Global stability and optimisation of a general impulsive biological control model

Ludovic Mailleret (1) , Frédéric Grognard (2)

1 URIH - Unité Recherches Intégrées en Horticulture
2 COMORE - Modeling and control of renewable resources

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 (\ie 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.

Mots clés: optimisation; Predator-prey dynamics; impulsive model; global stability; optimisation.

Domaines: Optimisation et contrôle [math.OC]; Ecosystèmes; Systèmes dynamiques [math.DS]

Ludovic Mailleret, Frédéric Grognard. Global stability and optimisation of a general impulsive biological control model. [Research Report] RR-6637, INRIA. 2008, pp.28. ⟨inria-00320578⟩