zbMATH — the first resource for mathematics

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

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