Fear-Induced Carry-Over Effects in Population Dynamics with Supplementary Food

The dynamic behaviour of an interacting species system with a hyperbolic functional response to fear and its carry-over effects is examined in this study. The primary objective is to analyze how fear influences system equilibrium, stability, and bifurcation through these effects. The existence and stability of equilibria and solution boundedness are investigated, with the Jacobian matrix used to assess equilibrium stability, revealing complex dynamics under feasible constraints. Parameter sensitivity is assessed in a model with Beverton-Holt-like food supplementation. Qualitative and quantitative analyses, including bifurcation theory, dynamical theory, and numerical simulations, are conducted, employing Sotomayor's theory to confirm the presence of codimension 1 bifurcations and the normal form theorem to determine Hopf bifurcation direction. The model also explores local and global codimension 2 bifurcations, indicating species coexistence and permanence under specific parameter constraints. Moreover, the system's intriguing dynamic bi-stability and bubbling phenomena are the focus of the present investigation. Analytical and numerical methods are integrated to provide a comprehensive understanding of the system’s dynamics, with numerical simulations further elucidating the system's sensitivity and complexity.