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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) u 1 /t=d 1 2 u 1 /x 2 +u 1 f 1 (u),u 2 /t=d 2 2 u 2 /x 2 +u 2 f 2 (u) in × + , u=(u 1 ,u 2 ). It is assumed that f 1 /u 2 <0 and f 2 /u 1 >0. One is interested in solutions of (1) which depend on the single variable ξ=x-ct, c=constant (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
MSC:
35K55Nonlinear parabolic equations
35B40Asymptotic behavior of solutions of PDE
92D40Ecology