The purpose of the current manuscript is to propose a generic method that causes the local extinction of a harmful invasive species. Eradication is achieved via introduction of phenotypically modified organisms into a target population. Here we propose a model without the logistic type term, of which the reaction terms may change sign, and so the solutions are not bounded a priori. We prove global existence of solutions via a Lyapunov function method, and show existence of a finite dimensional (L2(Ω);H2(Ω)) global attractor that supports states of extinction, improving current results in the literature. We also conduct numerical simulations to investigate the decay rate of the female species. Lastly we apply optimal control techniques to compare the effectiveness of various reaction terms on species extinction.

Keywords : Reaction Diffusion System, Global Existence, Global Attractor, Optimal Control, Invasive Species, Biological Control.