Dynamic Analysis of an Ecological Model with Fear Effect on Prey and Additional Food for Predator Using Ratio-Dependent Functional Response

Abstract

This study discusses the dynamic analysis of a predator-prey model that incorporates the fear effect on prey and supplemental food for predators, using a ratio-dependent functional response. The fear effect reduces the prey’s intrinsic growth rate due to behavioral changes under predation risk, while supplemental food enables the predator to survive even when prey density is low. The analysis begins with the formulation of the model equations, followed by the identification of equilibrium points, linearization of the system, and local stability analysis using eigenvalues. Numerical simulations are carried out using \textit{Matcont} and \textit{Pplane} to verify the analytical results and to illustrate the system’s qualitative behavior. The model parameters are based on the interaction between elk (\textit{ELK}) as prey and wolves (\textit{Canis lupus}) as predators. The results reveal four equilibrium points: $E_0=(0,0)$ is an unstable nodal source, $E_1=\left(0,\dfrac{nA(c\gamma - e\alpha)}{em}\right)$ and $E_2=\left(\dfrac{r-a}{b},0\right)$ are unstable saddle points, while the coexistence equilibrium $E_3=(x^*, y^*)$ is a stable spiral sink under certain parameter conditions. Bifurcation analysis with respect to the fear parameter $f$ and the supplemental food parameter $A$ reveals the occurrence of a transcritical bifurcation, where two equilibrium branches exchange stability. The system tends toward the equilibrium point $E_1$ when either $f$ or $A$ exceeds a critical threshold, indicating that predators can persist even as prey populations decline significantly. These findings suggest that predator survival is not solely dependent on prey availability but also influenced by the availability of alternative food sources and the intensity of the prey’s fear response.