Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect

Abstract

We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.

Unconventional criticality, scaling breakdown, and diverse universality classes in the Wilson-Cowan model of neural dynamics

“…( 38) diverges. The saddle-node and transcritical bifurcations collide into a saddle-node transcritical (SNT) bifurcation or tricritical point [62,63]. Observe that in Fig.…”

“…The tricritical point in case A (see Fig. 3a) corresponds to a saddle-node transcritical (SNT) bifurcation -i.e., where the lines of transcritical and saddle-node bifurcations intersect without further degeneracies [62,63]. Thus, in order to tune to this transition point one needs to set two parameters in the phase diagram (w EE , w EI ), as explicitly calculated in Appendix VI B.…”

Impact of the fear and Allee effect on a Holling type II prey–predator model

“…Allee effect is caused by many reasons, including inbreeding, depression [18], mating difficulty [19], low density social disorder [20] and so on. More and more scholars have studied Allee effect in recent years due to its biological significance [21][22][23][24][25][26][27][28][29]. For some endangered species, Allee effect is more likely to occur, so Allee effect is very important for the management of endangered species conservation, population development and utilization, as well as the introduction of species are very important.…”

Investigation on dynamics of an impulsive predator–prey system with generalized Holling type IV functional response and anti-predator behavior

“…Haldar et al [11] formulated a delay-induced ecoepidemic model using a reconstructed Leslie-Gower-type growth rate and then analyzed the dynamics of the system with infection in the prey population. A predator-prey model with the additive Allee effect and the fear effect in the prey was presented and investigated by Lai et al [12]. The bifurcation analysis of the model was explored by considering on the impact of these two effects.…”