Travelling wave solutions of diffusive Lotka-Volterra equations |
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Authors: | Steven R. Dunbar |
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Affiliation: | 1. Department of Mathematics, University of Utah, 84112, Salt Lake City, UT, USA
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Abstract: | We establish the existence of travelling wave solutions for two reaction diffusion systems based on the Lotka-Volterra model for predator and prey interactions. For simplicity, we consider only 1 space dimension. The waves are of transition front type, analogous to the travelling wave solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction diffusion equation. The waves discussed here are not necessarily monotone. For any speed c there is a travelling wave solution of transition front type. For one of the systems discussed here, there is a distinguished speed c* dividing the waves into two types, waves of speed c < c* being one type, waves of speed c ? c* being of the other type. We present numerical evidence that for this system the wave of speed c* is stable, and that c* is an asymptotic speed of propagation in some sense. For the other system, waves of all speeds are in some sense stable. The proof of existence uses a shooting argument and a Lyapunov function. We also discuss some possible biological implications of the existence of these waves. |
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