Résumé
Having in mind the issue of control of insects vectors or insects pests, we consider in this paper a metapopulation model with patches linearly interconnected, and explore the global effects of the (on purpose) increase of mortality in some of them. Based on previous results by Y. Takeuchi et al., we show that under appropriate conditions, the sign of the stability modulus of the Jacobian of the system at the origin determines the asymptotic behaviour of the solutions. If it is non-positive, then the population becomes extinct in every patch. Conversely, if it is positive, then there exists a unique nonnegative equilibrium, which is positive and globally asymptotically stable.In the latter case, given a subset of 'controlled' patches where human intervention is allowed, through mass-trapping for instance, we study whether the introduction of additional linear mortality in some of them can result in population elimination in every patch. We characterize this possibility by an algebraic property on the Jacobian at the origin of a so-called residual system. We then assess the minimal globally asymptotically stable equilibrium that may be attained in this way, and when elimination is possible, we study the optimization problem consisting in achieving this task while minimizing a certain cost function, chosen as a nondecreasing and convex function of the mortality rates added in the controlled patches. We show that such minimization problem admits a global minimizer, which is unique in the relevant cases. An interior point algorithm is proposed to compute the numerical solution.