Understanding the hydra effect in predator-dependent functional response models

The seemingly paradoxical increase of a species' means density in response to greater mortality is an interesting research topic in recent times, and such a type of counterintuitive response is known as the hydra effect. Identifying and predicting the consequences of increased mortality is crucial for conservation biology and population management. The existence of the hydra effect in Gause-type models is a known feature in mathematical ecology. However, the hydra effect is still not investigated in mathematical models of predator-dependent functional responses. The present paper addresses the presence of a hydra effect in either the prey and predator species of two-species unstructured predator-dependent functional response models with logistic prey growth and density-independent predator death rate. Our results show that the positive density-dependence of predators (grouping behavior or cooperation) in trophic functions shows a hydra effect in the prey species, whereas the negative density-dependence of predators (interference) shows a hydra effect in the predator species. Moreover, in the former case, prey always exhibits the hydra effect at a stable state, whether in the latter case, predator exhibits the hydra effect at a stable state when the prey has positive self-effects at the equilibrium. In our considered models, the prey's positive self-effects suggest that the intraspecific competition rate coefficient would be sufficiently small so that the species density would be positively related to its per capita growth rate. In the case of cyclic dynamics, the hydra effect is a typical feature of predator species for unstable systems with saturating functional responses.