In this paper, we investigate an initial-boundary value problem of a reaction–diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement method, where the boundary condition takes into account the inflow and outflow of individuals through the boundary. Using phase plane analysis, the present paper studies the existence and properties of non-constant steady-state solutions depending on several parameters. Then, we prove some sufficient conditions for their stability. We show that the long-time efficiency of this control method depends strongly on the size of the treated zone and the migration rate. To illustrate these theoretical results, we provide some numerical simulations in the framework of mosquito population control.

We study the bifurcation of steady-state solutions of a scalar reaction-diffusion equation in one space variable by modifying a “time map” technique introduced by J. Smoller and A. Wasserman. We count the exact number of steady-state solutions which are totally ordered in an order interval. We are then able to find their Conley indices and thus determine their stabilities.