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Dynamical analysis of a prey–predator model with Beddington–DeAngelis type function response incorporating a prey refuge

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Abstract

This paper discusses a prey–predator model with reserved area. The feeding rate of consumers (predators) per consumer (i.e., functional response) is considered to be Beddington–DeAngelis type. The Beddington–DeAngelis functional response is similar to the Holling type II functional response but contains an extra term describing mutual interference by predators. We investigate the role of reserved region and degree of mutual interference among predators in the dynamics of system. We obtain different conditions that affect the persistence of the system. We also discuss local and global asymptotic stability behavior of various equilibrium solutions to understand the dynamics of the model system. The global asymptotic stability of positive interior equilibrium solution is established using suitable Lyapunov functional. It is found that the Hopf bifurcation occurs when the parameter corresponding to reserved region (i.e., m) crosses some critical value. Our result indicates that the predator species exist so long as prey reserve value (m) does not cross a threshold value and after this value the predator species extinct. To mimic the real-world scenario, we also solve the inverse problem of estimation of model parameter (m) using the sampled data of the system. The results can also be interpreted in different contexts such as resource conservation, pest management and bio-economics of a renewable resource. At the end, we perform some numerical simulations to illustrate our analytical findings.

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References

  1. Lotka, A.: Elements of Mathematical Biology. Dover, New York (1956)

    MATH  Google Scholar 

  2. Ahmad, S., Rao, M.R.M.: Theory of Ordinary Differential Equations with Applications in Biology and Engineering. Affiliated East-West Press Private Limited, New Delhi (1999)

    Google Scholar 

  3. Beretta, E., Kuang, Y.: Global analysis in some delayed ratio-dependent predator–prey systems. Nonlinear Anal. Theory Methods Appl. 32(3), 381–408 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hsu, S.B., Hwang, T.W., Kuang, Y.: Global analysis of the Michaelis–Menten-type ratio-dependent predator–prey system. J. Math. Biol. 42, 489–506 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Tripathi, J.P., Abbas, S., Thakur, M.: Stability analysis of two prey one predator model. In: AIP Conference Proceedings, vol. 1479, pp. 905–909 (2012)

  6. Smith, J.: Models in Ecology. Cambridge University Press, Cambridge (1974)

    MATH  Google Scholar 

  7. Kar, T.K.: Stability analysis of a predator–prey model incorporating a prey refuge. Commun. Nonlinear Sci. Numer. Simul. 10, 681–691 (2006)

    Article  MathSciNet  Google Scholar 

  8. Liu, S., Beretta, E.: A stage-structured predator–prey model of Beddington–DeAngelis type. SIAM J. Appl. Math. 66(4), 1101–1129 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Li, H., Takeuchi, Y.: Dynamics of the density dependent predator–prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 374, 644–654 (2011)

  10. Wolkowicz, G.S.K.: Bifurcation analysis of a predator–prey system involving group defence. SIAM J. Appl. Math. 48, 592–606 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  11. Huang, Y., Chen, F., Zhong, L.: Stability analysis of a predator–prey model with Holling type III response function incorporating a prey refuge. Appl. Math. Comput. 182, 672–683 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ruan, S., Xiao, D.: Global analysis in a predator–prey system with non-monotonic functional response. SIAM J. Appl. Math. 61(4), 1445–1472 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Oaten, A., Murdoch, W.: Functional response and stability in predator–prey systems. Am. Nat. 109(967), 289–298 (1975)

    Article  Google Scholar 

  14. Berryman, A.A.: The origin and evolution of predator–prey theory. Ecol. Soc. Am. 73(5), 1530–1535 (1992)

    Google Scholar 

  15. DeAngelis, D.L., Goldstein, R.A., O’Neill, R.V.: A model for trophic interaction. Ecology 56(4), 881–892 (1975)

    Article  Google Scholar 

  16. Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385–398 (1959)

    Article  Google Scholar 

  17. Aziz-Alaoui, M.A., 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, 1069–1075 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Beddington, J.R.: Mutual interference between parasites or predators and it’s effect on searching efficiency. J. Anim. Ecol. 44(1), 331–340 (1975)

    Article  Google Scholar 

  19. Cosner, C., DeAngelis, D.L., Ault, J.S., Olson, D.B.: Effects of spatial grouping on the functional response of predators. Theor. Popul. Biol. 56, 65–75 (1999)

    Article  MATH  Google Scholar 

  20. Hassell, M.P.: Mutual interference between searching insect parasites. J. Anim. Ecol. 40, 473–486 (1971)

    Article  Google Scholar 

  21. Sklaski, G.T., Gilliam, J.F.: Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology 82(11), 3083–3092 (2001)

    Article  Google Scholar 

  22. Cosner, C., DeAngelis, D.L., Ault, J.S., Olson, D.B.: Effects of spatial grouping on the functional response of predators. Theor. Popul. Biol. 56, 65–75 (1999)

    Article  MATH  Google Scholar 

  23. Cantrell, R.S., Cosner, C.: On the dynamics of predator–prey models with the Beddington–DeAngelis functional response. J. Math. Anal. Appl. 257, 206–222 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sarwardi, S., Haque, M., Mandal, P.K.: Persistence and global stability of Bazykin predator–prey model with Beddington–DeAngelis response function. Commun. Nonlinear Sci. Numer. Simul. 19(1), 189–209 (2014)

    Article  MathSciNet  Google Scholar 

  25. Abbas, S., Banerjee, M., Hungerbuhler, N.: Existence, uniqueness and stability analysis of allelopathic stimulatory phytoplankton model. J. Math. Anal. Appl. 367, 249–259 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. Abbas, S., Sen, M., Banerjee, M.: Almost periodic solution of a non-autonomous model of phytoplankton allelopathy. Nonlinear Dyn. 67, 203–214 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Abbas, S., Mahto, L.: Existence of almost periodic solution of a model of phytoplankton allelopathy with delay. AIP Conference Proceedings, vol. 1479, pp. 900–904 (2012)

  28. Hutson, V., Schmitt, K.: Permanence and the dynamics of biological systems. Math. Biosci. 111, 1–71 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lv, Y., Yuan, R., Pei, Y.: A prey–predator model with harvesting for fishery resource with reserve area. Appl. Math. Model. 37, 3048–3062 (2013)

    Article  MathSciNet  Google Scholar 

  30. Hale, J.K.: Ordinary Differential Equations. Wiley, New York (1969)

    MATH  Google Scholar 

  31. Brauer, F., Chavez, C.C.: Mathematical Models in Population Biology and Epidemiology. Springer, New York (2001)

    MATH  Google Scholar 

  32. Butler, G., Freedman, H.I., Waltman, P.: Uniformly persistent systems. Proc. Am. Math. Soc. 96(3), 425–430 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  33. Gard, T.C., Halm, T.G.: Persistence in food webs—I Lotka–Volterra food chains. Bull. Math. Biol. 41, 877–891 (1979)

    MATH  MathSciNet  Google Scholar 

  34. Dubey, B., Chandra, P., Sinha, P.: A model for fishery resource with reserve area. Nonlinear Anal. Real World Appl. 4, 625–637 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Mukherjee, D.: Persistence in a generalized prey–predator model with prey reserve. Int. J. Nonlinear Sci. 14, 160–165 (2012)

    MATH  MathSciNet  Google Scholar 

  36. Chattopadhyay, J., Bairagi, N., Sarkar, R.R.: A prey–predator model with some cower on prey species. Nonlinear Phenom. Compl. Syst. 3(4), 407–420 (2004)

    MathSciNet  Google Scholar 

  37. Dubey, B.: A prey–predator model with a reserved area. Nonlinear Anal. Model. Control 12(4), 479–494 (2007)

    MATH  MathSciNet  Google Scholar 

  38. Hoy, M.A.: Almonds (California). In: Helle, W., Sabelis, M.W. (eds.) Spider Mites: Their Biology, Natural Enemies and Control, World Crop Pests, vol. 1B, pp. 229–310. Elsevier, Amsterdam (1985)

    Google Scholar 

  39. Kar, T.K., Misra, S.: Influence of prey reserve in a predator–prey fishery. Nonlinear Anal. 65, 1725–1735 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  40. Du, Y., Shi, J.: A diffusive predator–prey model with a protection zone. J. Differ. Equ. 63–91 (2006)

  41. Ashyraliyev, M., Nanfack, Y.F., Kaandrop, J.A., Blom, J.G.: Systems biology: parameter estimation for biochemical models. Febs J. 276, 886–902 (2009)

    Article  Google Scholar 

  42. Lillacci, G., Khammash, M.: Parameter estimation and model selection in computational biology. PLoS Comput. Biol. 6, e1000696 (2010)

    Article  MathSciNet  Google Scholar 

  43. Lawson, L.M., Spitz, Y.H., Hofmann, E.E., Long, R.B.: A data assimilation technique applied to a predator–prey model. Bull. Math. Biol. 57, 593–617 (1995)

    Article  MATH  Google Scholar 

  44. Walmag, J.M.B., Delhez, E.J.M.: A trust-region method applied to parameter identification of a simple prey–predator model. Appl. Math. Model. 29, 289–307 (2005)

  45. Thakur, M., Deep, K.: Data Assimilation of a Biological Model Using Genetic Algorithms Applications and Innovations in Intelligent Systems, vol. XIV, pp. 238–242. Springer, London (2007)

  46. Perko, L.: Differential Equations and Dynamical systems. Springer, New York (2001)

    MATH  Google Scholar 

  47. Hwang, T.W.: Uniqueness of limit cycle for Gauss type predator-prey systems. J. Math. Anal. Appl. 2380, 179–195 (1999)

    Article  Google Scholar 

  48. Hwang, T.W.: Uniqueness of limit cycles of the predator–prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 290, 113–122 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  49. Freedman, H.I., Waltman, P.: Persistence in models of three interacting predator–prey populations. Math. Biosci. 68, 213–231 (1984)

  50. Bengtsson, L., Ghil, M., Källén, E.: Dynamic Meteorology: Data Assimilation Methods. Springer, New York (2001)

  51. Fashman, M.J.R., Evans G.T.: The use of optimization technique to model marine ecosystems dynamics at JGOFS station at 47\(^{\circ }\)N 20\(^{\circ }\)W. Philos. Trans. R. Soc. Lond. 203–209 (1995)

  52. Huang, J., Gao, J., Liu, J., Zhang, Y.: State and parameter update of a hydrodynamic–phytoplankton model using ensemble Kalman filter. Ecol. Model. 263, 81–91 (2013)

    Article  Google Scholar 

  53. Kloppers, P.H., Greeff, J.C.: Lotka–Volterra model parameter estimation using experiential data. Appl. Math. Comput. 224, 817–825 (2013)

    Article  MathSciNet  Google Scholar 

  54. Kusum, D., Thakur, M.: A new mutation operator for real coded genetic algorithms. Appl. Math. Comput. 193, 211–230 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  55. Thakur, M., Meghwani, S.S., Jalota, H.: A modified real coded genetic algorithm for constrained optimization. Appl. Math. Comput. 235, 292–317 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We would like to express our gratitude to the reviewers for their comments and suggestions, which helped us to improve our manuscript considerably. We also would like to thank Manoj Dhiman and Suraj Meghwani for their helpful suggestions that improved the presentation of numerical section of the paper. The research work of first author (J. P. Tripathi ) is supported by the Council of Scientific and Industrial Research (CSIR) (No. 09/1058(0001)/2011-EMR-1), India.

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  1. School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, 1750-01, Himachal Pradesh, India

    Jai Prakash Tripathi, Syed Abbas & Manoj Thakur

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  1. Jai Prakash Tripathi
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Correspondence to Syed Abbas.

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Tripathi, J.P., Abbas, S. & Thakur, M. Dynamical analysis of a prey–predator model with Beddington–DeAngelis type function response incorporating a prey refuge. Nonlinear Dyn 80, 177–196 (2015). https://doi.org/10.1007/s11071-014-1859-2

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