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Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits. (English) Zbl 0617.92020

The reaction-diffusion system \[ U_ t=\alpha U(\gamma -U)- UW/(1+U),\quad W_ t=W_{xx}-W+\beta UW/(1+U) \] is considered under some natural assumptions on the positive parameters occuring. This system has drawn much attention even without the diffusive term [see also e.g. H. L. Smith, SIAM J. Appl. Math. 42, 27-43 (1982; Zbl 0489.92019), and the reviewer, Nonlinear Anal., Theory Methods Appl. 8, 1295-1309 (1984; Zbl 0561.92015)].
Assuming the existence of a traveling wave solution the author gets a three-dimensional autonomous system of ordinary differential equations to be satisfied by this solution. Establishing the Hopf bifurcation in the latter system he shows the occurrence of a traveling wave train solution of the original system.
He also shows the existence of a heteroclinic orbit joining an unstable equilibrium (absence of predators) to a positive equilibrium (in the pre- bifurcation situation) and to a periodic orbit (in the post-bifurcation situation).
Reviewer: M.Farkas

MSC:

92D25 Population dynamics (general)
35K57 Reaction-diffusion equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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