Résumé
The Sterile Insect Technique (SIT) is a biological control method used to reduce or eliminate pest populations or disease vectors. This technique involves releasing sterilized insects that, upon mating with the wild population, produce no offspring, leading to a decline or eventual eradication of the target species. We incorporate a spatial dimension by modeling the pest/vector population as being distributed across multiple patches, with both wild and released sterile insects migrating between these patches at predetermined rates. We mainly focus on a two-patch system. This study has two primary objectives: first, to derive sufficient conditions for achieving the elimination of the wild population through SIT, whether releases occur in one patch or in both patches. In particular, we provide an estimate of the minimal release rates to reach elimination thanks to the diffusion rates between patches. This is the first time that such a result is given in a general manner. Second, we study an optimal SIT control strategy, where we minimize the total amount of sterile insects to release, and show that, within one patch, it can successfully reduce the wild population in that patch to a desired level within a finite time frame, provided that the migration rates between patches are sufficiently low. Numerical simulations are employed to illustrate these results and further analyze the outcomes.