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

Carlos Andrés Trujillo-Salazar, Gerard Olivar-Tost () and Deissy Milena Sotelo-Castelblanco
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Carlos Andrés Trujillo-Salazar: Department of Mathematics and Statistics, Universidad Nacional de Colombia, Manizales 170004, Colombia
Gerard Olivar-Tost: Department of Natural Sciences and Technology, Universidad de Aysén, Coyhaique 5950000, Chile
Deissy Milena Sotelo-Castelblanco: Department of Mathematics and Statistics, Universidad Nacional de Colombia, Manizales 170004, Colombia

Mathematics, 2024, vol. 12, issue 11, 1-19

Abstract: One of the most important agricultural activities worldwide, coffee cultivation, is severely affected by the Coffee Berry Borer (CBB), Hypothenemus hampei , considered the primary coffee pest. The CBB is a tiny beetle that diminishes the quantity and quality of coffee beans by penetrating them to feed on the endosperm and deposit its eggs, continuing its life cycle. One strategy to combat CBBs is using biological control agents, such as certain species of ants. Here, a mathematical model (consisting of a system of nonlinear ordinary differential equations) is formulated to describe the prey–predator interaction between CBBs and an unspecified species of ants. From this mathematical perspective, the model allows us to determine conditions for the existence and stability of extinction, persistence or co-existence equilibria. Transitions among those equilibrium states are investigated through the maximum per capita consumption rate of the predator as a bifurcation parameter, allowing us to determine the existence of transcritical and saddle-node bifurcations. Phase portraits of the system are presented for different values of bifurcation parameter, to illustrate stability outcomes and the occurrence of bifurcations. It is concluded that an increase in bifurcation parameters significantly reduces the CBB population, suggesting that ant predation is an effective control strategy, at least theoretically.

Keywords: coffee berry borer; prey–predator model; nonhyperbolic equilibrium point; transcritical bifurcation; saddle-node bifurcation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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