Periodic traveling wave train and point-to-periodic traveling wave for a diffusive predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type scheme is investigated in this article. The existences of a small amplitude periodic traveling wave train Γ p and the traveling wave solution connecting the boundary equilibrium E u ( 1 , 0 ) to the periodic traveling wave Γ p are obtained. The existence of this point-to-periodic solution reveals that the predator invasion leads to the periodic population densities in the coexistence domain, and thus plays a mild role in the evolution of predator–prey communities. The techniques used here are the Hopf bifurcation theorem, the improved shooting method combining with the geometric singular perturbation method.