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Complex dynamics of sexually reproductive generalist predator and gestation delay in a food chain model: double Hopf-bifurcation to Chaos

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Abstract

Food uptake ability of higher trophic level species are more complicated and interesting due to their choice and availability of food and consequently their growth and also the effect of top predator interference on the dynamics of a tritrophic food chain model. In this paper, we consider the general framework for calculating the stability of equilibria, Hopf and double Hopf-bifurcation of a prey–predator system with Holling type IV and Beddington–DeAngelis type functional responses of intermediate and top predator respectively. The top predator is of generalist type and its growth is considered due to sexual reproduction. Firstly, we have shown feasibility and boundedness of the solutions of the considered model system, behavior of equilibria and the existence of Hopf-bifurcation. Conditions are determined under which the coexistence equilibrium point remains globally asymptotically stable. We identify the critical values of delay parameter for which stability switches and nature of the Hopf-bifurcation by using normal form theory and center manifold theorem. We investigate the occurence of double Hopf bifurcation at positive equilibrium point when we choose appropriate measure of the predator’s immunity or tolerance of the prey. Furthermore, some dynamic behaviors, such as stability switches, chaos, bifurcation and double Hopf-bifurcation scenarios are observed using numerical simulations. The chaotic behavior of the system is clarified by standard numerical tests.

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References

  1. Aziz-Alaoui, M.A.: Study of a Leslie–Gower-type tritrophic population model. Chaos Solitons Fractals 14, 12751293 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beddington, J.R.: Mutual interference between parasites or predators and its effects on searching efficiency. J. Anim. Ecol. 44, 331–340 (1975)

    Article  Google Scholar 

  3. Chen, Y.: Multiple periodic solutions of delayed predatorprey systems with type IV functional responses. Nonlinear Anal. 5, 45–53 (2004)

    Article  MathSciNet  Google Scholar 

  4. 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 

  5. DeAngelis, D.L., Goldstein, R.A., ONeil, R.V.: A model for trophic interaction. Ecology 56, 881–892 (1975)

    Article  Google Scholar 

  6. Ding, Y., Jiang, W.: Double Hopf bifurcation and chaos in liu system with delayed feedback. J. Appl. Anal. Comput. 1(3), 325–349 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Ding, Y., Jiang, W., Yu, P.: Double Hopf bifurcation in acontainer crane model with delayed position feedback. Appl. Math. Comput. 219, 9270–9281 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Erbe, L.H., Freedman, H.I., Sree Hari Rao, V.: Three-species food-chain models with mutual interference and time delays. Math. Biosci 80, 57–80 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feng, P.: Analysis of a delayed predator–prey model with ratio-dependent functional response and quadratic harvesting. J. Appl. Math. Comput. 44(1–2), 251–262 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fischer, B.M., Meyer, E., Maraun, M.: Positive correlation of trophic level and proportion of sexual taxa of oribatid mites (Acari: Oribatida) in alpine soil systems. Exp. Appl. Acarol. 63(4), 465–479 (2014)

    Article  Google Scholar 

  11. Freedman, H.I., Sree Hari Rao, V.: The trade-off between mutual interference and time lags in predator–prey systems. Bull. Math. Biol. 45(6), 991–1004 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Haile, D., Xie, Z.: Long-time behavior and Turing instability induced by cross-diffusion in a three species food chain model with a Holling type-II functional response. Math. Biosci. 267, 134–148 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Haque, M., Venturino, E.: Effect of parasitic infection in the Leslie–Gower predator–prey model. J. Biol. Syst. 16, 445–461 (2008)

    Article  MATH  Google Scholar 

  15. Hassard, B.D., Kazrinoff, N.D., Wan, W.H.: Theory and application of Hopf bifurcation. London math society lecture, vol. 41. Cambridge University Press, Cambridge (1981)

    Google Scholar 

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

    Article  Google Scholar 

  17. Huisman, G., De Boer, R.J.: A formal derivation of the “Beddington” functional response. J. Theor. Biol. 185, 389–400 (1997)

    Article  Google Scholar 

  18. Jana, D.: Chaotic dynamics of a discrete predator–prey system with prey refuge. Appl. Math. Comput. 224, 848–865 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Jana, D.: Stabilizing effect of prey refuge and predator’s interference on the dynamics of prey with delayed growth and generalist predator with delayed gestation. Int. J. Ecol. Article ID 429086, p. 12. doi: 10.1155/2014/429086 (2014)

  20. Jana, D., Agrawal, R., Upadhyay, R.K.: Top-predator interference and gestation delay as determinants of the dynamics of a realistic model food chain. Chaos Solitons Fractals 69, 50–63 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jana, D., Agrawal, R., Upadhyay, R.K.: Dynamics of generalist predator in a stochastic environment: effect of delayed growth and prey refuge. Appl. Math. Comput. 268, 1072–1094 (2015)

    MathSciNet  Google Scholar 

  22. Jiang, H., Zhang, T., Song, Y.: Delay-induced double Hopf bifurcations in a system of two delay-coupled van der Pol-duffling oscillators. Int. J. Bifurc. Chaos 25(4), 1550058 (2015)

    Article  MATH  Google Scholar 

  23. Kang, Y., Wedekin, L.: Dynamics of a intraguild predation model with generalist or specialist predator. J. Math. Biol. 67(5), 1227–1259 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Krebs, J.R., Davies, N.B.: An Introduction to Behavioural Ecology. Wiley, Ney York. ISBN 0-632-03546-3 (1993)

  25. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)

    MATH  Google Scholar 

  26. Liu, S., Beretta, E., Breda, D.: Predator–prey model of Beddington–DeAngelis type with maturation and gestation delays. Nonl. Anal. 11, 4072–4091 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mackey, M., Glass, L.: Oscillations and chaos in physiological control systems. Science 197, 287–289 (1997)

    Article  Google Scholar 

  28. Mandal, S., Jana, D., Roy, A.B., Majee, N.C.: Chaotic behavior of a class of neural network with discrete delays. Int. J. Modern Nonlinear Theory Appl. 2(1A), 97–101 (2013)

    Article  Google Scholar 

  29. Marwan, N., Romano, M.C., Thiel, M., Kurths, J.: Recurrence plots for the analysis of complex systems. Phys. Rep. 438, 237–329 (2007)

    Article  MathSciNet  Google Scholar 

  30. Matthiopoulos, J., Graham, K., Smout, S., Asseburg, C., Redpath, S., Thirgood, S., Hudson, P., Harwood, J.: Sensitivity to assumptions in models of generalist predation on cyclic prey. Ecology 88(10), 2576–2586 (2007)

    Article  Google Scholar 

  31. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)

    Book  MATH  Google Scholar 

  32. Nindjin, A.F., Aziz-Alaoui, M.A.: Analysis of a predator–prey model with modified Leslie–Gower and Holling type-II schemes with time delay. Nonlinear Anal. 7, 1104–1118 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rogers, D.J., Hassell, M.P.: General models for insect parasite and predator searching behavior: interference. J Anim. Ecol. 43, 239–253 (1974)

    Article  Google Scholar 

  34. Ruan, S.: On nonlinear dynamics of predator–prey models with disc rete delay. Math. Model. Nat. Phenom. 4(2), 140–188 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sen, M., Banerjee, M., Morozov, A.: A generalist predator regulating spread of a wildlife disease: exploring two infection transmission scenarios. Math. Model. Nat. Phenom. 7(2), 32–53 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Shen, C.: Permanence and global attractivity of the food-chain system with Holling IV type functional response. Appl. Math. Comput. 194(1), 179–185 (2007)

    MathSciNet  MATH  Google Scholar 

  37. Song, Z., Xu, J.: Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays. Cogn. Neurodyn. 7, 505–521 (2013)

    Article  Google Scholar 

  38. Strogatz, S.H.: Nonlinear Dynamics And Chaos: with Applications To Physics, Biology. Chemistry, and Engineering. Westview Press, Boulder (2009)

    MATH  Google Scholar 

  39. Temesgen, T.M.: Bifurcation analysis on the dynamics of a genralist predator–prey system. Int. J. Ecosyst. 2(3), 38–43 (2013)

    Google Scholar 

  40. Upadhyay, R.K., Iyengar, S.R.K., Rai, V.: Chaos: an ecological reality? Int. J. Bifur. Chaos 8, 1325–1333 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  41. Upadhyay, R.K., Kumari, N., Rai, V.: Wave of chaos and pattern formation in spatial predator–prey systems with Holling type IV predator response. Math. Model. Nat. Phenom. 3(4), 71–95 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Upadhyay, R.K., Raw, S.N.: Complex dynamics of a three species food-chain model with Holling type IV functional response. Nonlinear Anal. 16(3), 353–374 (2011)

    MathSciNet  MATH  Google Scholar 

  43. Upadhyay, R.K., Naji, R.K., Raw, S.N., Dubey, B.: The role of top predator interference on the dynamics of a food chain model. Commun. Nonlinear Sci. Numer. Simul. 18, 757–768 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Upadhyay, R.K., Iyengar, S.R.K.: Introduction to Mathematical Modelling and Chaotic Dynamics. Taylor and Francis, Boca Raton (2013)

    Google Scholar 

  45. Wang, W., Wang, H., Li, Z.: The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy. Chaos Solitons Fractals 32, 1772–1785 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wang, K., Wang, W., Pang, H., Liu, X.: Complex dynamical behavior in a viral model with delayed immune response. Physica D 226(20), 197–208 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  47. Xu, R., Ma, Z.: Stability and Hopf-bifurcation in a ratio-dependent predator–prey system with stage structure. Chaos Solitons Fractals 38, 669–684 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  48. Xu, R., Ma, Z., Gen, Q.: Stability and bifurcation in a Beddington–DeAngelis type predator-prey model with prey dispersal. J. Math. 38(5), 1761–1783 (2008)

    MathSciNet  MATH  Google Scholar 

  49. Xu, R., Gan, Q., Ma, Z.: Stability and bifurcation analysis on a ratio-dependent predator–prey model with time delay. J Comput. Appl. Math. 230(1), 187–203 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  50. Yafia, R., Adnani, F.F., Alaoui, H.: Limit cycle and numerical simulations for small and large delays in a predator–prey model with modified Leslie–Gower and Holling type-II schemes. Nonlinear Anal. 9, 2055–2067 (2008)

    Article  MATH  Google Scholar 

  51. Zhang, S., Wang, F., Chen, L.: A food chain model with impulsive perturbations and Holling IV functional response. Chaos Solitons Fractals 26(3), 855–866 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zhao, M., Songjuan, L.V.: Chaos in a three-species food chain model with a Beddington–DeAngelis functional response. Chaos Solitns Fractals 40(5), 2305–2316 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  53. Zhao, M., Yu, H., Zhu, J.: Effects of a population floor on the persistence of chaos in a mutual interference host-parasitoid model. Chaos Solitons Fractals 42(2), 1245–1250 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad, Jharkhand, 826 004, India

    Rashmi Agrawal & Ranjit Kumar Upadhyay

  2. Department of Mathematics, SRM University, Kattankulathur, Tamil Nadu, 603 203, India

    Debaldev Jana

  3. Foundation for Scientific Research and Technological Innovation, 13-405, Road No.14, Alakapuri, Hyderabad, 500 102, India

    V. Sree Hari Rao

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  1. Rashmi Agrawal
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  4. V. Sree Hari Rao
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Correspondence to Ranjit Kumar Upadhyay.

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Agrawal, R., Jana, D., Upadhyay, R.K. et al. Complex dynamics of sexually reproductive generalist predator and gestation delay in a food chain model: double Hopf-bifurcation to Chaos. J. Appl. Math. Comput. 55, 513–547 (2017). https://doi.org/10.1007/s12190-016-1048-1

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