Modeling the Allee effects induced by cost of predation fear and its carry-over effects

•We investigate the role of fear and its carry-over effects in a prey-predator model.•Predation-driven Allee effects can be observed in our model with type I response.•Single species model shows three types of growth (logistic, weak and strong Allee).•Fear and its carry-over effects have important role even with type I response.•Paradox of enrichment can be observed in our model with type I functional response. Predation-driven Allee effects play an important role in the dynamics of a small prey population; however, such effects cannot occur for the model with type I functional response. Predator-driven Allee effects generally occur when a generalist predator targets some specific prey. However, apart from the lethal effects of predation, there are some non-lethal effects in the presence of a predator. Due to the fear of predation, positive density dependence growth of prey may be observed at low population density because of reduced foraging activities. Moreover, such a non-lethal effect can be carried over through generations or seasons. In the present manuscript, we investigate the role of predation fear and its carry-over effects in the prey-predator model. First, we study the single-species model from a global perspective. We have shown that depending on the birth rate; our single-species model describes three types of growth dynamics, namely, strong Allee dynamics, weak Allee dynamics, and logistic dynamics. Then we include the explicit dynamics of the predator, with type I functional response. Basic dynamical properties, as well as the global stability of each equilibrium, have been discussed. From our analysis, we can observe that both the fear and its carry-over effects have a significant role in the stability of the coexistence equilibrium, even for the model with type I functional response. The phenomenon ‘paradox of enrichment’ can be observed in our model, which cannot be observed in the classical prey-predator model with type I functional response. However, we can see that such a phenomenon can be ruled out by choosing suitable non-lethal effect parameters. Therefore, our study shows how non-lethal effects change the dynamics of a prey-predator model and has powerful biological insights, especially for understanding the dynamics of small populations.