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


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Article Info

DATE

2021-08-10

AUTHORS

Deniu Yang, Minghuan Liu

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: ut(x,t)=d1uxx(x,t)+Au(x,t)(1-u(x,t)K)-φ(u(x,t))w(x,t),wt(x,t)=d2wxx(x,t)+w(x,t)(μφ(u(x,t))-C),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{l} u_{t}(x,t)=d_{1} u_{xx}(x,t)+Au(x,t)\big (1-\frac{u(x,t)}{K}\big )-\varphi (u(x,t))w(x,t),\\ w_{t}(x,t)=d_{2} w_{xx}(x,t)+w(x,t)\big (\mu \varphi (u(x,t))-C\big ), \end{array} \end{aligned}$$\end{document}where all parameters are positive which will be mentioned later. The traveling wave solutions are established in R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{R}^{4}$$\end{document}, 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. More... »

PAGES

1-22

References to SciGraph publications

  • 2012-02-25. Exact traveling wave solutions for diffusive Lotka–Volterra systems of two competing species in JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
  • 2001-07. How predation can slow, stop or reverse a prey invasion in BULLETIN OF MATHEMATICAL BIOLOGY
  • 1983-05. Travelling wave solutions of diffusive Lotka-Volterra equations in JOURNAL OF MATHEMATICAL BIOLOGY
  • 1996-02. A predator-prey reaction-diffusion system with nonlocal effects in JOURNAL OF MATHEMATICAL BIOLOGY
  • 2003-02. Existence of traveling wave solutions in a diffusive predator-prey model in JOURNAL OF MATHEMATICAL BIOLOGY
  • 1976. Periodic and traveling wave solutions to Volterra-Lotka equations with diffusion in BULLETIN OF MATHEMATICAL BIOLOGY
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    http://scigraph.springernature.com/pub.10.1007/s13160-021-00478-8

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    http://dx.doi.org/10.1007/s13160-021-00478-8

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