Avoiding and Leveraging Hormetic and Hydra Effects in a Discrete Predator–Prey Model with Pesticide Application

Improper pesticide use can induce hormetic or hydra effects and trigger pest outbreaks. We consider a discrete model for a pest and its natural enemy under pesticide application and predation. When pesticide is applied at the end of each generation, threshold conditions for fixed-point existence and local stability, together with the associated Flip and Neimark–Sacker bifurcations, are derived, and if the killing rates are increased they do not generate hormetic or hydra responses in either population. When pesticide is applied within a generation, one- and two-parameter bifurcation diagrams show how timing, predation, and the relation between the two killing rates reorganize fixed-point existence and stability and generate bubble structures, shrimp-like periodic windows, quasiperiodicity, and chaos. These transitions delimit parameter regions in which the pest exhibits hormetic and hydra responses and the natural enemy does so under suitable predation. High predation weakens these responses in pests, whereas low predation can induce or amplify them in natural enemies. Early application tends to raise population density in the pest, whereas intermediate application is more often associated with stable dynamics. The results clarify how application timing and relative pesticide sensitivity shift paradoxical responses between pests and natural enemies.