Optimal Control of Coffee Berry Borers: Synergy Between Bio-insecticide and Traps

Abstract

The coffee berry borer (CBB), Hypothenemus hampei, is the most destructive insect pest affecting coffee plantations in most coffee-producing countries, hence causing major economic losses worldwide. The cryptic life cycle of CBB inside coffee berries makes their control extremely difficult. To tackle this problem, we use a dynamical model describing the plant–pest interactions during a cropping season, which includes a berry age structure to account for CBB preference for mature berries. We introduce two environmentally friendly control methods, consisting in applying a bio-insecticide to reduce berry infestation and in trapping the colonising CBB. Our objective is to maximise the profit generated by the harvest of healthy coffee berries, while minimising the CBB population for the next cropping season. The existence of an optimal control strategy is provided, and necessary optimality conditions are established. Finally, the optimal control problem is solved numerically and simulations are provided. They show that combining the two control methods is a cost-effective strategy to protect coffee berries from CBB infestation.

Breakdown of the Optimal Control Impact

Figures 5 and  are obtained by numerically solving optimal control Problem 3.1 to obtain the optimal control pair $(u^{\star },v^{\star })$, but then applying only one of the two controls and setting the other one to zero to integrate system (3). We hence observe the impact of the bio-insecticide ($u^{\star }$) alone and the traps ($v^{\star }$) alone on the crop–pest dynamics.

Fig. 5

a–c Simulation of system (3 when only applying control $u^{\star }$ ($v=0$, plain magenta curves) or control $v^{\star }$ ($u=0$, plain orange curves) from optimal control pair $(u^{\star },v^{\star })$, without controls ($u=v=0$, dashed blue curves) and without pest ($y(0)=z(0)=0$, dash-dotted black curve). d Evolution of controls $u^{\star }$ representing the bio-insecticide and $v^{\star }$ representing traps. The parameter values are given in Table 1. Zero initial conditions are set, except for colonising CBB: $y(0)=10^{4}$ females

Fig. 6

Age distribution of the healthy coffee berries and their price at the end of the simulation ($t=t_f$) with optimal control pair $(u^{\star },v^{\star })$ (plain red curves), when only biopesticide control $u^{\star }$ ($v=0$, plain magenta curves) or trap control $v^{\star }$ ($u=0$, plain orange curves) is applied, without control ($u=v=0$, dashed blue curves) and without pest ($y(0)=z(0)=0$, dash-dotted black curves). Cases without control and with constant infestation rates $\beta$ are also represented (blue-shaded areas delimited by $\beta _{\min }$ and $\beta _{\max }=\beta _{\min }+\beta _a$, plain blue curves for mean value ${\bar{\beta }}$). The parameter values are given in Table 1. Zero initial conditions are set, except for colonising CBB: $y(0)=10^{4}$ females