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Zbl 1221.35421
Lee, Eunkyoung; Sasi, Sarath; Shivaji, R.
S-shaped bifurcation curves in ecosystems.
(English)
[J] J. Math. Anal. Appl. 381, No. 2, 732-741 (2011). ISSN 0022-247X

Summary: We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form: $$\cases -\Delta u=\lambda\left(u-{u^2\over K}- c{u^2\over 1+ u^2}\right),\quad & x\in\Omega,\\ u= 0,\quad & x\in\partial\Omega.\endcases$$ Here $\Delta u= \text{div}(\nabla u)$ is the Laplacian of $u$, ${1\over\lambda}$ is the diffusion coefficient, $K$ and $c$ are positive constants and $\Omega\subset\bbfR^N$ is a smooth bounded region with $\partial\Omega$ 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.
MSC 2000:
*35Q92
35J62
35J25 Second order elliptic equations, boundary value problems
92D40 Ecology
92D25 Population dynamics
35J20 Second order elliptic equations, variational methods

Keywords: ecological systems; S-shaped bifurcation curves; sub-supersolutions

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