Abstract
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.
Highlights
The Lotka-Volterra system has been widely used in the models of predation ever since Lotka and Volterra did two independent studies [17, 20] near one century ago
We assume that it has a positive equilibrium (U, V ), where V = rb(U )/μ and U is a positive root of b(U ) − f (U, rb(U )/μ) = 0
A traveling wave solution connecting the predator-free equilibrium to the positive equilibrium is a solution of the special form (u(x + ct), v(x + ct)) with c > 0 such that (u(−∞), v(−∞)) = (K, 0) and (u(∞), v(∞)) = (U, V )
Summary
The Lotka-Volterra system has been widely used in the models of predation ever since Lotka and Volterra did two independent studies [17, 20] near one century ago. The objective of this paper is to relax this sublinearity condition and establish existence theory of travelling waves for a diffusive model of cooperative predation. The nonlinear predation rate f (u, v) = puv + quv does not satisfy the sublinearity condition (1.3) and the results in the existing literature do not apply To overcome this difficulty, we will modify the shooting method which was introduced by Dunbar [6, 7], and later developed by Hosono and Ilyas [8, 9] and by Huang [12, 13]. We will apply an extension of the technique introduced by Huang [12, 13] to prove the existence of a positive traveling wave connecting the predator-prey equilibrium.
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