Mathematical Model for the Biological Control of the Coffee Berry Borer Hypothenemus hampei through Ant Predation

Abstract

Coffee is a relevant agricultural product in the global economy, with the amount and quality of the bean being seriously affected by the coffee berry borer Hypothenemus hampei (Ferrari), CBB, its principal pest. One of the ways to deal with this beetle is through biological control agents, like ants (Hymenoptera: Formicidae), some of which are characterized by naturally inhabiting coffee plantations and feeding on CBB in all their life stages. Our paper considers a predator–prey interaction between these two insects through a novel mathematical model based on ordinary differential equations, where the state variables correspond to adult CBBs, immature CBBs, and ants from one species, without specifying whether preying on the CBB is among their feeding habits, in both adult and immature stages. Through this new mathematical model, we could qualitatively predict the different dynamics present in the system as some meaningful parameters were varied, filling the existing gap in the literature and envisioning ways to manage pests. Mathematically, the system’s equilibrium points were determined, and its stability was studied through qualitative theory. Bifurcation theory and numerical simulations were applied to illustrate the stability of the results, which were interpreted as conditions of the coexistence of the species, as well as conditions for eradicating the pest, at least theoretically, through biocontrol action in combination with other actions focused on eliminating only adult CBBs.

Bifurcation Analysis in a Coffee Berry-Borer-and-Ants Prey–Predator Model

“…It is also assumed that the CBB population is reduced by the predatory effect of ants. According to works such as [4,5,7,10,13,15], different ant species feed on CBB adults and/or immatures (eggs, larvae, and pupae). In this case, it is assumed that the number of "consumptions" is proportional to the average number of CBB, regardless of their state, but we also take into account the satiation of predators as prey density increases.…”

“…For the ant population, denoted by the letter R, logistic growth is also assumed, with r representing the intrinsic growth rate and k representing the carrying capacity. Furthermore, an additional increase in the intrinsic growth of ants, attributed to predation, is assumed-a hypothesis incorporated in population dynamics works such as [24][25][26][27] in the context of Aedes aegypti and in [5] within the context of CBB. As is customary in prey-predator models, it should be noted that not all prey biomass is converted into predator biomass, so a biomass conversion rate denoted by ε is considered.…”

“…Unfortunately, coffee cultivation faces a global threat: namely, the Coffee Berry Borer (CBB), Hypothenemus hampei (Ferrari, 1867) (Coleoptera: Curculionidae: Scolytidae), a beetle about the size of a pinhead that penetrates coffee beans to feed and reproduce within them, thereby reducing both the quantity and the quality of production. Its presence and impact on coffee crops worldwide designate it the primary coffee pest and pose a significant concern to the coffee industry [3][4][5]. In fact, ref.…”

“…In [5], the population dynamics of the CBB when preyed upon by ants were studied. In particular, modifications in dynamics were investigated based on variations in the death rate of adult CBBs due to factors other than predation-factors such as cultural and/or chemical control methods.…”

“…This study, along with the article by [5], represents a novelty in modeling the interaction of the CBB with a beneficial predator for coffee growers, such as ants. However, it is important to highlight that the focus of this work is on analyzing the effects of predation by ants while excluding external factors.…”