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

Zhan-Ping Ma|Shao-Wen Yao
대한수학회 (The Korean Mathematical Society)|2020.05|대한수학회보 (Bulletin of the KMS)|제57권 제3호|pp. 677-690|DOI 10.4134/BKMS.b190416

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키워드

bifurcationpositive solutionsdegree theoryReaction-diffusionfood chain model
주제분류수학

초록

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.

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.

참고문헌 (27)

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