Constrained control of gene-flow models

In ecology and population dynamics, gene-flow refers to the transfer of a trait (e.g. genetic material) from one another. This phenomenon is great relevance in studying spread diseases or evolution social features, such as languages. From mathematical point view, modelled using bistable reaction-diffusion equations. The unknown proportion $p$ that possesses certain trait, within an overall $N$. models, taken into account by assuming density $N$ depends either on (if corresponds fitter individuals) location $x$ some zones domain can carry more individuals). Recent applications stemming mosquito-borne disease control problems study bilingualism have called for investigation controllability properties these models. At level, this boundary and, since we are working with proportions, $u$ has satisfy constraints $0\leq u \leq 1$. article, provide thorough analysis influence effect properties. We prove that, when only $p$, geometry criterion be considered. then tackle case densities varying $x$. first varies slowly narrow enough, always holds. result proved robust perturbation method. consider sharp fluctuations $N$: give examples may fail. Conversely, heterogeneities will guaranteed: other words equation very strongly influenced variations All negative results showing existence non-trivial stationary states, which act barriers. solutions methods proof independent interest. Our article completed several numerical experiments confirm our analysis.