S-shaped bifurcation curves in ecosystems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:381

Article

Lee, Eunkyoung²  Sasi, Sarath¹  Shivaji, R.¹ 

[1] Mississippi State Univ, Ctr Computat Sci, Dept Math & Stat, Mississippi State, MS 39759 USA

[2] Pusan Natl Univ, Dept Math, Pusan 609735, South Korea

关键词: Ecological systems;    S-shaped bifurcation curves;    Sub-supersolutions;   

DOI  :  10.1016/j.jmaa.2011.03.048

来源: Elsevier

【 摘 要 】

We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form: {-Delta u = lambda[u - u(2)/K - cu(2)/1 + u(2)], x is an element of Omega, u = 0, x is an element of partial derivative Omega. Here Delta u = div(del u) is the Laplacian of u, 1/lambda is the diffusion coefficient, K and c are positive constants and Omega subset of R(N) is a smooth bounded region with partial derivative Omega in C(2). This model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. In this paper we discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve. We prove our results by the method of sub-supersolutions. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

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