Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion

Ke Li; Hongmei Cheng

doi:doi:10.11648/j.ajam.20200805.11

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Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion

Ke Li, Hongmei Cheng

Published in American Journal of Applied Mathematics (Volume 8, Issue 5)

Received: 18 February 2020 Accepted: 14 August 2020 Published: 25 August 2020

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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.

Published in	American Journal of Applied Mathematics (Volume 8, Issue 5) This article belongs to the Special Issue Application of Nonlinear Analysis
DOI	10.11648/j.ajam.20200805.11
Page(s)	236-246
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Traveling Wave Solution, Predator-prey Model, Nonlocal Diffusion, Ratio-dependent Functional Response, Schauder’s Fixed Point Theorem, Comparison Principle

References

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APA Style

Ke Li, Hongmei Cheng. (2020). Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion. American Journal of Applied Mathematics, 8(5), 236-246. https://doi.org/10.11648/j.ajam.20200805.11

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ACS Style

Ke Li; Hongmei Cheng. Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion. Am. J. Appl. Math. 2020, 8(5), 236-246. doi: 10.11648/j.ajam.20200805.11

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AMA Style

Ke Li, Hongmei Cheng. Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion. Am J Appl Math. 2020;8(5):236-246. doi: 10.11648/j.ajam.20200805.11

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@article{10.11648/j.ajam.20200805.11,
  author = {Ke Li and Hongmei Cheng},
  title = {Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {5},
  pages = {236-246},
  doi = {10.11648/j.ajam.20200805.11},
  url = {https://doi.org/10.11648/j.ajam.20200805.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200805.11},
  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.},
 year = {2020}
}

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TY  - JOUR
T1  - Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion
AU  - Ke Li
AU  - Hongmei Cheng
Y1  - 2020/08/25
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200805.11
DO  - 10.11648/j.ajam.20200805.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 236
EP  - 246
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200805.11
AB  - 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.
VL  - 8
IS  - 5
ER  -

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Author Information

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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Plain Text BibTeX RIS

APA Style

Ke Li, Hongmei Cheng. (2020). Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion. American Journal of Applied Mathematics, 8(5), 236-246. https://doi.org/10.11648/j.ajam.20200805.11

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ACS Style

Ke Li; Hongmei Cheng. Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion. Am. J. Appl. Math. 2020, 8(5), 236-246. doi: 10.11648/j.ajam.20200805.11

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AMA Style

Ke Li, Hongmei Cheng. Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion. Am J Appl Math. 2020;8(5):236-246. doi: 10.11648/j.ajam.20200805.11

Copy | Download

@article{10.11648/j.ajam.20200805.11,
  author = {Ke Li and Hongmei Cheng},
  title = {Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {5},
  pages = {236-246},
  doi = {10.11648/j.ajam.20200805.11},
  url = {https://doi.org/10.11648/j.ajam.20200805.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200805.11},
  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.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion
AU  - Ke Li
AU  - Hongmei Cheng
Y1  - 2020/08/25
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200805.11
DO  - 10.11648/j.ajam.20200805.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 236
EP  - 246
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200805.11
AB  - 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.
VL  - 8
IS  - 5
ER  -

Copy | Download