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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Analysis of an Eco-Epidemic Predator-Prey Model with Nonlinear Prey Refuges and Predator Harvesting

Journal of Applied Nonlinear Dynamics 12(3) (2023) 465--483 | DOI:10.5890/JAND.2023.09.004

Md Sarijul Islam, Sahabuddin Sarwardi

Department of Mathematics and Statistics, Aliah University, IIA/27, New Town, Kolkata - 700 160, India

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Abstract

In this article, we have studied a mathematical model for one prey and two predator with disease in predator. Here the total population is divided into three classes, namely, prey, sound predator and infected predator. We consider two different nonlinear prey refuge coefficients against the sound predator and infected predator respectively. Also both the sound predator and infected predator are harvested in the proposed dynamical system. The possibility of existence of bionomic equilibrium has been considered. The optimal harvesting policy is studied by using Pontryagin's maximal principle. The optimal harvesting efforts corresponding to the optimal solution have been derived. The positive invariance and boundedness of the solutions of the system are shown. The existence of feasible equilibria and their stability analyses are performed. It is found that the zero equilibrium point is unstable, while other equilibrium points are asymptotically stable under certain conditions. We have investigated that harvesting parameters bear an important role to control the spread of infection. Moreover, the increment of predation rates of sound and infected predator change the stability of prey only equilibrium point to infected predator-free equilibrium point and sound predator-free equilibrium point respectively. Suitable graphical representation with proper discussions are performed in the Numerical simulation section to support the proposed dynamical system.

Acknowledgments

The researchers would like to thank the Department of Mathematics and Statistics, Aliah University for providing facilities to complete this research work.

References

  1. [1]  Lotka, A.J. (1956), Elements of Mathematical Biology, Dover: New York.
  2. [2]  Volterra, V. and D'Ancona U. (1931), La concorrenza vitale tra le specie dell'ambiente marino, Vlle Congr. Int. acquicult et de pêche, Paris, 1-14.
  3. [3]  Holling, C.S. (1959), The components of predation as revealed by a study of small-mammal predation of the European Pine Sawfly, The Canadian Entomologist, 91(5), 293-320.
  4. [4]  Holling, C.S. (1965), The functional response of predators to prey density and its role in mimicry and population regulation, The Memoirs of the Entomological Society of Canada, 97(S45), 5-60.
  5. [5]  Ghosh, U., Mondal, B., Rahaman, M.S., and Sarkar, S. (2021), Stability analysis of three species food chain model with linear functional response via imprecise and parametric approach, Journal of Computational Science, 54, 101423.
  6. [6]  Molla, H., Rahman, M.S., and Sarwardi, S. (2019), Dynamics of a predator-prey model with Holling type II functional response incorporating a prey refuge depending on both the species, International Journal of Nonlinear Sciences and Numerical Simulation, 20(1), 89-104.
  7. [7]  Hassell, M.P. (2020), The Dynamics of Arthopod Predator-Prey Systems, (MPB-13), Volume 13, Princeton University Press.
  8. [8]  McNair, J.N. (1986), The effects of refuges on predator-prey interactions: a reconsideration, Theoretical Population Biology, 29(1), 38-63.
  9. [9]  Mukherjee, D. (2016), The effect of refuge and immigration in a predator-prey system in the presence of a competitor for the prey, Nonlinear Analysis: Real World Applications, 31, 277-287.
  10. [10]  Haque, M., Rahman, M.S., Venturino, E., and Li, B.L. (2014), Effect of a functional response-dependent prey refuge in a predator-prey model, Ecological Complexity, 20, 248-256.
  11. [11]  Sk, N., Tiwari, P.K., Kang, Y., and Pal, S. (2021), A nonautonomous model for the interactive effects of fear, refuge and additional food in a prey-predator system, Journal of Biological Systems, 29(01), 107--145.
  12. [12]  Barman, D., Roy, J., and Alam, S. (2021), Dynamical behaviour of an infected predator-prey model with fear effect, Iranian Journal of Science and Technology, Transactions A: Science, 45(1), 309-325.
  13. [13]  Huang, Y., Zhu, Z., and Li, Z. (2020), Modeling the Allee effect and fear effect in predator-prey system incorporating a prey refuge, Advances in Difference Equations, 2020(1), 1-13.
  14. [14]  Barman, D., Roy, J., Alrabaiah, H., Panja, P., Mondal, S.P., and Alam, S. (2021), Impact of predator incited fear and prey refuge in a fractional order prey predator model, Chaos, Solitons & Fractals, 142, 110420.
  15. [15]  Roy, J., Barman, D., and Alam, S. (2020), Role of fear in a predator-prey system with ratio-dependent functional response in deterministic and stochastic environment, Biosystems, 197, 104176.
  16. [16]  Molla, H., Rahman, M.S., and Sarwardi, S. (2020), Incorporating prey refuge in a prey-predator model with Beddington-Deangelis type functional response: a comparative study on intra-specific competition, Discontinuity, Nonlinearity, and Complexity, 9(3), 395-419.
  17. [17]  Manarul Haque, M. and Sarwardi, S. (2018), Dynamics of a harvested prey-predator model with prey refuge dependent on both species, International Journal of Bifurcation and Chaos, 28(12), 1830040.
  18. [18]  González-Olivares, E. and Ramos-Jiliberto, R. (2012), Comments to ``The effect of prey refuge in a simple predator-prey model"[Ecol. Model. 222 (September (18))(2011) 3453-3454], Ecological Modelling, 232, 158-160.
  19. [19]  Thota, S. (2020), A mathematical study on a diseased prey-predator model with predator harvesting, Asian Journal of Fuzzy and Applied Mathematics, (ISSN: 2321-564X), 8(2).
  20. [20]  Pal, D. and Mahapatra, G.S. (2014), A bioeconomic modeling of two-prey and one-predator fishery model with optimal harvesting policy through hybridization approach, Applied Mathematics and Computation, 242, 748-763.
  21. [21]  Das, H., Shaikh, A.A., and Sarwardi, S. (2020), Mathematical analysis of an eco-epidemic model with different functional responses of healthy and infected predators on prey species, Journal of Applied Nonlinear Dynamics, 9(4), 667-684.
  22. [22]  Bhattacharyya, R. and Mukhopadhyay, B. (2010), On an eco-epidemiological model with prey harvesting and predator switching: local and global perspectives, Nonlinear Analysis: Real World Applications, 11(5), 3824-3833.
  23. [23]  Pal, D., Mahaptra, G.S., and Samanta, G.P. (2013), Optimal harvesting of prey-predator system with interval biological parameters: a bioeconomic model, Mathematical biosciences, 241(2), 181-187.
  24. [24]  Liu, X. and Huang, Q. (2020), Analysis of optimal harvesting of a predator-prey model with Holling type IV functional response, Ecological Complexity, 42, 100816.
  25. [25]  Jana, S., Guria, S., Das, U., Kar, T.K., and Ghorai, A. (2015), Effect of harvesting and infection on predator in a prey-predator system, Nonlinear Dynamics, 81(1), 917-930.
  26. [26]  Kar, T.K., Chattopadhyay, S.K., and Pati, C.K. (2009), A bio-economic model of two-prey one-predator system, Journal of applied mathematics & informatics, 27(5/6), 1411-1427.
  27. [27]  Kar, T.K., Misra, S., and Mukhopadhyay, B. (2006), A bioeconomic model of a ratio-dependent predator-prey system and optimal harvesting, Journal of Applied Mathematics and Computing, 22(1), 387-401.
  28. [28]  Anderson, R.M. and May, R.M. (1981), The population dynamics of microparasites and their invertebrate hosts, Philosophical Transactions of the Royal Society of London. B, Biological Sciences, 291(1054), 451-524.
  29. [29]  Haque, M. and Venturino, E. (2007), An ecoepidemiological model with disease in predator: the ratio dependent case, Mathematical methods in the Applied Sciences, 30(14), 1791-1809.
  30. [30]  Sarwardi, S., Haque, M., and Venturino, E. (2011), A Leslie-Gower Holling-type II ecoepidemic model, Journal of Applied Mathematics and Computing, 35(1), 263-280.
  31. [31]  Rahman, M.S. and Chakravarty, S. (2013), A predator-prey model with disease in prey, Nonlinear Analysis: Modelling and Control, 18(2), 191-209.
  32. [32]  Sieber, M., Malchow, H., and Hilker, F.M. (2014), Disease-induced modification of prey competition in eco-epidemiological models, Ecological Complexity, 18, 74-82.
  33. [33]  Melese, D., Muhye, O., and Sahu, S.K. (2020), Dynamical Behavior of an Eco-epidemiological Model Incorporating Prey Refuge and Prey Harvesting, Applications & Applied Mathematics, 15(2).
  34. [34]  Sarwardi, S., Haque, M., and Venturino, E. (2011), Global stability and persistence in LG-Holling type II diseased predator ecosystems, Journal of Biological Physics, 37(1), 91-106.
  35. [35]  Peng, M., Zhang, Z., Lim, C.W., and Wang, X. (2018), Hopf bifurcation and hybrid control of a delayed ecoepidemiological model with nonlinear incidence rate and Holling type II functional response, Mathematical Problems in Engineering, 2018.
  36. [36]  Bezabih, A.F., Edessa, G.K., and Koya, P.R. (2020), Mathematical eco-epidemiological model on prey-predator system, Mathematical Modeling and Applications, 5(3), 183-190.
  37. [37]  Haque, M., Rahman, M.S., and Venturino, E. (2013), Comparing functional responses in predator-infected eco-epidemics models, BioSystems, 114(2), 98-117.
  38. [38]  Huang, C., Zhang, H., Cao, J., and Hu, H. (2019), Stability and Hopf bifurcation of a delayed prey-predator model with disease in the predator, International Journal of Bifurcation and Chaos, 29(07), 1950091.
  39. [39]  Amalia, R.D. and Arif, D.K. (2018), Optimal control of predator-prey mathematical model with infection and harvesting on prey, In Journal of Physics: Conference Series, 974(1), p. 012050, IOP Publishing.
  40. [40]  Sowndarrajan, P.T., Nyamoradi, N., Shangerganesh, L., and Manimaran, J. (2020), Mathematical analysis of an optimal control problem for the predator-prey model with disease in prey, Optimal Control Applications and Methods, 41(5), 1495-1509.
  41. [41]  Shaikh, A.A., Das, H., and Ali, N. (2018), Study of LG-Holling type III predator-prey model with disease in predator, Journal of Applied Mathematics and Computing, 58(1), 235-255.
  42. [42]  Pusawidjayanti, K., Suryanto, A., and Wibowo, R.B.E. (2015), Dynamics of a predator-prey model incorporating prey refuge, predator infection and harvesting, Applied Mathematical Sciences, 9(76), 3751-3760.
  43. [43]  Birkhoff, G. and Rota, G. (1978), Ordinary differential equations, John Wiley and Sons, New York: 342 QA372 B, 58.
  44. [44]  Hale, J. (1989), Ordinary Differential Equation, Klieger Publishing Company, Malabar.
  45. [45]  Gard, T.C. and Hallam T.G. (1979), Persistence in Food web-1, Lotka-Volterra food chains, Bulletin of Mathematical Biology, 41(6), 877-891
  46. [46]  Anderson, R.M. and May, R.M. (1981), The population dynamics of microparasites and their invertebrate hosts, Philosophical Transactions of the Royal Society of London. B, Biological Sciences, 291(1054), 451-524.
  47. [47]  Diekmann, O., Heesterbeek, J.A.P., and Metz, J.A. (1990), On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28(4), 365-382.
  48. [48]  Sotomayor, J. (1973), Generic bifurcations of dynamical systems: Dynamical systems, Academic Press, 549-560.
  49. [49]  Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F. (1962), The Mathematical Theory of Optimal Process, Wiley, New York.

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