Complex dynamics of a tri-trophic predator-prey system with double fear effects

Abstract

In this paper, we investigate a tri-trophic predator-prey system with double fear effects, where the prey is affected by fear induced by both the intermediate predator and the top predator. The top predator is assumed to feed on both the basal prey and the intermediate predator. We establish the positivity and boundedness of solutions and derive sufficient conditions for the global asymptotic stability of axial equilibria. The possible interior equilibria are characterized algebraically, and their local stability is analyzed. We further derive the Hopf bifurcation conditions and compute the first Lyapunov coefficient to determine the direction of the bifurcation and the stability of the bifurcating periodic orbit. Numerical simulations reveal several forms of complex dynamics, including a period-three orbit in the associated return dynamics, the coexistence of a stable equilibrium and a stable limit cycle, the coexistence of two stable limit cycles, and chaotic behavior. The corresponding multistability is further illustrated by basin-of-attraction diagrams. Numerical evidence for chaotic attractors in parameter regions admitting different numbers of interior equilibria is provided by positive largest Lyapunov exponents, parameter bifurcation diagrams, and bounded long-time trajectories. These results show that, within the parameter regimes examined, the interaction between the two fear mechanisms and the tri-trophic feeding structure can generate pronounced dynamical transitions, multistability, and strong sensitivity to initial population densities.