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Global bifurcation of coexistence state for the competition model in the chemostat. (English) Zbl 0940.35114
The paper derives the global structure for the coexistence and investigates the steady state solutions of the following reaction-diffusion system in \(N\)-dimensional case: \[ S_t=\Delta S-auf_1(S)-bvf_2(S),\quad x\in\Omega, t>0, \] \[ u_t=\Delta u+auf_1(S),\quad x\in\Omega, t>0, \] \[ v_t=\Delta v+bvf_2(S),\quad x\in\Omega, t>0, \] with boundary conditions \[ {\partial S\over\partial n}+b(x) S=h(x),\quad x\in\partial\Omega, t>0, \] \[ {\partial u\over\partial n}+b(x)u=0,\quad {\partial v\over\partial n}+b(x)v=0,\quad x\in\partial\Omega, t>0, \] where \(\Omega\) is a bounded region in \(\mathbb{R}^N\) (\(N\geq 1\)) with smooth boundary \(\partial\Omega\), \(f_i(S)=S/(a_i+S)\), \(a>0\), \(b>0\) are the maximum growth rates, and \(a_i>0\) is the Michaelis-Menten constant. The system models the competing organism in a chemostat, \(S\) is the concentration of nutrient, \(u\) and \(v\) are the concentrations of the competing species. Monotone method, generalized maximum principle, Sturm-type eigenvalue and a theorem for the global bifurcation are used as tools.

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B32 Bifurcations in context of PDEs
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
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