Minimal Wave Speed in a Competitive Integrodifference System without Comparison Principle

Luping Li,Shugui Kang,Huiqin Chen,Lili Kong

doi:10.3390/math7070571

Abstract

We investigate the traveling wave solutions of a competitive integrodifference system without comparison principle. In the earlier conclusions, a threshold of wave speed is defined while the existence or nonexistence of traveling wave solutions remains open when the wave speed is the threshold. By constructing generalized upper and lower solutions, we confirm the existence of traveling wave solutions when the wave speed is the threshold. Our conclusion completes the known results and shows the different decay behavior of traveling wave solutions compared with the case of large wave speeds.

Highlights

Spatial propagation thresholds of population models have attracted much attention since these thresholds may characterize the spatial expansion of individuals [1,2], and such a constant invasion speed is coincident with some history data; see some examples by Murray ([1], Chapter 13)
A traveling wave solution is a special entire solution, and similar propagation phenomena have been widely observed in different fields ([28], Chapter 1)
The minimal wave speed of traveling wave solutions is important since it may characterize the propagation threshold, and it has been widely studied for monotone systems

Summary

Introduction

Spatial propagation thresholds of population models have attracted much attention since these thresholds may characterize the spatial expansion of individuals [1,2], and such a constant invasion speed is coincident with some history data; see some examples by Murray ([1], Chapter 13). When spatial propagation dynamics are considered, we may refer to some results on the propagation dynamics of non-monotone integrodifference systems by Hsu and Zhao [12], Li et al [13], Lin [14], Pan and Lin [15], Pan and Zhang [16], and very recent papers [17,18] and references cited therein for other non-monotone diffusion systems In these works, to establish the minimal wave speed, a general recipe is to pass to a limit function from the results of large wave speeds. In this paper, motivated by [14,26], we construct proper generalized upper and lower solutions to prove the existence of traveling wave solutions if the wave speed c = c∗ , which may complete the conclusion in [22].

Preliminaries

Main Result

Concluding Remarks