Abstract
In this paper, we study the traveling waves for the ratio-dependent predator-prey model with nonlocal diffusion, which is devoted to the existence and nonexistence of traveling wave solution. This model incorporates the ratio-dependent functional response into the Lotka-Volterra type system, and both species obey the logistic growth. Firstly, we construct a nice pair of upper and lower solutions when the wave speed is greater than the minimal wave speed. Then by applying Schauder's fixed point theorem with the help of suitable upper and lower solutions, we can obtain the existence of traveling waves when the wave speed is greater than the minimal wave speed. Moreover, in order to prove the limit behavior of the traveling waves at infinity, we construct a sequence that converges to the coexistence state. Finally, by using the comparison principle, we obtain the nonexistence of the traveling waves when the wave speed is greater than 0 and less than the minimal wave speed. The difficulty of this paper is to construct a suitable upper and lower solution, which is also the novelty of this paper. Under certain restricted condition, this paper concludes the existence and the nonexistence of the traveling waves for the ratio-dependent predator-prey model with nonlocal diffusion.
Highlights
The interaction between the predator and the prey constitutes a dynamic relationship that has been one of the main topics in ecological research, which is important for studying the distribution of organisms and the balance of the environment
This relationship can be described by the dynamic behavior of some mathematical models
We mainly study the existence of the traveling wave solution which connects the predator free state 1,0 with the coexistence state T, T of the system (4), where T = 1 − E > 0, when 0 < 1 < '
Summary
The interaction between the predator and the prey constitutes a dynamic relationship that has been one of the main topics in ecological research, which is important for studying the distribution of organisms and the balance of the environment. #, ) 1 − #, , 237 Ke Li and Hongmei Cheng: Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion the two species move randomly along a one-dimensional region R, and the parameter $ > 0 is a rescaled diffusion coefficient of the prey species while the diffusion coefficient for the predator is rescaled to be 1. Many researchers study the properties of the traveling wave solution for the reaction-diffusion systems with nonlocal diffusion term, see [2, 4, 5, 6, 7, 9] Inspired by these results, we consider the ratio-dependent predator-prey model with nonlocal diffusion, that is. We obtain the nonexistence of the traveling waves by the comparison principle
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