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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:

$\left\{\begin{array}{cc}-{\Delta }u=\lambda \left(u-\frac{{u}^{2}}{K}-c\frac{{u}^{2}}{1+{u}^{2}}\right),\phantom{\rule{1.em}{0ex}}\hfill & x\in {\Omega },\hfill \\ u=0,\phantom{\rule{1.em}{0ex}}\hfill & x\in \partial {\Omega }·\hfill \end{array}\right\$

Here ${\Delta }u=\text{div}\left(\nabla u\right)$ is the Laplacian of $u$, $\frac{1}{\lambda }$ is the diffusion coefficient, $K$ and $c$ are positive constants and ${\Omega }\subset {ℝ}^{N}$ 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 and other natural sciences 35J62 Quasilinear elliptic equations 35J25 Second order elliptic equations, boundary value problems 92D40 Ecology 92D25 Population dynamics (general) 35J20 Second order elliptic equations, variational methods