Optimal pest control for banana production under varying infestation density

Abstract

Banana and plantain production faces significant caused by many pests, the most harmful being Radopholus similis. This nematode infests and spends most of its life in the plant roots, which makes it difficult to control. Conventional methods face efficiency, cost, and environmental limitations, making it crucial to design optimized strategies. Our aim is to tackle this issue by formulating an optimal control problem on an original model of banana-pest interactions. We propose a four-compartment model describing the host and pest dynamics. Its originality is that both infected roots and infesting nematodes are considered, which allows variable nematode densities in the roots. By employing time-scale separation, we reduce the model dimension and reveal a scenario where both stable pest-free and endemic equilibria coexist (backward bifurcation). We then include a control variable which reduces the root infestation rate. We formulate an optimal control problem aiming at maximizing the profit during a cropping season. The existence of an optimal control and necessary optimality conditions are established. We solve the optimal control problem by a direct method implemented in Bocop. For numerical settings of interest, the optimal solutions exhibit a bang-bang quasi-periodic structure. We extensively test whether candidate suboptimal strategies, based on periodic or constant control, can approximate the optimal performance. Numerical comparisons reveal that constant control performs poorly, whereas periodic control closely approaches the optimal solution. This confirms the overyielding effect and supports the realistic implementation of periodic control.