TRAVELING WAVES IN COOPERATIVE PREDATION: RELAXATION OF SUBLINEARITY

In this paper, we investigate traveling wave solutions of a diffusive predator-prey model which takes into consideration hunting cooperation. Sublinearity condition is violated for the function of cooperative predation. When the basic reproduction number for the diffusion-free model is greater than one, we find a critical wave speed below which no positive traveling wave solution shall exist. On the other hand, if the wave speed exceeds this critical value, we prove the existence of a positive traveling wave solution connecting the predator-free equilibrium to the unique positive equilibrium under a technical assumption of weak cooperative predation. The key idea of the proof contains two major steps: (i) we construct a suitable pentahedron and find inside it a trajectory connecting the predator-free equilibrium; and (ii) we construct a suitable Lyapunov function and use LaSalle invariance principle to prove that the trajectory also connects the positive equilibrium. In the end of this paper, we propose five open problems related to traveling wave solutions in cooperative predation.

Automated summary

This paper investigates traveling wave solutions of a diffusive predator-prey model with hunting cooperation, where a critical wave speed is identified below which no positive traveling wave solution exists if the basic reproduction number for the diffusion-free model is greater than one; on the other hand, if the wave speed exceeds this critical value, the existence of a positive traveling wave solution connecting predator-free equilibrium to unique positive equilibrium is proven under a technical assumption of weak cooperative predation, with key steps involving the construction of a suitable pentahedron and Lyapunov function. Five open problems related to traveling wave solutions in cooperative predation are proposed at the end of the paper.