Threshold Dynamics of the Hydra and Hormetic Effects in a Discrete Population Model

Externally induced overcompensation can generate paradoxical population responses, including hydra and hormetic effects, in which external disturbances increase equilibrium population size or produce nonmonotone response curves. In this study, we formulate a single-population discrete-time model with heterogeneous harvesting and alternative placements of compensatory feedback. The model divides the population update into two parts: individuals or biomass that persist directly to the next generation, and the remaining component that contributes through density-dependent recruitment. This formulation allows us to compare how different compensatory assumptions affect equilibrium responses. When the positive equilibrium is locally stable, we derive threshold conditions for hydra and hormetic effects and show that their occurrence depends on where compensation is introduced in the population update. We also examine the local stability and bifurcation behavior of the positive equilibrium as harvesting intensity changes. The equilibrium formulas are further fitted to two published nonmonotone response datasets. The fitted curves show good agreement with the observed U-shaped and inverted U-shaped patterns.