Basic reproduction number of a deterministic and stochastic predator–prey model with novel fear functions

In predator–prey models, the basic reproduction number serves as a critical threshold parameter that determines whether predators and prey can coexist stably or whether predators will go extinct, thereby guiding ecological management strategies to control pest outbreaks or maintain biodiversity. In this paper, we formulate two predator–prey models with prey refuge under the novel fear functions. For the deterministic model, we investigate the basic reproduction numbers R0 that determines the existence and stability of equilibria, which is shown that (i): extinction equilibrium always exists and is an unstable saddle; (ii): when R0<1, predator-free equilibrium is always existence and stable; (iii): when R0>1, predator-free equilibrium becomes unstable, while a coexistence equilibrium appears and remains stable; (iv) when R0=1, this model undergoes a transcritical bifurcation at predator-free equilibrium. For the stochastic model, we obtain two basic reproduction numbers R1 and R0˜, which are shown that (i): R1 determines the persistence and extinction of prey population, when R1>0, prey population is persistent in the mean, otherwise it is extinct; (ii): R0˜ determines the persistence and extinction of predator population, when R0˜<1, predator population is extinct, otherwise it is persistent in the mean; (iii): R0˜≤R0; (iv): when R0˜>1, we have the existence and uniqueness of stationary distribution; (v): when R0>1, we get the explicit expression of the stationary probability density and the stochastic stability and Hopf bifurcation may occur. Our results reveal that (i): the fear effect is independent of the basic reproduction number and can only influence the population size; (ii): High-intensity environmental noise is more prone to destabilize predator–prey coexistence, thereby increasing the risk of predator extinction or prey population outbreaks.