Impact of Fear on Death Rate of Prey Species: Codimension-1 Bifurcations and Strong Resonances in a Discrete Predator–Prey Model

Predator–prey interaction is considered a natural phenomenon in the ecological system. Empirical research conducted on vertebrates has demonstrated that the presence of predators can have a significant impact on the survival rates and reproductive capabilities of prey populations. Recently, there has been research on mathematical models of predator–prey systems that include different predator functional responses and fear effects. These studies have overlooked the influence of fear on the death rates of species that are hunted. From the given findings, we present a mathematical model of predator–prey systems that includes fear costs impacting the rates of reproduction and death in prey population. By reducing the discrete model into different normal forms, we prove that there exists a set of codimension-1 and codimension-2 bifurcations, which include transcritical, flip, Neimark–Sacker bifurcations, 1:2 and 1:4 strong resonance bifurcations. These findings indicate that, compared with the system without fear effect, the increase of the fear effect parameter [Formula: see text] that affects the birth rate of prey and the fear effect parameter [Formula: see text] that affects the death rate of prey will strengthen the oscillation of prey population and reduce the oscillation of predator population. In addition, the increases of [Formula: see text] and [Formula: see text] have no effect on the density of the prey population but reduce the density of the predator population. When the fear effect [Formula: see text] and other parameter values remain the same, the system generates an expanding limit circle as [Formula: see text] increases, indicating that the effect of fear effects on the death rate enhances the stability of the system.