Traveling wave solutions in predator–prey models with competition

Abstract

This paper studies the minimal wave speed of traveling wave solutions in predator–prey models, in which there are several groups of predators that compete among different groups. We investigate the existence and nonexistence of traveling wave solutions modeling the invasion of predators and coexistence of these species. When the positive solution of the corresponding kinetic system converges to the unique positive steady state, a threshold that is the minimal wave speed of traveling wave solutions is obtained. To finish the proof, we construct contracting rectangles and upper–lower solutions and apply the asymptotic spreading theory of scalar equations. Moreover, multiple propagation thresholds in the corresponding initial value problem are presented by numerical examples, and one threshold may be the minimal wave speed of traveling wave solutions.