Non-smooth dynamics of a fishery model with a two-threshold harvesting policy

Abstract

The non-linear dynamical systems theory helps implement regulatory measures to control the growth and evolution of various populations. While invasion by alien fish species is an emerging threat to native fish species in marine ecosystems, a suitable fishery management protocol needs to be incorporated in marine protected areas (MPAs) to mitigate the problem. We propose a policy of selective harvesting whenever the density of a species crosses a predefined threshold. An ecosystem with such a harvesting policy is modelled by a piece-wise smooth prey-predator type fishery model, where a control function defined by two thresholds evokes signals to cease or commence harvesting. This work reports the dynamics and bifurcations of a Filippov system, including the possibility of sliding mode dynamics. We show that the variations in the growth rate and the harvesting rate of the invasive fish species can lead to hysteresis through saddle–node and transcritical bifurcations and the appearance of a stable homoclinic orbit. We obtain the conditions for sliding domains, regular or virtual equilibria, boundary equilibria, pseudo-equilibria, and tangent points. Depending on the parameter choice, the system can stabilize at a regular equilibrium, a pseudo-equilibrium, or a pseudo-attractor, exhibiting multistability. We report the occurrence of regular or virtual equilibrium bifurcation, boundary node bifurcation and pseudo-saddle–node bifurcation. Our findings indicate that proper harvesting strategies can prevent the proliferation of invasive fish species in MPAs.