Positive solutions of a reaction-diffusion system with Dirichlet boundary condition
Bull. Korean Math. Soc. 2020 Vol. 57, No. 3, 677-690
https://doi.org/10.4134/BKMS.b190416
Published online May 31, 2020
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.
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