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Diffusive logistic equation with constant yield harvesting. I: Steady states. (English) Zbl 1109.35049

Summary: We consider a reaction-diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth \[ u_t=\Delta u +au -bu^2 -c h(x) \text{ in } \Omega, \] subject to \(u=0\) on \(\partial\Omega,\) where \(a\), \(b\), \(c\) are positive constants and \(h \in C^\alpha(\overline{\Omega}) \) is positive in \(\Omega\) and zero on \(\partial \Omega\). We prove the existence, uniqueness and stability of the maximal steady state solutions under certain conditions, and we also classify all steady state solutions under more restricted conditions. Exact global bifurcation diagrams are obtained in the latter case. Our method is a combination of comparison arguments and bifurcation theory.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35B32 Bifurcations in context of PDEs
92D25 Population dynamics (general)
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