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S-shaped bifurcation curves in ecosystems. (English) Zbl 1221.35421

Summary: We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form: \[ \begin{cases} -\Delta u=\lambda\left(u-{u^2\over K}- c{u^2\over 1+ u^2}\right),\quad & x\in\Omega,\\ u= 0,\quad & x\in\partial\Omega.\end{cases} \] 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 \(\Omega\subset\mathbb{R}^N\) 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:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35J62 Quasilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
92D40 Ecology
92D25 Population dynamics (general)
35J20 Variational methods for second-order elliptic equations
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