Global analysis and patch-induced bifurcations in a spatially heterogeneous fishery system with price feedback and taxis

This paper investigates the spatio-temporal dynamics of a spatially heterogeneous fishery system governed by a novel tripartite coupled partial differential equation-ordinary differential equation (PDE-ODE) model, integrating fish biomass logistic growth, profit-driven fleet aggregation (taxis) toward high-resource gradients, and a global supply-demand mechanism regulating market price. We establish the global existence of nonnegative weak solutions by employing regularization techniques and deriving uniform-in- ϵ estimates, thereby addressing the analytical challenges posed by the cross-diffusive taxis coupling and the low spatial regularity of the catchability function—specifically in the presence of sharply defined Marine Protected Areas (MPAs). Furthermore, by analyzing the spectral properties of the linearized steady-state operators and applying the Crandall-Rabinowitz theorem, we rigorously prove the existence of transcritical-type bifurcations driven by harvesting intensity and the MPA radius, identifying the precise thresholds that trigger a regime shift from fishery extinction to stable, spatially non-homogeneous coexistence. Numerical simulations complement the analytical results by revealing the Fishing the Line phenomenon, where harvesting effort sharply aggregates along protected boundaries to exploit biomass spillover, and by quantitatively validating the theoretical critical radius R ⁎ beyond which the fishery collapses under spatial constraints. These findings elucidate how the interplay between directed harvester movement, market feedback, and spatial protection modulates the long-term sustainability and spatial structure of coupled bio-socio-economic systems.