Impact of insecticides on bee ecosystems: A mathematical approach using piecewise smooth dynamical systems

Honey bees (Apis mellifera) are key pollinators in agroecological systems and are crucial to ensuring global food security. However, apiculture is currently facing a crisis due to the proliferation of pests and diseases, with the ectoparasitic mite Varroa destructor posing the most significant threat. By feeding on the hemolymph of honey bees, this mite reduces their lifespan and acts as a vector for lethal viruses, potentially leading to the collapse of the entire beekeeping ecosystem. In this context, we propose a novel mathematical model based on a nonlinear system of ordinary differential equations to describe the interaction between the honey bee population and the Varroa destructor mite population, along with the temporal evolution of honey production. The model identifies the equilibrium states and examines their biological feasibility and stability in relation to key ecological thresholds. Numerical simulations are then conducted to illustrate the system's dynamics and to interpret, from an agroecological perspective, the impact of chemical control applied to crops situated in close proximity to the beekeeping environment. Subsequently, a switching mechanism is incorporated into the mite population equation, transforming the system in a piecewise smooth model. This mechanism activates or deactivates chemical control, depending on whether the ratio of mites to bees exceeds a predefined critical threshold. This approach captures the abrupt transitions induced by technical monitoring in apicultural practice, providing a more realistic framework for representing discontinuous control strategies. This work thus enables the evaluation of the differential impact of chemical pressure arising from neighboring crops and from targeted mite control, demonstrating, within the framework of ecological theory, how such interventions influence a population's viability and the long-term sustainability of the apicultural ecosystem.