Dynamical Analysis of a Trophic Model on Guano, Invertebrates, and Fish in Cave Ecosystems

Abstract

This study investigates the dynamical interaction between guano biomass density (x), invertebrate biomass density (y), and fish biomass density (z) through a three-compartment trophic model representing a nutrient-based cave ecosystem. The analysis identifies three equilibrium points: the consumer-free equilibrium E₀, the predator-free equilibrium E₁, and the coexistence equilibrium E₂. Local stability analysis shows that the coexistence equilibrium is asymptotically stable, characterized by eigenvalues with negative real parts (λ₁ = -0.519706 and λ_2,3 = -0.085385 ± 0.188169i). Numerical simulations using the fourth–fifth order Runge–Kutta method (RK45) support these analytical results, showing trajectories that exhibit damped oscillations before converging to the steady state. Furthermore, a bifurcation analysis reveals a critical Branching Point (BP) at the predation rate b₂ ≈ 0.043956. This threshold signifies a transcritical bifurcation where the system transitions from a predator-extinction regime to a stable coexistence regime, highlighting the sensitivity of the food web to energy transfer efficiency. These findings suggest that under the assumed parameter set, the interaction between guano nutrients, invertebrates, and fish can maintain a stable ecological balance through top-down control and nutrient-dependent dynamics.