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Nonlinear Dynamics

, Volume 98, Issue 2, pp 1299–1314 | Cite as

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

  • Joydeb BhattacharyyaEmail author
  • Daniel L. Roelke
  • Samares Pal
  • Soumitro Banerjee
Original paper
  • 102 Downloads

Abstract

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.

Keywords

Discontinuous harvesting policy Modified Leslie–Gower scheme Filippov’s system Sliding bifurcation Selective harvesting 

Mathematics Subject Classification

92B05 92D25 92D40 

Notes

Acknowledgements

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.

Informed consent

Not applicable.

References

  1. 1.
    Albins, M.A.: Effects of invasive pacific red lionfish Pterois volitans versus a native predator on bahamian coral-reef fish communities. Biol. Invasions 15(1), 29–43 (2013)CrossRefGoogle Scholar
  2. 2.
    Almany, G.R., Hamilton, R.J., Bode, M., Matawai, M., Potuku, T., Saenz-Agudelo, P., Planes, S., Berumen, M.L., Rhodes, K.L., Thorrold, S.R., et al.: Dispersal of grouper larvae drives local resource sharing in a coral reef fishery. Curr. Biol. 23(7), 626–630 (2013)CrossRefGoogle Scholar
  3. 3.
    Aziz-Alaoui, M., Okiye, M.D.: Boundedness and global stability for a predator–prey model with modified Leslie–Gower and Holling-type ii schemes. Appl. Math. Lett. 16(7), 1069–1075 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barbour, A.B., Allen, M.S., Frazer, T.K., Sherman, K.D.: Evaluating the potential efficacy of invasive lionfish (Pterois volitans) removals. PLoS ONE 6(5), e19666 (2011)CrossRefGoogle Scholar
  5. 5.
    Blackwood, J.C., Hastings, A., Mumby, P.J.: The effect of fishing on hysteresis in Caribbean coral reefs. Theor. Ecol. 5(1), 105–114 (2012)CrossRefGoogle Scholar
  6. 6.
    Boukal, D.S., Krivan, V.: Lyapunov functions for Lotka–Volterra predator–prey models with optimal foraging behavior. J. Math. Biol. 39(6), 493–517 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cao, J., Yuan, R.: Bifurcation analysis in a modified Lesile–Gower model with Holling type ii functional response and delay. Nonlinear Dyn. 84(3), 1341–1352 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Christie, P.: Marine protected areas as biological successes and social failures in Southeast Asia. In: American Fisheries Society Symposium, vol. 42, pp. 155–164. Citeseer (2004)Google Scholar
  9. 9.
    Cinner, J.E., Marnane, M.J., McClanahan, T.R.: Conservation and community benefits from traditional coral reef management at Ahus Island, Papua New Guinea. Conserv. Biol. 19(6), 1714–1723 (2005)CrossRefGoogle Scholar
  10. 10.
    Cohen, P.J., Foale, S.J.: Sustaining small-scale fisheries with periodically harvested marine reserves. Mar. Policy 37, 278–287 (2013)CrossRefGoogle Scholar
  11. 11.
    Côté, I.M., Green, S.J., Hixon, M.A.: Predatory fish invaders: insights from Indo-Pacific lionfish in the western Atlantic and Caribbean. Biol. Conserv. 164, 50–61 (2013)CrossRefGoogle Scholar
  12. 12.
    Dahl, K.A., Patterson III, W.F.: Habitat-specific density and diet of rapidly expanding invasive red lionfish, Pterois volitans, populations in the northern Gulf of Mexico. PLoS ONE 9(8), e105852 (2014)CrossRefGoogle Scholar
  13. 13.
    Drakunov, S.V., Utkin, V.I.: Sliding mode control in dynamic systems. Int. J. Control 55(4), 1029–1037 (1992)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Edwards, C.B., Friedlander, A., Green, A., Hardt, M., Sala, E., Sweatman, H., Williams, I., Zgliczynski, B., Sandin, S., Smith, J.: Global assessment of the status of coral reef herbivorous fishes: evidence for fishing effects. Proc. R. Soc. B Biol. Sci. 281(1774), 20131835 (2014)CrossRefGoogle Scholar
  15. 15.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides: Control Systems, vol. 18. Springer, Berlin (2013)Google Scholar
  16. 16.
    Fogarty, M.J.: Essential habitat, marine reserves and fishery management. Trends Ecol. Evol. 14(4), 133–134 (1999)CrossRefGoogle Scholar
  17. 17.
    Frazer, T.K., Jacoby, C.A., Edwards, M.A., Barry, S.C., Manfrino, C.M.: Coping with the lionfish invasion: Can targeted removals yield beneficial effects? Rev. Fish. Sci. 20(4), 185–191 (2012)CrossRefGoogle Scholar
  18. 18.
    Game, E.T., Bode, M., McDonald-Madden, E., Grantham, H.S., Possingham, H.P.: Dynamic marine protected areas can improve the resilience of coral reef systems. Ecol. Lett. 12(12), 1336–1346 (2009)CrossRefGoogle Scholar
  19. 19.
    Hall, S.J.: Closed areas for fisheries management—the case consolidates. Trends Ecol. Evol. 13(8), 297–298 (1998)CrossRefGoogle Scholar
  20. 20.
    Jana, S., Guria, S., Das, U., Kar, T., Ghorai, A.: Effect of harvesting and infection on predator in a prey–predator system. Nonlinear Dyn. 81(1–2), 917–930 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kellner, J.B., Litvin, S.Y., Hastings, A., Micheli, F., Mumby, P.J.: Disentangling trophic interactions inside a caribbean marine reserve. Ecol. Appl. 20(7), 1979–1992 (2010)CrossRefGoogle Scholar
  22. 22.
    Layman, C.A., Allgeier, J.E.: Characterizing trophic ecology of generalist consumers: a case study of the invasive lionfish in The Bahamas. Mar. Ecol. Prog. Ser. 448, 131–141 (2012)CrossRefGoogle Scholar
  23. 23.
    Leine, R., Van Campen, D., Van de Vrande, B.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23(2), 105–164 (2000)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lv, Y., Yuan, R., Pei, Y.: Dynamics in two nonsmooth predator–prey models with threshold harvesting. Nonlinear Dyn. 74(1–2), 107–132 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Micheli, F., Halpern, B.S.: Low functional redundancy in coastal marine assemblages. Ecol. Lett. 8(4), 391–400 (2005)CrossRefGoogle Scholar
  26. 26.
    Mumby, P.J.: Stratifying herbivore fisheries by habitat to avoid ecosystem overfishing of coral reefs. Fish Fish. 17(1), 266–278 (2016)CrossRefGoogle Scholar
  27. 27.
    Russ, G.R., Questel, S.L.A., Rizzari, J.R., Alcala, A.C.: The parrotfish–coral relationship: refuting the ubiquity of a prevailing paradigm. Mar. Biol. 162(10), 2029–2045 (2015)CrossRefGoogle Scholar
  28. 28.
    Utkin, V.I.: Sliding Modes in Control and Optimization. Springer, Berlin (2013)Google Scholar
  29. 29.
    Vallès, H., Oxenford, H.A.: Parrotfish size: A simple yet useful alternative indicator of fishing effects on caribbean reefs? PLoS ONE 9(1), e86291 (2014)CrossRefGoogle Scholar
  30. 30.
    Wang, Y., Jiang, W., Wang, H.: Stability and global Hopf bifurcation in toxic phytoplankton–zooplankton model with delay and selective harvesting. Nonlinear Dyn. 73(1–2), 881–896 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Xu, J., Tian, Y., Guo, H., Song, X.: Dynamical analysis of a pest management Leslie–Gower model with ratio-dependent functional response. Nonlinear Dyn. 93, 705–720 (2018)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department MathematicsKarimpur Pannadevi CollegeNadiaIndia
  2. 2.Department of Marine BiologyTexas A&M University at GalvestonGalvestonUSA
  3. 3.Department of MathematicsUniversity of KalyaniNadiaIndia
  4. 4.Department of Physical SciencesIndian Institute of Science Education and Research, KolkataNadiaIndia

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