Advertisement

Nonlinear Dynamics

, Volume 96, Issue 4, pp 2653–2679 | Cite as

Dynamics of prey–predator model with stage structure in prey including maturation and gestation delays

  • Balram DubeyEmail author
  • Ankit Kumar
Original Paper

Abstract

This study proposes a three-dimensional prey–predator model with stage structure in prey (immature and mature) including maturation delay in prey population and gestation delay in predator population. It is assumed that the immature prey population is consumed by predators with Holling type I functional response and the interaction between mature prey and predator species is followed by Crowley–Martin-type functional response. We analyzed the equilibrium points, local and global asymptotic behavior of interior equilibrium point for the non-delayed system. Hopf-bifurcation with respect to different parameters has also been studied for the system. Further, the existence of periodic solutions through Hopf-bifurcation is shown with respect to both the delays. Our model analysis shows that time delay plays a vital role in governing the dynamics of the system. It changes the stability behavior of the system into instability, even with the switching of stability. The direction and stability of Hopf-bifurcation are also studied by using normal form method and center manifold theorem. Finally, computer simulation and graphical illustrations have been carried out to support our theoretical investigations.

Keywords

Prey–predator Stage structure Hopf-bifurcation Maturation delay Gestation delay Switching stability 

Notes

Acknowledgements

Authors are thankful to anonymous reviewers for careful reading and constructive suggestions that improved the quality and presentation of the paper. The author (AK) acknowledges the Junior Research Fellowship received from University Grant Commission, New Delhi, India.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Aiello, W.G., Freedman, H.I.: A time-delay model of single-species growth with stage structure. Math. Biosci. 101(2), 139–153 (1990)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aiello, W.G., Freedman, H.I., Wu, J.: Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM J. Appl. Math. 52(3), 855–869 (1992)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arino, J., Wang, L., Wolkowicz, G.S.K.: An alternative formulation for a delayed logistic equation. J. Theor. Biol. 241(1), 109–119 (2006)MathSciNetGoogle Scholar
  4. 4.
    Bairagi, N., Jana, D.: On the stability and Hopf bifurcation of a delay-induced predator–prey system with habitat complexity. Appl. Math. Model. 35(7), 3255–3267 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bairagi, N., Jana, D.: Age-structured predator–prey model with habitat complexity: oscillations and control. Dyn. Syst. 27(4), 475–499 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bellman, R., Cooke, K.L.: Differential Difference Equations. Academic Press, New York (1963)zbMATHGoogle Scholar
  7. 7.
    Beretta, E., Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33(5), 1144–1165 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Berryman, A.A.: The orgins and evolution of predator–prey theory. Ecology 73(5), 1530–1535 (1992)Google Scholar
  9. 9.
    Bosch, F.V.D., Gabriel, W.: Cannibalism in an age-structured predator–prey system. Bull. Math. Biol. 59(3), 551–567 (1997)zbMATHGoogle Scholar
  10. 10.
    Chakraborty, K., Haldar, S., Kar, T.K.: Global stability and bifurcation analysis of a delay induced prey–predator system with stage structure. Nonlinear Dyn. 73(3), 1307–1325 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Crowley, P.H., Martin, E.K.: Functional responses and interference within and between year classes of a dragonfly population. J. N. Am. Benth. Soc. 8(3), 211–221 (1989)Google Scholar
  12. 12.
    Dai, G., Tang, M.: Coexistence region and global dynamics of a harvested predator–prey system. SIAM J. Appl. Math. 58(1), 193–210 (1998)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Deng, L., Wang, X., Peng, M.: Hopf bifurcation analysis for a ratio-dependent predator–prey system with two delays and stage structure for the predator. Appl. Math. Comput. 231, 214–230 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Devi, S.: Effects of prey refuge on a ratio-dependent predator–prey model with stage-structure of prey population. Appl. Math. Model. 37(6), 4337–4349 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dong, Q., Ma, W., Sun, M.: The asymptotic behavior of a chemostat model with Crowley–Martin type functional response and time delays. J. Math. Chem. 51(5), 1231–1248 (2013)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gakkhar, S., Singh, A.: Complex dynamics in a prey–predator system with multiple delays. Commun. Nonlinear Sci. Numer. Simul. 17(2), 914–929 (2012)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gámez, M., Martínez, C.: Persistence and global stability in a predator–prey system with delay. Int. J. Bifur. Chaos 16(10), 2915–2922 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Gourley, S.A., Kuang, Y.: A stage structured predator–prey model and its dependence on maturation delay and death rate. J. Math. Biol. 49(2), 188–200 (2004)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Guin, L.N., Mandal, P.K.: Spatial pattern in a diffusive predator–prey model with sigmoid ratio-dependent functional response. Int. J. Biomath. 7(05), 1450047 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Gupta, R.P., Chandra, P.: Bifurcation analysis of modified Leslie–Gower predator–prey model with Michaelis-Menten type prey harvesting. J. Math. Anal. Appl. 398(1), 278–295 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University, Cambridge (1981)zbMATHGoogle Scholar
  22. 22.
    Huang, J., Gong, Y., Chen, J.: Multiple bifurcations in a predator–prey system of Holling and Leslie type with constant-yield prey harvesting. Int. J. Bifur. Chaos 23(10), 1350164 (2013)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Jana, D., Agrawal, R., Upadhyay, R.K., Samanta, G.P.: Ecological dynamics of age selective harvesting of fish population: maximum sustainable yield and its control strategy. Chaos Solitons Fract. 93, 111–122 (2016)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kar, T.K., Matsuda, H.: Controllability of a harvested prey–predator system with time delay. J. Biol. Syst. 14(02), 243–254 (2006)zbMATHGoogle Scholar
  25. 25.
    Kuang, Y., Takeuchi, Y.: Predator–prey dynamics in models of prey dispersal in two-patch environments. Math. Biosci. 120(1), 77–98 (1994)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Landahl, H.D., Hansen, B.D.: A three stage population model with cannibalism. Bull. Math. Biol. 37, 11–17 (1975)zbMATHGoogle Scholar
  27. 27.
    Li, F., Li, H.: Hopf bifurcation of a predator–prey model with time delay and stage structure for the prey. Math. Comput. Model. 55(3–4), 672–679 (2012)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Li, H., Takeuchi, Y.: Dynamics of the density dependent predator–prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 374(2), 644–654 (2011)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Li, K., Wei, J.: Stability and Hopf bifurcation analysis of a prey–predator system with two delays. Chaos Solitons Fract. 42(5), 2606–2613 (2009)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Liao, M., Tang, X., Xu, C.: Bifurcation analysis for a three-species predator–prey system with two delays. Commun. Nonlinear Sci. Numer. Simul. 17(1), 183–194 (2012)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Liu, S., Beretta, E.: A stage-structured predator–prey model of Beddington–DeAngelis type. SIAM J. Appl. Math. 66(4), 1101–1129 (2006)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Liu, Y., Zhang, X., Zhou, T.: Multiple periodic solutions of a delayed predator–prey model with non-monotonic functional response and stage structure. J. Biol. Dyn. 8(1), 145–160 (2014)MathSciNetGoogle Scholar
  33. 33.
    Maiti, A.P., Dubey, B.: Stability and bifurcation of a fishery model with Crowley–Martin functional response. Int. J. Bifur. Chaos 27(11), 1750174 (2017)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Maiti, A.P., Dubey, B., Tushar, J.: A delayed prey–predator model with Crowley–Martin-type functional response including prey refuge. Math. Methods Appl. Sci. 40(16), 5792–5809 (2017)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Martin, A., Ruan, S.: Predator–prey models with delay and prey harvesting. J. Math. Biol. 43(3), 247–267 (2001)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Misra, A.K., Dubey, B.: A ratio-dependent predator–prey model with delay and harvesting. J. Biol. Syst. 18(02), 437–453 (2010)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Murray, J.D.: Mathematical Biology I. An Introduction. Springer, New York (2002)zbMATHGoogle Scholar
  38. 38.
    Nakaoka, S., Saito, Y., Takeuchi, Y.: Stability, delay, and chaotic behavior in a Lotka–Volterra predator–prey system. Math. Biosci. Eng. 3(1), 173 (2006)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Pathak, S., Maiti, A., Bera, S.P.: Effect of time-delay on a prey–predator model with microparasite infection in the predator. J. Biol. Syst. 19(02), 365–387 (2011)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Smith, H.L.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Bull. Am. Math. Soc. 33, 203–209 (1996)Google Scholar
  41. 41.
    Song, Y., Wei, J.: Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos. Chaos Solitons Fract. 22(1), 75–91 (2004)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Tian-Wei-Tian, Z.: Multiplicity of positive almost periodic solutions in a delayed Hassell–Varley-type predator–prey model with harvesting on prey. Math. Methods Appl. Sci. 37(5), 686–697 (2014)MathSciNetzbMATHGoogle Scholar
  43. 43.
    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(1–2), 177–196 (2015)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Tripathi, J.P., Tyagi, S., Abbas, S.: Global analysis of a delayed density dependent predator–prey model with Crowley–Martin functional response. Commun. Nonlinear Sci. Numer. Simul. 30(1), 45–69 (2016)MathSciNetGoogle Scholar
  45. 45.
    Upadhyay, R.K., Naji, R.K.: Dynamics of a three species food chain model with Crowley–Martin type functional response. Chaos Solitons Fract. 42(3), 1337–1346 (2009)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Wang, X., Liu, X.: Stability and Hopf bifurcation of a delayed ratio-dependent eco-epidemiological model with two time delays and Holling type III functional response. Int. J. Nonlinear Sci. 23(2), 102–108 (2017)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Wang, Y., Zhou, Y., Brauer, F., Heffernan, J.M.: Viral dynamics model with CTL immune response incorporating antiretroviral therapy. J. Math. Biol. 67(4), 901–934 (2013)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Wei, F., Fu, Q.: Hopf bifurcation and stability for predator–prey systems with Beddington–DeAngelis type functional response and stage structure for prey incorporating refuge. Appl. Math. Model. 40(1), 126–134 (2016)MathSciNetGoogle Scholar
  49. 49.
    Wood, S.N., Blythe, S.P., Gurney, W.S.C., Nisbet, R.M.: Instability in mortality estimation schemes related to stage-structure population models. Math. Med. Biol. 6(1), 47–68 (1989)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Xu, C., Li, P.: Dynamical analysis in a delayed predator-prey model with two delays. Discrete Dyn. Nat. Soc. 2012, 652947 (2012).  https://doi.org/10.1155/2012/652947 MathSciNetzbMATHGoogle Scholar
  51. 51.
    Yan, X.P., Chu, Y.D.: Stability and bifurcation analysis for a delayed Lotka–Volterra predator–prey system. J. Comput. Appl. Math. 196(1), 198–210 (2006)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Zhang, W., Liu, H., Xu, C.: Bifurcation analysis for a Leslie–Gower predator–prey system with time delay. Int. J. Nonlinear Sci. 15(1), 35–44 (2013)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Zhang, Y., Zhang, Q.: Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting. Nonlinear Dyn. 66(1–2), 231–245 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsBITS Pilani, Pilani CampusPilaniIndia

Personalised recommendations