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In this paper, we investigate an initial-boundary-value problem of areaction-diffusion equation in a bounded domain with a Robin boundary conditionand introduce some particular parameters to consider the non-zero flux on theboundary. This problem arises in the study of mosquito populations under theintervention of the population replacement method, where the boundary conditiontakes into account the inflow and outflow of individuals through the boundary.Using phase-plane analysis, the present paper studies the existence andproperties of non-constant steady-state solutions depending on severalparameters. Then, we use the principle of linearized stability to prove somesufficient conditions for their stability. We show that the long-timeefficiency of this control method depends strongly on the size of the treatedzone and the migration rate. To illustrate these theoretical results, weprovide some numerical simulations in the framework of mosquito populationcontrol.

Steady-state solutions for a reaction-diffusion equation with Robin boundary conditions: Application to the control of dengue vectors