Eradicating a pest population via a predator in a seasonal environment

We examined the problem of eliminating a pest population in a seasonal environment by introducing a predator species that preys on it. This was formulated as a zero-stabilization problem with state constraints. The dynamics of the two populations were modeled using a prey-predator system with nonlocal reaction terms and time-periodic rates. Our primary focus was finding a regional control that acts on the predators within a specific habitat region. For a fixed subregion $ \omega $ (of the whole domain $ \Omega $), we established both a necessary and a sufficient optimality condition for the eradication (zero-stabilization) of the pest population, expressed in terms of the sign of the principal eigenvalue of a non-self-adjoint operator.