Dynamical system of quokka population depicting Fennecaphobia by Vulpes vulpes

A spatio-temporal prey-predator (quokka and red fox interaction) model with the fear effect, Holling type Ⅱ functional response, and a generalist predator is proposed. The existence of equilibrium points and their corresponding stability are analyzed under certain conditions to explore the system's dynamics. The occurrence of a Hopf bifurcation, a saddle-node bifurcation, and a Bogdanov-Takens bifurcation are confirmed. The partial rank correlation coefficient method is performed for the sensitivity analysis. Furthermore, the cross-diffusion is incorporated in the formulated model system to identify the spatio-temporal dynamics of the system. All theoretical results are validated through a numerical simulation. The outcome of the temporal model shows a decrease in the fear effect due to the predation by the red fox helps to increase the quokka population. The spatio-temporal model indicates that as the diffusion coefficient and fear parameters vary, the pattern changes from isolated spots to stripes, and again from stripes to spots. This represents the variation in spatial interactions and aggregation. The dispersion of predators and prey increases with an increased diffusion; however, the group formation is restricted by a stronger fear effect that scatters prey.