Advertisement

Dynamical Behavior for a Stochastic Predator–Prey Model with HV Type Functional Response

  • Bo DuEmail author
  • Maolin Hu
  • Xiuguo Lian
Article

Abstract

This paper establishes the existence-and-uniqueness theorem of a stochastic delayed predator–prey model with Hassell–Varley type functional response and examines stochastically ultimate boundedness, extinction and global asymptotic stability of this solution. It is interesting to note that the results are based on time-varying delay, which is different from the previous work (the results are delay-independent). Some numerical simulations are introduced to support the analytical findings.

Keywords

Hassell–Varley type functional response Delay Existence Stability 

Mathematics Subject Classification

34C57 58E05 

References

  1. 1.
    Hassell, M., Varley, G.: New inductive population model for insect parasites and its bearing on biological control. Nature 223, 1133–1136 (1969)CrossRefGoogle Scholar
  2. 2.
    Cosner, C., DeAngelis, D., Ault, J., Olson, D.: Effects of spatial grouping on the functional response of predators. Theor. Popul. Biol. 56, 65–75 (1999)CrossRefzbMATHGoogle Scholar
  3. 3.
    Hsu, S.B., Hwang, T.W., Kuang, Y.: Global dynamics of a predator–prey model with Hassell–Varley type functional response. Discret. Contin. Dyn. Syst. B 10(4), 857–871 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Kim, H., Baek, H.: The dynamical complexity of a predator–prey system with Hassell–Varley functional response and impulsive effect. Math. Comput. Simul. 94, 1–14 (2013)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Liu, X.: Impulsive periodic oscillation for a predator–prey model with Hassell–Varley–Holling functional response. Appl. Math. Model. 38, 1482–1494 (2014)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Macdonald, N.: Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  7. 7.
    Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publisher, Boston (1992)CrossRefzbMATHGoogle Scholar
  8. 8.
    Fan, M., Wang, K., Jiang, D.: Existence and global attractivity of positive periodic solutions of periodic n-species Lotka–Volterra competition systems with several deviating arguments. Math. Biosci. 160, 47–61 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Xu, R., Chaplain, M., Davidson, F.: Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments. Nonlinear Anal. RWA 5, 183–206 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Egami, C., Hirano, N.: Periodic solutions in a class of periodic delay predator-prey systems. Yokohama Math. J. 51, 45–61 (2004)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Mohamad, S., Gopalsamy, K.: Dynamics of a class of discrete-time neural networks and their continuous-time counterparts. Math. Comput. Simul. 53, 1–39 (2000)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Mohamad, S., Gopalsamy, K.: Exponential stability of continuous-time and discrete-time cellular neural networks with delays. Appl. Math. Comput. 135(1), 17–38 (2003)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Pao, C.: Global asymptotic stability of Lotka–Volterra 3-species reaction–diffusion systems with time delays. J. Math. Anal. Appl. 281, 186–204 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Liang, J., Wang, Z., Liu, Y., Liu, X.: Global synchronization control of general delayed discrete-time networks with stochastic coupling and disturbances. IEEE Trans. Syst. Man Cybern. Part B 38, 1073–1083 (2008)CrossRefGoogle Scholar
  15. 15.
    Liu, Y., Wang, Z., Liu, X.: Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Netw. 19, 667–675 (2006)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)zbMATHGoogle Scholar
  17. 17.
    Wang, K.: Periodic solutions to a delayed predator–prey model with Hassell–Varley type functional response. Nonlinear Anal. 12, 137–145 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Gard, T.: Persistence in stochastic food web models. Bull. Math. Biol. 46, 357–370 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Gard, T.: Stability for multispecies population models in random environments. Nonlinear Anal. 10, 411–419 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (2001)zbMATHGoogle Scholar
  21. 21.
    Arditi, R., Saiah, H.: Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 73, 1544–1551 (1992)CrossRefGoogle Scholar
  22. 22.
    Li, X., Mao, X.: Population dynamical behavior of non-autonomous Lotka–Volterra competitive system with random perturbation. Discret. Contin. Dyn. Syst. 24, 523–545 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Liu, M., Wang, K.: Population dynamical behavior of Lotka–Volterra cooperative systems with random perturbations. Discret. Contin. Dyn. Syst. 33, 2495–2522 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Li, X., Mao, X.: Population dynamical behavior of non-autonomous Lotka–Volterra competitive system with random perturbation. Discret. Contin. Dyn. Syst. 24, 523–545 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Qiu, H., Liu, M., Wang, K., Wang, Y.: Dynamics of a stochastic predator-prey system with Beddington–DeAngelis functional response. Appl. Math. Comput. 219, 2303–2312 (2012)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Vasilova, M., Jovanovic, M.: Stochastic Gilpin–Ayala competition model with infinite delay. Appl. Math. Comput. 217, 4944–4959 (2011)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Bao, J., Yuan, C.: Stochastic population dynamics driven by Lévy noise. J. Math. Anal. Appl. 391, 363–375 (2012)CrossRefzbMATHGoogle Scholar
  28. 28.
    Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)zbMATHGoogle Scholar
  29. 29.
    Rao, F., Jiang, S., Li, Y., Liu Hao: Stochastic Analysis of a Hassell–Varley Type Predation Model, Abstract and Applied Analysis Volume 2013. Article ID 738342Google Scholar
  30. 30.
    Bahar, A., Mao, X.: Stochastic delay Lotka–Volterra model. J. Math. Anal. Appl. 335, 1207–1218 (2007)CrossRefzbMATHGoogle Scholar
  31. 31.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer-Verlag, Berlin (1991)zbMATHGoogle Scholar
  32. 32.
    Barbalat, I.: Systems dequations differential d’osci nonlineaires. Revue Roumaine de Mathematiques Pures et Appliquees 4, 267–270 (1959)MathSciNetGoogle Scholar
  33. 33.
    Kloeden, P.E., Shardlow, T.: The Milstein scheme for stochastic delay differential equations without using anticipative calculus. Stoch. Anal. Appl. 30, 181–202 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2016

Authors and Affiliations

  1. 1.Department of MathematicsHuaiyin Normal UniversityHuaianPeople’s Republic of China

Personalised recommendations