Minimal wave speed in a dispersal predator–prey system with delays

Abstract

This paper is concerned with the minimal wave speed in a nonlocal dispersal predator-prey system with delays. We define a threshold. By presenting the existence and nonexistence of traveling wave solutions, we confirm that the threshold is the minimal wave speed, which completes the known results.

Spreading Speed in A Nonmonotone Equation with Dispersal and Delay

“…In this paper, we shall construct a proper auxiliary equation and estimate the spreading speed of u(x, t), which is motivated by Lin [12] for a delayed reaction-diffusion equation and depends on the conclusions in Jin and Zhao [13] for a monotone dispersal model. The spreading speed in this paper equals the minimal wave speed of traveling wave solutions in Li et al [14] when the kernel function J has nonempty compact support. By passing to a limit function, we also present the existence of traveling wave solutions when the wave speed is the spreading speed, during which (J) holds.…”

“…In this paper, we obtained a threshold of asymptotic spreading for nonmonotone Equation (1), which is also the minimal wave speed in known results. For the existence of traveling wave solutions, we further presented a result under weaker conditions than that in [14].…”

Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle

“…In this paper, we show that c * may be the spreading speed of u 1 or u 2 , which is the minimal wave speed in [17]. It is possible to study the asymptotic spreading of the model in [31] by our idea.…”

“…is a predator-prey system in population dynamics. In Li et al [17], Yu and Yuan [41], Zhang et al [42], the authors investigated its traveling wave solutions connecting (0, 0) with the positive steady state, which reflect the process that these two species invade a new habitat from the viewpoint of biology invasion. In particular, Li et al [17] obtained the minimal wave speed defined by…”

“…From the viewpoint of initial value problem, let any fixed time be the initial time, the traveling wave solutions in [17,41,42] indicate the initial size of habitat of both species is infinite, which contradicts to some natural phenomena because the initial invasion often begins in finite domain. The purpose of this paper is to explore the dynamics when the initial habitats of two invaders are finite and investigate the long time behavior of (2)…”