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Traveling wave solutions in a diffusive predator–prey system with Holling type-III functional response

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Abstract

This work concerns with the existence of traveling wave solutions for the following diffusive predator–prey type system with Holling type-III functional response:

$$\begin{aligned} \begin{array}{l} u_{t}(x,t)=d_{1} u_{xx}(x,t)+Au(x,t)\big (1-\frac{u(x,t)}{K}\big )-\varphi (u(x,t))w(x,t),\\ w_{t}(x,t)=d_{2} w_{xx}(x,t)+w(x,t)\big (\mu \varphi (u(x,t))-C\big ), \end{array} \end{aligned}$$

where all parameters are positive which will be mentioned later. The traveling wave solutions are established in \(\varvec{R}^{4}\), which is a heteroclinic orbit connecting the boundary equilibrium and the positive equilibrium. Applying the methods of Wazewski Theorem and LaSalle’s Invariance Principle, and constructing a Liapunov function, we obtain the existence of traveling wave solutions. We also discuss some possible biological implications of the existence of these waves.

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Funding

Science and Technology Project Founded by the Education Department of Jiangxi Province(No. GJJ191645).

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Authors and Affiliations

  1. School of Mathematical Sciences, Sichuan Normal University, Chengdu, 610068, China

    Deniu Yang

  2. College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang, 330063, China

    Minghuan Liu

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  1. Deniu Yang
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  2. Minghuan Liu
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Correspondence to Minghuan Liu.

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Yang, D., Liu, M. Traveling wave solutions in a diffusive predator–prey system with Holling type-III functional response. Japan J. Indust. Appl. Math. 39, 97–118 (2022). https://doi.org/10.1007/s13160-021-00478-8

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  • DOI: https://doi.org/10.1007/s13160-021-00478-8

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