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On the Cauchy problem for Volterra-Lotka’s competition equations with migration effect and its travelling wave like solutions. (English) Zbl 0551.92011
The author considers the predator-prey-equations on the real line with general interaction terms: $u_ t+\mu u_ x=(a_ 1-b_{11}u- b_{12}v)u+\epsilon F(u,v),$ $v_ t+\nu v_ x=(a_ 2-b_{21}u- b_{22}v)v+\epsilon G(u,v),$ $u(x,0)=\phi (x),\quad v(x,0)=\psi (x).$ Here $$\epsilon$$ $$\mu$$ $$\neq \nu \in R$$; $$a_ i$$, $$b_{ij}\geq 0$$; $$\phi$$, $$\psi\geq 0$$ and bounded together with their derivatives; F, G real analytic functions in $$0<| u|$$, $$| v| <\rho_ 0\leq \infty.$$
He gives a unique solution of this system for $$x\in R$$, $$0\leq t\leq T$$, if there is $$r_ 0<\rho_ 0$$ such that sup $$\phi$$, sup $$\psi$$, $$a_ 1/b_{11}$$ and $$a_ 2/b_{22}<r_ 0$$, and if $$\epsilon <\epsilon_ T$$. This solution is continuous together with its derivative and has an absolutely convergent power series representation $$u(x,t)=\sum^{\infty}_{n=0}u_ n(x,t)\epsilon^ n.$$ Moreover, for a special choice of $$\phi$$, $$\psi$$ and for $$\epsilon$$ small enough, the system has traveling wave like solutions.
Reviewer: R.Repges

##### MSC:
 92D25 Population dynamics (general) 35F25 Initial value problems for nonlinear first-order PDEs