that \(\partial f_ 1/\partial u_ 2<0\) and \(\partial f_ 2/\partial u_ 1>0\). 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


35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
92D40 Ecology
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