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ISSN:2325-6192 (print)
ISSN:2325-6206 (online)
Journal of Environmental Accounting and Management
António Mendes Lopes (editor), Jiazhong Zhang(editor)
António Mendes Lopes (editor)

University of Porto, Portugal

Email: aml@fe.up.pt

Jiazhong Zhang (editor)

School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi Province 710049, China

Fax: +86 29 82668723 Email: jzzhang@mail.xjtu.edu.cn

An Eco-Epidemiological Model with Harvesting and Discrete Delay: A Mathematical Study

Journal of Environmental Accounting and Management 11(4) (2023) 441--465 | DOI:10.5890/JEAM.2023.12.006

Manish Sarkar$^1$, Ashok Mondal$^2$, Anindita Bhattacharyya$^1$, A. K. Pal$^3$

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Abstract

The present study deals with the study of the dynamics of an eco-epidemiological prey-predator mathematical system. The model that is proposed is an eco-epidemiological system of prey-predator association in consideration of discrete time delay where the prey population suffers from a disease which is modelled by the Susceptible-Infected (SI) epidemic scheme. We have assumed that the predator devours the vulnerable prey along with the diseased prey population in accordance with the modified Holling type II functional response. Harvesting of susceptible prey and predator have been considered and their effect has been analyzed also. To avoid the extinction of the prey population, refuge of prey is considered and its effect is observed on the stability of the system. Numerical simulation of the models reveals that the dynamical system undergoes Hopf-bifurcation due to the encounter rate, presence of refuge population and infection rate. Ecologically, the study discloses that harvesting, refugia and the rate of infection can be effectively used as a modulating parameter and may explain the reason behind the coexistence of prey-predator species and can remarkably affect the equilibrium of the dynamical system.

References

  1. [1]  Anderson, R.M. and May, R.M. (1981), The population dynamics of microparasites and their invertebrate hosts, Philosophical Transactions of The Royal Society B Biological Sciences, 291(1054), 451-524.
  2. [2]  Venturino, E. (1994), The influence of diseases on Lotka-Volterra systems, The Rocky Mountain Journal of Mathematics, 24, 381–402.
  3. [3]  Xiao, Y.N. and Chen, L.S. (2001), Analysis of a three species eco-epidemiological model, Journal of Mathematical Analysis and Applications, 258(2), 733–754.
  4. [4]  Hethcote, H., Wang, W., Han, L., and Ma, Z. (2004), A predator–prey model with infected prey, Theoretical Population Biology, 66(3), 259–268.
  5. [5]  Haque, M. and Venturino, E. (2009), Modelling Disease Spreading in Symbiotic Communities, Wildlife Destruction, Conservation and Biodiversity, Nova Science Publishers, New York.
  6. [6]  Zhou, X.Y., Shi, X.Y., and Song, X.Y. (2009), Analysis of a delay prey–predator model with disease in the prey species only, Journal of the Korean Mathematical Society, 46(4), 713–731.
  7. [7]  Tewa, J.J., Djeumen, V.Y., and Bowong, S. (2013), Predator–prey model with Holling response function of type II and SIS infectious disease, Applied Mathematical Modelling, 37, 4825–41.
  8. [8]  Singh, H., Srivastava, H.M., Hammouch, Z., and Nisar, K.S. (2021), Numerical simulation and stability analysis for the fractional-order dynamics of COVID-19, Results in Physics, 20, 103722. https://doi.org/10.1016/j.rinp.2020.103722.
  9. [9]  Farman, M., Akgül, A., Nisar, K.S, Ahmad, D., Ahmad, A., and Kamangar, S. (2022), Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel, AIMS Mathematics 2022, 7(1), 756-783. DOI:10.3934/math.2022046.
  10. [10]  Pal, A.K., Bhattacharyya, A., Mondal, A., and Pal, S. (2022), Qualitative analysis of an eco-epidemiological model with a role of prey and predator harvesting, Zeitschrift für Naturforschung A, 77(7)a, 629–645. https://doi.org/10.1515/zna-2021-0333.
  11. [11]  Naji, R.K. and Mustafa, A.N. (2012), The dynamics of an eco-epidemiological model with nonlinear incidence rate, Journal of Applied Mathematics, 2012, 1-24.
  12. [12]  Dutta, H. (2017), Insights into the impacts of four current environmental problems on flying birds, Energy, Ecology and Environment, 2(5),329–349
  13. [13]  Ang, T.K., Safuan, M.H., and Kavikumar, J. (2018), The impacts of harvesting activities on prey-predator fishery model in the presence of Toxin, Journal of Science and Technology: Special issue on the Application of Science and Mathematics, 10(2).
  14. [14]  Abdulghafour, A.S. and Naji R.K. (2018), A study of a diseased prey-predator model with refuge in prey and harvesting from predator, Journal of Applied Mathematics, 2018, 1-17.
  15. [15]  Xie, J., Liu, H., and Luo, D. (2021), The effects of harvesting on the dynamics of a Leslie–Gower model, Discrete Dynamics in Nature and Society, 2021, 5520758.
  16. [16]  Bezabih, A.F., Edessa, G.K., and Rao, K.P. (2021), Ecoepidemiological model and analysis of prey-predator system, Science Journal of Applied Mathematics and Statistics, 9(1), 1-14.
  17. [17]  Hassell, M.P. and May, R.M. (1973), Stability in insect host–parasite models, The Journal of Animal Ecology, 42, 693–726.
  18. [18]  Maynard, S.J. (1974), Models in ecology, Cambridge University,Cambridge, Agricultural Systems, Elsevier, 1(2), 164-165.
  19. [19]  Hassell, M.P. (1978), The Dynamics of Arthropod Predator–Prey Systems, Princeton University Press, Princeton, 13.
  20. [20]  Hoy, M.A., Helle, W., and Sabelis, M.W. (1985), Spider Mites: Their Biology, Natural Enemies and Control, World Crop Pests, Elesvier, Amsterdam, 1B, 229–310.
  21. [21]  McNair, J.N. (1986), The effects of refuges on predator–prey interactions:a reconsideration., Theoretical Population Biology, 29, 38–63.
  22. [22]  Gause, G.F., Smaragdova, N.P., and Witt, A.A. (1936), Further studies of interaction between predators and prey, The Journal of Animal Ecology, 5, 1–18.
  23. [23]  Krivan, V. (2011), On the Gause predator–prey model with a refuge: a fresh look at the history, Journal of Theoretical Biology, 274(1), 67–73.
  24. [24]  Sih, A. (1987), Prey refuges and predator–prey stability, Theoretical Population Biology, 31(1), 1–12.
  25. [25]  Collings, J.B. (1995), Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, Bulletin of Mathematical Biology, 57, 63–76.
  26. [26]  Hochberg, M.E. and Holt, R.D. (1995), Refuge evolution and the population dynamics of coupled host–parasitoid associations, Evolutionary Ecology, 9, 633–661.
  27. [27]  Olivares, E.G. and Jiliberto, R.R. (2003), Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological Modelling, 166, 135–146.
  28. [28]  Huang, Y., Chen, F., and Zhong, L. (2006), Stability analysis of a prey–predator model with Holling type III response function incorporating a prey refuge, Applied Mathematics and Computation, 182, 672–683.
  29. [29]  Mukherjee, D. (2014), The effect of prey refuges on a three species food chain model, Differential Equations and Dynamical Systems, 22(4), 413–426.
  30. [30]  Mondal, R., Kesh, D., and Mukherjee, D. (2018), Influence of induced plant volatile and refuge in tritrophic model, Energy, Ecology and Environment, 3(3), 171–184.
  31. [31]  Pal, A.K., Bhattacharyya, A., and Pal, S. (2022), Study of delay induced eco-epidemiological model incorporating a prey refuge, Filomat, 36(2), 557–578. https://doi.org/10.2298/FIL2202557P.
  32. [32]  Ruxton, G.D. (1995), Short term refuge use and stability of predator–prey models, Theoretical Population Biology, 47, 1–17.
  33. [33]  Berryman, A.A., Lima, M., and Hawkins, B.A. (2002), Population regulation, emergent properties, and a requiem for density dependence, Oikos, 99, 600–606.
  34. [34]  Macdonald, N. (1978), Time delays in Biological Models, Spring, Heidelberg.
  35. [35]  Freedman, H. and Rao, V.S.H. (1983), The trade-off between mutual interference and tome lags in predator-prey systems, Bulletin of Mathematical Biology, 45, 991-1004.
  36. [36]  Gopalsamy, K. (1992), Stability and Oscillation in Delay-Differential Equations of Population Dynamics, Springer Science \& Business Media.
  37. [37]  Kuang, Y. (1993), Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 191, 3-12.
  38. [38]  Mezouaghi, A., Djillali, S., Zeb, A., and Nisar, K.S. (2022), Global proprieties of a delayed epidemic model with partial susceptible protection, Mathematical Biosciences and Engineering, 19(1), 209-224. DOI:10.3934/mbe.2022011.
  39. [39]  Zhou, X. and Cui, J. (2010), Stability and Hopf bifurcation of a delay eco-epidemiological model with nonlinear incidence rate, Mathematical Modelling and Analysis, 15(4), 547–569
  40. [40]  Mao, S., Xu, R., Li, Z., and Li, Y. (2011), Global stability of an eco-epidemiological model with time delay and saturation incidence, Discrete Dynamics in Nature and Society, 2011, 730783.
  41. [41]  Ghosh, K., Sudip, S., Santanu, B., Sourav, R., ELmojtaba, I.M., Kesh, D.K., and Chattopadhyay, J. (2016), Stability and bifurcation analysis of an eco-epidemiological model with multiple delays, Non-linear Studies, 23(2), 167-208.
  42. [42]  Huang, C., Cao, J., Wen, F., and Yang, X. (2016), Stability analysis of sir model with distributed delay on complex networks, Plos One, Public Library of Science, 11(8),1-22.
  43. [43]  Biswas, S., Sasmal, S.K., Samanta, S., Saifuddin, Md., Pal, N., and Chattopadhyay, J. (2017), Optimal harvesting and complex dynamics in a delayed eco-epidemiological model with weak Allee effects, Nonlinear Dynamics, 87,1553–1573.
  44. [44]  Mondal, A., Pal, A.K., and Samanta, G.P. (2019), Analysis of a delayed eco-epidemiological pest–plant model with infected pest, Biophysical Reviews and Letters, 14(03), 141-170.
  45. [45]  Babakordi, N., Zangeneh, R.Z.H., and Ghahfarokhi, M.M. (2019), Multiple bifurcation analysis in a diffusive eco-epidemiological model with time delay, International Journal of Bifurcation and Chaos, 29(03), 1950033.
  46. [46]  Thakur, N.K., Srivastava, S.C., and Ojha, A. (2021), Dynamical study of an eco-epidemiological delay model for Plankton system with toxicity, Iranian Journal of Science and Technology, Transactions A: Science, 45, 283–304.
  47. [47]  Biswas, S., Tiwari, P.K., and Pal, S. (2021), Delay-induced chaos and its possible control in a seasonally forced eco-epidemiological model with fear effect and predator switching, Nonlinear Dynamics, 104, 2901–2930.
  48. [48]  Yan, P. and Liu, S. (2005), SEIR Epidemic Model with Delay, ANZIAMJ, 48, 119-134.
  49. [49]  Hale, J.K. (1969), Ordinary Differential Equations, Willey, New York.
  50. [50]  Hale, J.K. (1969), Theory of Functional Differential Equations, Springer, Heidelberg.
  51. [51]  Yang, X., Chen, L., and Chen, J. (1996), Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Computers \& Mathematics with Applications, 32(4), 109–116.
  52. [52]  Ruan, S. and Wei, J. (2003), On the zeros of transcendental functions with application to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Syatems, Series A: Mathematical Analysis, 10, 863-874.
  53. [53]  Erbe, L.H., Rao, V.S.H., and Freedman, H. (1986), Three species food chain models with mutual interference and time delays, Mathematical Biosciences, 80(1), 57-80.

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