Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2020; 57(3): 677-690

Online first article February 13, 2020      Printed May 31, 2020

https://doi.org/10.4134/BKMS.b190416

Copyright © The Korean Mathematical Society.

Positive solutions of a reaction-diffusion system with Dirichlet boundary condition

Zhan-Ping Ma, Shao-Wen Yao

Henan Polytechnic University; Henan Polytechnic University

Abstract

In this article, we study a reaction-diffusion system with homogeneous Dirichlet boundary conditions, which describing a three-species food chain model. Under some conditions, the predator-prey subsystem ($u_{1}\equiv0 $) has a unique positive solution $\left( \overline{u_{2}},\overline{u_{3}} \right).$ By using the birth rate of the prey $r_{1}$ as a bifurcation parameter, a connected set of positive solutions of our system bifurcating from semi-trivial solution set $\left( r_{1},\left( 0,\overline{ u_{2}},\overline{u_{3}}\right) \right) $ is obtained. Results are obtained by the use of degree theory in cones and sub and super solution techniques.

Keywords: Reaction-diffusion, food chain model, positive solutions, bifurcation, degree theory

MSC numbers: 35K57, 35B32, 92D25

Supported by: This work was financially supported by the NSF of China 11701243.