Global existence and asymptotic behavior of a model for biological control of invasive species via supermale introduction

R. D. Parshad, S. Kouachi and J. B. Gutierrez,  Communications in Mathematical Sciences,  11:971-992. 2013.

The purpose of this manuscript is to propose a model for the biological control of invasive species, via introduction of phenotypically modified organisms into a target population. We are inspired by the earlier Trojan Y Chromosome model [J.B. Gutierrez, J.L. Teem, J. Theo. Bio., 241(22), 333-341, 2006]. However, in the current work, we remove the assumption of logistic growth rate, and do not consider the addition of sex-reversed supermales. Also the constant birth and death coefficients, considered earlier, are replaced by functionally dependent ones. In this case the nonlinearities present serious difficulties since they change sign, and the components of the solution are not a priori bounded, in some LP-space for p large, to permit the application of the well known regularizing effect principle. Thus functional methods to deduce the global existence in time, for the system in question, are not applicable. Our techniques are based on the Lyapunov functional method. We prove global existence of solutions, as well as existence of a finite dimensional global attractor, that supports states of extinction. Our analytical finding are in accordance with numerical simulations, which we also present.