# POSITIVE SOLUTIONS OF A REACTION-DIFFUSION SYSTEM WITH DIRICHLET BOUNDARY CONDITION

• Ma, Zhan-Ping (School of Mathematics and Information Science Henan Polytechnic University) ;
• Yao, Shao-Wen (School of Mathematics and Information Science Henan Polytechnic University)
• Accepted : 2019.09.19
• Published : 2020.05.31

#### 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 (u1 ≡ 0) has a unique positive solution (${\bar{u_2}}$, ${\bar{u_3}}$). By using the birth rate of the prey r1 as a bifurcation parameter, a connected set of positive solutions of our system bifurcating from semi-trivial solution set (r1, (0, ${\bar{u_2}}$, ${\bar{u_3}}$)) is obtained. Results are obtained by the use of degree theory in cones and sub and super solution techniques.

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