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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Research supported in part by a grant from the University of Utah Research Fund
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Dunbar, S.R. Travelling wave solutions of diffusive Lotka-Volterra equations. J. Math. Biology 17, 11–32 (1983). https://doi.org/10.1007/BF00276112
- Reaction diffusion equations
- Travelling waves
- Diffusive Lotka-Volterra equations