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Stability and bifurcation in a generalized delay prey–predator model

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Abstract

The present paper considers a generalized prey–predator model with time delay. It studies the stability of the nontrivial positive equilibrium and the existence of Hopf bifurcation for this system by choosing delay as a bifurcation parameter and analyzes the associated characteristic equation. The researcher investigates the direction of this bifurcation by using an explicit algorithm. Eventually, some numerical simulations are carried out to support the analytical results.

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References

  1. Beckerman, S.: The equations of war. J. Curr. Anthropol. 32(5), 636–640 (1991)

    Article  Google Scholar 

  2. Bianca, C., Ferrara, M., Guerrini, L.: Qualitative analysis of a retarded mathematical frame work with applications to living systems. Abstr. Appl. Anal. Article ID 7360528, 14 (2013)

  3. Bianca, C., Guerrini, L.: Existence of limit cycles in the Solow model with delayed-logistic population growth. Sci. World J. Article ID 207806 (2014)

  4. Bianca, C., Ferrara, M., Guerrini, L.: The Cai model with time delay: existence of periodic solutions and asymptotic analysis. Appl. Math. Inf. Sci. 7, 21–27 (2013)

    Article  MathSciNet  Google Scholar 

  5. Bruce Nauman, E.: Chemical Reactor Design, Optimization and Scaleup. Wiley, Hoboken (2008)

    Book  Google Scholar 

  6. Diekman, O., Van Giles, S., Verduyn Lunel, S., Walter, H.: Delay Equations. Springer, New York (1995)

    Book  Google Scholar 

  7. Das, T., Mukherjee, R.N., Chaudhuri, K.S.: Harvesting of a prey-predator fishery in the presence of toxicity. Appl. Math. Model. 33, 2282–2292 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Faria, T.: Stability and bifurcation for a delayed predator–prey model and the effect of diffusion. J. Math. Anal. Appl. 254, 433–463 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gottman, J.M., Murray, J.D., Swanson, C.C., Tyson, R., Swanson, R.K.: The Mathematics of Marriage Dynamic Nonlinear Model. The MIT Press, Cambridge (2002)

    MATH  Google Scholar 

  10. Grabner, C., Hahn, H., Leopold-Wildburger, U., Pickl, S.: Analyzing the sustainability of harvesting behavior and the relationship to personality traits in a simulated Lotka–Volterra biotope. Eur. J. Oper. Res. 193, 761–767 (2009)

    Article  MATH  Google Scholar 

  11. Hale, J.: Theory of functional differential equation. Int. J. Appl. Math. Mech. 8(1), 50–65 (1977)

    MathSciNet  Google Scholar 

  12. Hale, J., Lunel, S.V.: Introduction to Functional Differential Equations. Springer, New York (1993)

    Book  MATH  Google Scholar 

  13. Kaddar, A., Talibi Alaoui, H.: Local Hopf bifurcation and stability of limit cycle in a delayed Kaldor–Kalecki model. Nonlinear Anal. Model. Control 14(3), 333–343 (2009)

    MATH  MathSciNet  Google Scholar 

  14. Khalil, H.K.: Nonlinear Systems. Prentice Hall, New Jersey (2001)

    Google Scholar 

  15. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)

    MATH  Google Scholar 

  16. Lang, S.: Real and Functional Analysis. Springer, New York (1991)

    Google Scholar 

  17. Liu, Z.J., Zhong, S.M., Liu, X.Y.: Permanence and periodic solutions for an impulsive reaction–diffusion food-chain system with ratio-dependent functional response. Commun. Nonlinear Sci. Numer. Simul. 19, 173–188 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ma, Z.P., Huo, H.F., Liu, C.Y.: Stability and Hopf bifurcation on a predator–prey model with discrete and distributed delays. Nonlinear Anal. Real World Appl. 10, 1160–1172 (2009)

  19. May, R.M.: Time delay versus stability in population models with two and three trophic levels. Ecology 4, 315–325 (1973)

    Article  Google Scholar 

  20. Perko, L.: Differential Equations and Dynamical System. Springer, New York (2000)

    Google Scholar 

  21. Randy, L., Haupt, L., Haupt, S.E.: Practical Genetic Algorithms. Wiley, Hoboken (2004)

    MATH  Google Scholar 

  22. Ruan, S.: Absolute stability, conditional stability and Hopf bifurcation in Kolmogorov-type predator–prey systems with discrete delays. Q. Appl. Math. 59, 159–173 (2001)

    Article  MATH  Google Scholar 

  23. Ruan, S.G., Wei, J.J.: On the zero of some transcendental functions with applications to stability of delay differential equations with two delays. Dyn. Contin. Discret. Impuls. Syst. Ser. A Math. Anal. 10, 863–874 (2003)

    MATH  Google Scholar 

  24. Song, Y.L., Yuan, S.L.: Bifurcation analysis in a predator–prey system with time delay. Nonlinear Anal. Real World Appl. 7, 265–284 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Yan, X.P., Zhang, C.H.: Hopf bifurcation in a delayed Lokta–Volterra predator–prey system. Nonlinear Anal. Real World Appl. 9, 114–127 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Esfarayen University of Technology, Esfarayen, North Khorasan, Iran

    Atena Ghasemabadi

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  1. Atena Ghasemabadi
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Correspondence to Atena Ghasemabadi.

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Ghasemabadi, A. Stability and bifurcation in a generalized delay prey–predator model. Nonlinear Dyn 90, 2239–2251 (2017). https://doi.org/10.1007/s11071-017-3431-3

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  • DOI: https://doi.org/10.1007/s11071-017-3431-3

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