Dynamics of a Leslie-Gower model with weak Allee effect on prey and fear effect on predator (Q6537599)

The paper proposes and studies a Leslie-Gower model with weak Allee effect on the prey and fear effect on the predator (not the usual fear effect on the prey, but rather the damaging effect on the catching ability of the predator due to its fear from other predators higher in the food chain). The prey equation is \(\frac{dx}{dt}=rx\left(1-\frac{x}{K}\right)A(x)-\frac{cxy}{1+ky}\) with weak Allee effects term \(A(x)=\frac{x}{\beta +x}\) and the predator equation is \(\frac{dy}{dt}=\frac{sy}{1+ky}\left(1-\frac{y}{nx}\right)\), all parameters being positive. The same model without the fear effect (\(k=0\)) was studied in [\textit{K. Fang} et al., Qual. Theory Dyn. Syst. 21, No. 3, Paper No. 86, 19 p. (2022; Zbl 1502.34058)]. A similar model, including the fear effect on the predator, but with strong (instead of weak) Allee effects term \(A(x)=x-m\), was studied in [\textit{T. Liu} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 32, No. 6, Article ID 2250082, 24 p. (2022; Zbl 1501.34045)].\N\NThis paper complements these two earlier papers. By studying and comparing the richer range of possible dynamical behaviours of its model (including two positive equilibria, stable limit cycles, and saddle-node, Hopf and Bogdanv-Takens bifurcations) with those of the previous models, emphasizes the impact of weak vs strong Allee effects and the impact of predator's fear. The dynamical behaviour of the model is studied, using a dimension-free equivalent version of the model and its parameters, mostly by analytical methods and occasionally by numerical methods. Figures and phase portraits illustrate the different possible behaviours. An interesting feature is that the fear effect can permit coexistence of both species for intermediate values of the Allee parameter \(\beta\).