On a reaction-diffusion model for sterile insect release method with release on the boundary

Xin Li,Xingfu Zou

doi:10.3934/dcdsb.2012.17.2509

Abstract

We consider a partial differential equation model that describes the sterile insect release method (SIRM) in a bounded 1-dimensional domain (interval). Unlike everywhere-releasing in the domain as considered in previous works [17] and [14] , we propose the mechanism of releasing on the boundary only. We show existence of the fertile-free steady state and prove its stability under some conditions. By using the upper-lower solution method, we also show that under some other conditions there may exist a coexistence steady state. Biological implications of our mathematical results are that the SIRM with releasing only on the boundary can successfully eradicate the fertile insects as long as the strength of the sterile releasing is reasonably large, while the method may also fail if the releasing is not sufficient.

Highlights

Among various biological control methods for insects is the Sterile Insect Release Method (SIRM) which was originally suggested by Knipling [15]
We have considered a reaction diffusion system to model the the Sterile Insect Releasing Method (SIRM)
We have proved the well-posedness of this alternated model, and obtained conditions under which the fertile-free steady state exists and is globally asymptotically stable, accounting for success of the sterile insect release method (SIRM)

Summary

Introduction

Among various biological control methods for insects is the Sterile Insect Release Method (SIRM) which was originally suggested by Knipling [15]. We adopt the two partial differential equations in (1.1) to describe the interaction of the fertile and sterile female insects, but we confine the spatial variable x to a bounded interval Ω = (− , ); instead of releasing the sterile insects everywhere in Ω, we consider releasing the sterile insects only at the boundary of Ω, i.e., ∂Ω = {− , }, with the releasing amount proportional to the gradient of the sterile insects at the two end points These considerations lead to the following model in the form of Initial-Boundary-Value problem: ut = d1uxx + u nt = d2nxx − a2n − 2g n(u + n),. N, x ∈ Ω is positively invariant for (1.5)

Boundary steady state

Existence or non-existence of coexistence steady state

Conclusion and discussion