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Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion
American Journal of Applied Mathematics
Volume 8, Issue 5, October 2020, Pages: 236-246
Received: Feb. 18, 2020; Accepted: Aug. 14, 2020; Published: Aug. 25, 2020
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Ke Li, School of Mathematics and Statistics, Shandong Normal University, Jinan, P.R. China
Hongmei Cheng, School of Mathematics and Statistics, Shandong Normal University, Jinan, P.R. China
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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.
Traveling Wave Solution, Predator-prey Model, Nonlocal Diffusion, Ratio-dependent Functional Response, Schauder’s Fixed Point Theorem, Comparison Principle
To cite this article
Ke Li, Hongmei Cheng, Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion, American Journal of Applied Mathematics. Special Issue: Application of Nonlinear Analysis. Vol. 8, No. 5, 2020, pp. 236-246. doi: 10.11648/j.ajam.20200805.11
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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