Global stability and optimisation of a general impulsive biological control model

TL;DR

This paper analyzes a predator-prey model with periodic predator releases, establishing conditions for prey eradication and optimizing predator deployment frequency to minimize invasion time, highlighting that more frequent releases are optimal.

Contribution

It introduces a general impulsive biological control model, proves the existence and stability of prey eradication solutions, and formulates an optimization problem for predator release strategies.

Findings

Existence of an invariant periodic solution for prey eradication.

Global asymptotic stability condition for the prey-free solution.

Optimal strategy involves the most frequent predator releases.

Abstract

An impulsive model of augmentative biological control consisting of a general continuous predator-prey model in ordinary differential equations augmented by a discrete part describing periodic introductions of predators is considered. It is shown that there exists an invariant periodic solution that corresponds to prey eradication and a condition ensuring its global asymptotic stability is given. An optimisation problem related to the preemptive use of augmentative biological control is then considered. It is assumed that the per time unit budget of biological control (i.e. the number of predators to be released) is fixed and the best deployment of this budget is sought after in terms of release frequency. The cost function to be minimised is the time taken to reduce an unforeseen prey (pest) invasion under some harmless level. The analysis shows that the optimisation problem admits a countable infinite number of solutions. An argumentation considering the required robustness of the optimisation result is then conducted and it is shown that the best deployment is to use as frequent (and thus as small) as possible predator introductions.