Fear, refuge, and adaptive harvesting stabilise a five-dimensional tri-trophic predator–prey model exhibiting a transcritical bifurcation

Abstract We propose and analyse a five-dimensional continuous-time tri-trophic predator–prey model comprising a prey species, an intermediate predator, a top predator, and two adaptive harvesting efforts. The model integrates three ecologically relevant mechanisms: fear-induced reduction in prey growth, prey refuge, and dynamic harvesting effort governed by economic payoffs. Positivity and uniform boundedness of all solutions are rigorously established. All biologically feasible equilibrium points are identified, and their local asymptotic stability conditions are derived via Jacobian linearisation and the Routh–Hurwitz criterion for the interior coexistence equilibrium. A Lyapunov candidate is analysed, and extensive numerical integration from a wide range of initial conditions indicates that the interior equilibrium is globally attracting whenever it exists. The system undergoes a transcritical bifurcation at the top-predator-free planar equilibrium with respect to the predator attack rate: the top predator invades and a stable interior coexistence equilibrium emerges once the attack rate exceeds a critical threshold, established analytically through a transversal eigenvalue-crossing argument (exchange of stability) and located numerically. Extensive simulations confirm this analysis: below the threshold the top predator is excluded and the system settles to the top-predator-free state, while above it all trajectories converge to coexistence. Within the parameter ranges studied, the interior equilibrium remains asymptotically stable, with no Hopf bifurcation or sustained oscillation observed, indicating that fear, refuge, and adaptive harvesting jointly stabilise the tri-trophic chain. We further quantify how fear intensity, refuge level, and harvesting effort modulate coexistence densities and the invasion threshold.