Sliding mode dynamics on a prey-predator system with intermittent harvesting policy

Unsustainable fishing has been identified as one of the most important direct drivers of the degradation of coral reef ecosystems. Herbivorous reef fish, which prevent excessive accumulation of coral-suffocating algae, are in a steady decline due to the presence of invasive predators together with overfishing in many coral reef systems. Here, a two-dimensional predator-prey model is proposed that enables exploration of intermittent harvesting policies, where a control function in the model is evoked that signals when harvesting should cease and commence to maintain a herbivorous fish density above a critical threshold necessary for population stability. We analyze the model by using Filippov's regularization method, which is ideal for discontinuous dynamic systems, such as the system explored here with intermittent harvesting. We obtain the sliding segment of the Filippov's system and its domains together with the conditions for the existence and stability of the regular, virtual and pseudo-equilibria. We show that regular equilibria and pseudo-equilibrium can coexist. Further, we show that intermittent harvesting, with time lags between fishery harvesting decisions and implementations considered, can lead to a bounded oscillation about pseudo-equilibria, or in other words, a sustainable herbivorous fishery. Finally, using a selective harvesting policy, where the invasive predator fish is targeted (here lionfish), we show that a longer herbivorous fish harvesting period can be achieved without changing the critical threshold for harvesting.