Journal of Mathematical Biology

, Volume 17, Issue 1, pp 11–32 | Cite as

Travelling wave solutions of diffusive Lotka-Volterra equations

  • Steven R. Dunbar


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.

Key words

Reaction diffusion equations Travelling waves Diffusive Lotka-Volterra equations 


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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Steven R. Dunbar
    • 1
  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA

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