On a reaction-diffusion model for sterile insect release method on a bounded domain

We consider a system of partial differential equations that describes the interaction of the sterile and fertile species undergoing the sterile insect release method (SIRM). Unlike in the previous work [M. A. Lewis and P. van den Driessche, Waves of extinction from sterile insect release, Math. Biosci. 5 (1992) 221-247] where the habitat is assumed to be the one-dimensional whole space R, we consider this system in a bounded one-dimensional domain (interval). Our goal is to derive sufficient conditions for success of the SIRM. We show the existence of the fertile-free steady state and prove its stability. Using the releasing rate as the parameter, and by a saddle-node bifurcation analysis, we obtain conditions for existence of two co-persistence steady states, one stable and the other unstable. Biological implications of our mathematical results are that: (i) when the fertile population is at low level, the SIRM, even with small releasing rate, can successfully eradicate the fertile insects; (ii) when the fertile population is at a higher level, the SIRM can succeed as long as the strength of the sterile releasing is large enough, while the method may also fail if the releasing is not sufficient.