A mathematical model for wild and sterile species in competition: Immigration

We consider a mathematical model of eradication by competition between wild and sterile species when immigration is present. We discuss the case with constant and random immigration flux. In the first case, there is a threshold of eradication M_c related to the number M of sterile individuals. This threshold is evaluated analytically and depends on the niche capacity for instance. Under this threshold (M < M_c) and depending on the immigration flux I₀ there is a discontinuous transition between the low- and the high-population phase corresponding to the niche colonization. In the random immigration case, there is a finite probability to occupy the ecological niche depending on the values of M and I₀. By analogy with some mathematical bistable models, we define the niche colonization time for the wild species. This model is inspired by the fruit flies (Ceratitis capitata) eradication and frontiers control program carried out in different regions of the world by the corresponding agricultural services.