Traveling Wave Solutions for a Class of Predator–Prey Systems

Abstract

We use a shooting method to show the existence of traveling wave fronts and to obtain an explicit expression of minimum wave speed for a class of diffusive predator–prey systems. The existence of traveling wave fronts indicates the existence of a transition zone from a boundary equilibrium to a co-existence steady state and the minimum wave speed measures the asymptotic speed of population spread in some sense. Our approach is a significant improvement of techniques introduced by Dunbar. The advantage of our method is that it does not need the notion of Wazewski’s set and LaSalle’s invariance principle used in Dunbar’s approach. In our approach, we convert the equations for traveling wave solutions to a system of first order equations by a “non-traditional transformation”. With this converted new system, we are able to construct a Liapunov function, which gives an immediate implication of the boundedness and convergence of the relevant class of heteroclinic orbits. Our method provides a more efficient way to study the existence of traveling wave solutions for general predator–prey systems.