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Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate. (English) Zbl 1422.35045
Summary: The paper is concerned with a predator-prey diffusive dynamics subject to homogeneous Dirichlet boundary conditions, in which the growth rate of the the predator is nonlinear. Taking $$m$$ as the main parameter, we show the existence, stability and exact number of positive solution when $$m$$ is large, and some numerical simulations are done to complement the analytical results. The main tools used here include the fixed point index theory, the super-sub solution method, the bifurcation theory and the perturbation technique.

##### MSC:
 35J57 Boundary value problems for second-order elliptic systems 35K57 Reaction-diffusion equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35B09 Positive solutions to PDEs 35B35 Stability in context of PDEs 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 92D25 Population dynamics (general)
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