Sliding mode dynamics on a prey–predator system with intermittent harvesting policy
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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.
KeywordsDiscontinuous harvesting policy Modified Leslie–Gower scheme Filippov’s system Sliding bifurcation Selective harvesting
Mathematics Subject Classification92B05 92D25 92D40
JB is supported by the grants from Science and Engineering Research Board (SERB), Govt. of India (File No. TAR/2018/000283). SB acknowledges financial support in the form of J C Bose Fellowship by the SERB, Govt. of India, no. SB/S2/JCB-023/2015.
Compliance with Ethical Standards
Conflict of Interest
The authors declare that they have no conflict of interest.
Human Participants and/or Animal
Research involving Human Participants and/or Animal: Not applicable.
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