Biological control, the use of an organism to reduce the population density of another organism, is an alternative technique to pesticide for crop protection against pests. Its efficiency is achieved though predation, parasitism, pathogenicity,... Biological control  has the upper-hand on pesticides from health, environmental and durability point of views. However, it needs to be optimized in order to rival pesticides from economic and efficiency point of views. In the present work, we will focus on predator-prey relationships and we consider augmentative biological control, that is a technique that consists in periodically releasing biological control agents for pest control. We consider that the area to be protected is divided into patches, either because it is naturally so (eg. separate fields) or because a single field is artificially divided into areas into which separate agent releases are achieved. Starting with a two patch model with dispersal of both species between the two patches, we attempt to give answers to two dual questions : 1) how to release the predators to achieve a given prey decreasing rate with minimal effort and 2) how to spread a given predator release rate between the patches to achieve the fastest prey decay.

In this study, we first examine the effectiveness of impulsive release of biological agents into a two patch environment through mathematical modeling. We develop continuous predator-prey models with dispersal of the species in both patches between which the predator-prey dynamics can be quite different due to differing environmental conditions. Predator releases are then modelled through instantaneous impulses taking place periodically in each patch. Three parameters are then used to analyze this problem : the total predator release rate (which needs to be minimzed in question 1, but is fixed in question 2), the release period, and the predator release distribution between the patches. We first consider Lotka-Volterra models with linear dispersal (Yang & Tang, 2009), for which we show that, most of the time, it is best to split the release between the two patches to achieve pest eradication optimally. A model with fitness-based dispersal is then considered (Cressmann & Krivan, 2013) so that the impact of population dispersal modes on biological control strategies can be analyzed.

References :

Cressman, R., Krivan, V.  2013. Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds. Journal of Mathematical Biology 67:329-358. DOI 10.1007/s00285-012-0548-3.

Yang, J., Tang S. 2009. Effects of population dispersal and impulsive control tactics on pest management. Nonlinear Analysis: Hybrid Systems 3 :487-500.