Invasive speed for a competition-diffusion system with three species

Abstract

<p style='text-indent:20px;'>Competition stems from the fact that resources are limited. When multiple competitive species are involved with spatial diffusion, the dynamics becomes even complex and challenging. In this paper, we investigate the invasive speed to a diffusive three species competition system of Lotka-Volterra type. We first show that multiple species share a common spreading speed when initial data are compactly supported. By transforming the competitive system into a cooperative system, the determinacy of the invasive speed is studied by the upper-lower solution method. In our work, for linearly predicting the invasive speed, we concentrate on finding upper solutions only, and don't care about the existence of lower solutions. Similarly, for nonlinear selection of the spreading speed, we focus only on the construction of lower solutions with fast decay rate. This greatly develops and simplifies the ideas of past references in this topic.</p>

Speed determinacy of the traveling waves for a three species time‐periodic Lotka–Volterra competition system

“…Within the context of three species constant‐coefficient Lotka–Volterra competition system, the issue about the linear selection has been studied and some sufficient conditions were established in Guo et al 34 and Pan et al 35 . Here, we only want to compare our results with the ones obtained in Pan et al, 35 since the works of Pan et al 35 extend the previous ones of Guo et al 34 …”

“…In this paper, by using the upper and lower solution method, the linear and nonlinear selections of the minimal wave speed of three species time-periodic Lotka-Volterra competition system are studied. Within the context of three species constant-coefficient Lotka-Volterra competition system, the issue about the linear selection has been studied and some sufficient conditions were established in Guo et al 34 and Pan et al 35 Here, we only want to compare our results with the ones obtained in Pan et al, 35 since the works of Pan et al 35 extend the previous ones of Guo et al 34 The results obtained in Pan et al 35 can be summarized as follows:…”

“…However, much less is known for the speed selection for a three species Lotka-Volterra competition system even with constant coefficients except for the works. 34,35 More precisely, Guo et al 34 provided some conditions on the parameters of the three species Lotka-Volterra competition system with constant coefficients such that the linear determinacy of the minimal wave speed is assured. The sufficient conditions of linear selection and nonlinear selection of the minimal wave speed are further derived in Pan et al 35 Motivated by the models of two species time-periodic Lotka-Volterra system and three species Lotka-Volterra competition system with constant coefficients, we shall investigate the competition system of Lotka-Volterra type…”

“…For such a topic, we refer the reader to other studies 7,15,16 for a diffusive Lotka–Volterra model with constant coefficients, 17 for an integro‐differential model, 18 for cooperative models, 19,20 for lattice systems, 21 and for a spatial periodic Lotka–Volterra competition model. However, much less is known for the speed selection for a three species Lotka–Volterra competition system even with constant coefficients except for the works 34,35 . More precisely, Guo et al 34 provided some conditions on the parameters of the three species Lotka–Volterra competition system with constant coefficients such that the linear determinacy of the minimal wave speed is assured.…”

“…More precisely, Guo et al 34 provided some conditions on the parameters of the three species Lotka–Volterra competition system with constant coefficients such that the linear determinacy of the minimal wave speed is assured. The sufficient conditions of linear selection and nonlinear selection of the minimal wave speed are further derived in Pan et al 35 …”