Predator-prey dynamics of four fish species in the Black Sea with analytical and numerical analysis

Abstract

In this study, a four-species predator–prey model representing a mesopredator-driven trophic structure in the Black Sea ecosystem, including the harvesting process, is developed and analyzed. The model involves anchovy ($A$), horse mackerel ($H$), bluefish ($B$), and Atlantic bonito ($P$). The model incorporates intraguild predation, logistic prey regulation, predator-dependent persistence, and proportional harvesting at all trophic levels within a seasonal time window. Subsequent to non-dimensionalization, positivity, forward invariance, and uniform boundedness of solutions are established. The explicit derivation of all biologically feasible boundary and interior equilibria is a key highlight of the study. Local stability conditions are obtained via Jacobian spectral analysis and Routh–Hurwitz criteria, and a Volterra-type logarithmic Lyapunov function is constructed to establish asymptotic stability of the coexistence equilibrium within a positively invariant ordering region. In order to establish a connection between theoretical concepts and practical applications, annual harvesting data are converted into dimensionless pressure indices, and a constrained numerical calibration is performed. The calibrated regime is characterized by a strictly positive and locally asymptotically stable coexistence equilibrium. Continuation analysis with respect to harvesting intensities reveals asymmetric trophic sensitivity: while the anchovy–horse mackerel axis remains structurally robust, bluefish and bonito exhibit significantly larger long-run variability under harvest perturbations. The results establish explicit stability thresholds for multi-species fisheries under harvesting and show that selective exploitation can drive trophic redistribution without destabilizing coexistence. The framework provides a rigorous, management-relevant approach to modeling harvested marine ecosystems.