*(English)*Zbl 0617.92020

The reaction-diffusion system

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).

##### MSC:

92D25 | Population dynamics (general) |

35K57 | Reaction-diffusion equations |

37G99 | Local and nonlocal bifurcation theory |

34C05 | Location of integral curves, singular points, limit cycles (ODE) |