Why aphids are not pests in cacao? An approach based on a predator-prey model with aging

We studied a mean-field predator-prey model with aging to simulate the \mbox{interaction} between aphids (\textit{Toxoptera aurantii}) and syrphid larvae in \mbox{cacao} farms in Ilheus, Bahia. Based on the classical predator-prey model, we \mbox{propose} a system of differential equations with three rate equations. \mbox{Unlike} the original Lotka-Volterra model, our model includes two aphid population classes: juveniles (non-breeding) and adult females (asexually breeding). We obtained steady-state solutions for juvenile and adult populations by \mbox{analyzing} the stability of the fixed points as a function of model \mbox{parameters}. The results show that the absorbing state (zero prey population) is always possible, but not consistently stable. A nonzero stationary solution is achievable with appropriate parameter values. Using phase diagrams, we analyzed the \mbox{stationary} solution, providing a comprehensive understanding of the \mbox{dynamics} involved. Simulations on complete graphs yielded \mbox{results} closely matching the differential equations. We also \mbox{performed} simulations on \mbox{random} networks to highlight the influence of \mbox{network} topology on \mbox{system} behavior. Our findings highlight the critical role of life-stage structure, \mbox{predation}, and spatial variation in stabilizing predator-prey \mbox{systems}. This emphasizes the importance of network effects in population dynamics and refines the framework for biological pest control in agriculture. Ultimately, our research contributes to sustainable agricultural practices.