Traveling Waves in Diffusive Predator–Prey Equations: Periodic Orbits and Point-to-Periodic Heteroclinic Orbits

We demonstrate the existence of periodic traveling wave train and traveling front solutions for a diffusive predator–prey system. The analysis is in the three-dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator–prey system in the traveling wave variable. The analysis shows the existence of periodic orbits, heteroclinic orbits and a heteroclinic connection of a point and a periodic orbit. The proof uses shooting techniques, invariant manifold theory, and the qualitative theory of ordinary differential equations.