Skip to main content

Advertisement

SpringerLink
Log in

Dynamics of an infected prey–generalist predator system with the effects of fear, refuge and harvesting: deterministic and stochastic approach

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

A predator–prey model with infection in the prey population, including fear, refuge and harvesting factors, is proposed. An expression for basic reproduction number is obtained. Depending on reproduction number, global stability of disease-free and endemic equilibria is established. We have found that disease can be eradicated from the system by controlling the value of basic reproduction number less than one. In addition, deterministic model is analyzed for transcritical and Hopf bifurcations. Thresholds of parameters are identified which determine when the equilibrium is disease-free and when it becomes endemic. An extension is made in the deterministic model by including environmental noise. Sufficient conditions for extinction of species are derived based on environmental noise. We observed that the stochastic system fluctuates around the solutions of corresponding deterministic system when the intensity of white noise is low, whereas high intensity drive the populations to extinction. For justification of analytical results, we have performed numerical simulations and presented the results, which manifest the reliability of the model from ecological viewpoint.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Data availability statements

All data generated or analyzed during this study are included in this article.

References

  1. W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics. II. The problem of endemicity. Proc. R. Soc. Lond. Ser. A 138(834), 55–83 (1932)

    Article  ADS  MATH  Google Scholar 

  2. H.I. Freedman, A model of predator-prey dynamics as modified by the action of a parasite. Math. Biosci. 99(2), 143–155 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Y. Xiao, L. Chen, Modeling and analysis of a predator–prey model with disease in the prey. Math. Biosci. 171(1), 59–82 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Haque, E. Venturino, The role of transmissible diseases in the Holling–Tanner predator–prey model. Theor. Popul. Biol. 70(3), 273–288 (2006)

    Article  MATH  Google Scholar 

  5. K.P. Hadeler, H.I. Freedman, Predator–prey populations with parasitic infection. J. Math. Biol. 27(6), 609–631 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Mondal, A. Lahiri, N. Bairagi, Analysis of a fractional order eco-epidemiological model with prey infection and type 2 functional response. Math. Methods Appl. Sci. 40(18), 6776–6789 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. S. Biswas, P.K. Tiwari, S. Pal, Delay-induced chaos and its possible control in a seasonally forced eco-epidemiological model with fear effect and predator switching. Nonlinear Dyn. 104(3), 2901–2930 (2021)

    Article  Google Scholar 

  8. C. Gokila, M. Sambath, K. Balachandran, Y.K. Ma, Analysis of stochastic predator–prey model with disease in the prey and Holling Type II functional response. Adv. Math. Phys. (2020)

  9. W. Cresswell, Predation in bird populations. J. Ornithol. 152(1), 251–263 (2011)

    Article  Google Scholar 

  10. X. Wang, L. Zanette, X. Zou, Modelling the fear effect in predator–prey interactions. J. Math. Biol. 73(5), 1179–1204 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. G.C. Trussell, P.J. Ewanchuk, C.M. Matassa, The fear of being eaten reduces energy transfer in a simple food chain. Ecology 87(12), 2979–2984 (2006)

    Article  Google Scholar 

  12. S.J. McCauley, L. Rowe, M.J. Fortin, The deadly effects of nonlethal predators. Ecology 92(11), 2043–2048 (2011)

    Article  Google Scholar 

  13. L. Vesel, D.S. Boukal, M. Buric, P. Kozk, A. Kouba, A. Sentis, Effects of prey density, temperature and predator diversity on nonconsumptive predator-driven mortality in a freshwater food web. Sci. Rep. 7(1), 1–9 (2017)

    Google Scholar 

  14. G.F. Gause, The Struggle for Existence: A Classic of Mathematical Biology and Ecology (Courier Dover Publications, New York, 2019)

    Google Scholar 

  15. R.A. Dolbeer, W.R. Clark, Population ecology of snowshoe hares in the central Rocky Mountains. J. Wildl. Manag. 39, 535–549 (1975)

    Article  Google Scholar 

  16. F. Chen, L. Chen, X. Xie, On a Leslie-Gower predator–prey model incorporating a prey refuge. Nonlinear Anal. Real World Appl. 10(5), 2905–2908 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. N. Sk, P.K. Tiwari, S. Pal, M. Martcheva, A delay non-autonomous model for the combined effects of fear, prey refuge and additional food for predator. J. Biol. Dyn. 15(1), 580–622 (2021)

    Article  MathSciNet  Google Scholar 

  18. S. Samanta, R. Dhar, I.M. Elmojtaba, J. Chattopadhyay, The role of additional food in a predator–prey model with a prey refuge. J. Biol. Syst. 24(02n03), 345–365 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. N. Sk, P.K. Tiwari, Y. Kang, S. Pal, A nonautonomous model for the interactive effects of fear, refuge and additional food in a prey–predator system. J. Biol. Syst. 29(01), 107–145 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Hanski, L. Hansson, H. Henttonen, Specialist predators, generalist predators, and the microtine rodent cycle. J. Anim. Ecol. 60, 353–367 (1991)

    Article  Google Scholar 

  21. C. Magal, C. Cosner, S. Ruan, J. Casas, Control of invasive hosts by generalist parasitoids. Math. Med. Biol. J. IMA 25(1), 1–20 (2008)

    Article  MATH  Google Scholar 

  22. A. Erbach, F. Lutscher, G. Seo, Bistability and limit cycles in generalist predator-prey dynamics. Ecol. Complex. 14, 48–55 (2013)

    Article  Google Scholar 

  23. D. Sen, S. Petrovskii, S. Ghorai, M. Banerjee, Rich bifurcation structure of prey–predator model induced by the Allee effect in the growth of generalist predator. Int. J. Bifurc. Chaos 30(06), 2050084 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. V. Weide Rodrigues, D. Cristina Mistro, L.A. Diaz Rodrigues, Pattern formation and bistability in a generalist predator–prey model. Mathematics 8(1), 20 (2020)

    Article  Google Scholar 

  25. D. Hu, H. Cao, Stability and bifurcation analysis in a predator–prey system with Michaelis–Menten type predator harvesting. Nonlinear Anal. Real World Appl. 33, 58–82 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. C.W. Clark, Bioeconomic Modelling and Fisheries Management (Wiley, New York, 1985)

    Google Scholar 

  27. X.A. Zhang, L. Chen, A.U. Neumann, The stage-structured predator–prey model and optimal harvesting policy. Math. Biosci. 168(2), 201–210 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Y. Zhang, S. Chen, S. Gao, X. Wei, Stochastic periodic solution for a perturbed non-autonomous predator–prey model with generalized nonlinear harvesting and impulses. Physica A 486, 347–366 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  29. B. Sahoo, B. Das, S. Samanta, Dynamics of harvested-predator–prey model: role of alternative resources. Model. Earth Syst. Environ. 2(3), 1–12 (2016)

    Article  Google Scholar 

  30. T.C. Gard, Persistence in stochastic food web models. Bull. Math. Biol. 46(3), 357–370 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  31. R.M. May, Stability and Complexity in Model Ecosystems (Princeton University Press, Princeton, 2019)

    Book  Google Scholar 

  32. S. Zhang, X. Meng, T. Feng, T. Zhang, Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects. Nonlinear Anal. Hybrid Syst. 26, 19–37 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Das, G.P. Samanta, Stochastic prey–predator model with additional food for predator. Physica A 512, 121–141 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  34. J. Roy, S. Alam, Fear factor in a prey–predator system in deterministic and stochastic environment. Physica A 541, 123359 (2020)

    Article  MathSciNet  Google Scholar 

  35. B. Mondal, U. Ghosh, M.S. Rahman, P. Saha, S. Sarkar, Studies of different types of bifurcations analyses of an imprecise two species food chain model with fear effect and non-linear harvesting. Math. Comput. Simul. 192, 111–135 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  36. P.S. Mandal, M. Banerjee, Stochastic persistence and stationary distribution in a Holling–Tanner type prey–predator model. Physica A 391(4), 1216–1233 (2012)

    Article  ADS  Google Scholar 

  37. O. Ovaskainen, B. Meerson, Stochastic models of population extinction. Trends Ecol. Evol. 25(11), 643–652 (2010)

    Article  Google Scholar 

  38. N.V. Devi, D. Jana, Impact of stochastic perturbation on the persistence and extinction risk of a multi-delayed prey–predator system in non-autonomous environment. Model. Earth Syst. Environ. 7(2), 1241–1267 (2021)

    Article  Google Scholar 

  39. Y. Xia, S. Yuan, Survival analysis of a stochastic predator-prey model with prey refuge and fear effect. J. Biol. Dyn. 14(1), 871–892 (2020)

    Article  MathSciNet  Google Scholar 

  40. K.D. Lafferty, A.K. Morris, Altered behavior of parasitized killifish increases susceptibility to predation by bird final hosts. Ecology 77(5), 1390–1397 (1996)

    Article  Google Scholar 

  41. D. Greenhalgh, Q.J. Khan, F.A. Al-Kharousi, Eco-epidemiological model with fatal disease in the prey. Nonlinear Anal. Real World Appl. 53, 103072 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  42. W.D. Hamilton, R. Axelrod, R. Tanese, Sexual reproduction as an adaptation to resist parasite. Proc. Natl. Acad. Sci. USA 87, 3566–3573 (1990)

    Article  ADS  Google Scholar 

  43. D. Mukherjee, Role of fear in predator-prey system with intraspecific competition. Math. Comput. Simul. 177, 263–275 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  44. J. Wang, Y. Cai, S. Fu, W. Wang, The effect of the fear factor on the dynamics of a predator–prey model incorporating the prey refuge. Chaos 29(8), 083109 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. J. Huang, Y. Gong, S. Ruan, Bifurcation analysis in a predator–prey model with constant-yield predator harvesting. Discrete Contin. Dyn. Syst. B 18(8), 2101 (2013)

    MathSciNet  MATH  Google Scholar 

  46. V. Lakshmikantham, S. Leela, A.A. Martynyuk, Stability Analysis of Nonlinear Systems (Springer, Switzerland, 1989)

    MATH  Google Scholar 

  47. P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  48. Z. Shuai, P. Van den Driessche, Global stability of infectious disease models using Lyapunov functions. SIAM J. Appl. Math. 73(4), 1513–1532 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. L. Perko, Differential Equations and Dynamical Systems, 3rd edn. (Springer, New York, 2001)

    Book  MATH  Google Scholar 

  50. E. Allen, Modeling with Itô Stochastic Differential Equations, vol. 22 (Springer, Berlin, 2007)

    MATH  Google Scholar 

  51. A.K. Misra, P.K. Tiwari, E. Venturino, Modeling the impact of awareness on the mitigation of algal bloom in a lake. J. Biol. Phys. 42(1), 147–165 (2016)

    Article  Google Scholar 

Download references

Acknowledgements

The authors express their gratitude to the reviewers whose comments and suggestions have helped the improvements of this paper.

Funding

The research of Samares Pal is partially supported by Science and Engineering Research Board, Government of India (Grant No. CRG/2019/003248).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kalyani, Kalyani, 741235, India

    Nazmul Sk & Samares Pal

Authors
  1. Nazmul Sk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samares Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this article.

Ethical standard

The authors state that this research complies with ethical standards. This research does not involve either human participants or animals.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sk, N., Pal, S. Dynamics of an infected prey–generalist predator system with the effects of fear, refuge and harvesting: deterministic and stochastic approach. Eur. Phys. J. Plus 137, 138 (2022). https://doi.org/10.1140/epjp/s13360-022-02348-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-022-02348-9

Search

Navigation