Stability Switches in a Logistic Population Model with Mixed Instantaneous and Delayed Density Dependence



The local asymptotic stability and stability switches of the positive equilibrium in a logistic population model with mixed instantaneous and delayed density dependence is analyzed. It is shown that when the delayed dependence is more dominant, either the positive equilibrium becomes unstable for all large delay values, or the stability of equilibrium switches back and force several times as the delay value increases. Compared with the logistic model with the instantaneous term and a delayed term, our finding here is that the incorporation of another delayed term can lead to the occurrence of multiple stability switches.


Logistic model Instantaneous and delayed density dependence Stability switches Hopf bifurcation 

Mathematics Subject Classification

34K08 34K18 34K20 35R10 92E20 


  1. 1.
    Ahlfors, L.: Complex Analysis. McGraw-Hill, New York (1966)MATHGoogle Scholar
  2. 2.
    Bélair, J., Campbell, S.A.: Stability and bifurcations of equilibria in a multiple-delayed differential equation. SIAM J. Appl. Math. 54(5), 1402–1424 (1994)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, S.S., Shi, J.P., Wei, J.J.: Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems. J. Nonlinear Sci. 23(1), 1–38 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cooke, K., Grossman, Z.: Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86(2), 592–627 (1982)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cooke, K.L., Huang, W.Z.: A theorem of George Seifert and an equation with state-dependent delay. Delay and Differential Equations (Ames. IA, 1991), pp. 65–77. World Scientific Publishing, River Edge (1992)Google Scholar
  6. 6.
    Cushing, J.M.: Integrodifferential equations and delay models in population dynamics. Lecture Notes in Biomathematics, vol. 20. Springer-Verlag, Berlin (1977)Google Scholar
  7. 7.
    Erneux, T.: Applied delay differential equations. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 3. Springer, New York (2009)Google Scholar
  8. 8.
    Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics. Mathematics and its Applications, vol. 74. Kluwer Academic Publishers Group, Dordrecht (1992)Google Scholar
  9. 9.
    Hale, J.K., Huang, W.Z.: Global geometry of the stable regions for two delay differential equations. J. Math. Anal. Appl. 178(2), 344–362 (1993)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to functional-differential equations. Applied Mathematical Sciences, vol. 99. Springer-Verlag, New York (1993)Google Scholar
  11. 11.
    Hu, X., Shu, H.Y., Wang, L., Watmough, J.: Delay induced stability switch, multitype bistability and chaos in an intraguild predation model (submitted) (2014)Google Scholar
  12. 12.
    Huang, W.Z.: Global stability analysis for the second order linear delay differential equations. J. Anhui University Special Edition Math. (1985)Google Scholar
  13. 13.
    Huang, W.Z.: Global dynamics for a reaction-diffusion equation with time delay. J. Differ. Equ. 143(2), 293–326 (1998)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kuang, Y.: Delay differential equations with applications in population dynamics. Mathematics in Science and Engineering, vol. 191. Academic Press Inc., Boston (1993)Google Scholar
  15. 15.
    Lenhart, S.M., Travis, C.C.: Global stability of a biological model with time delay. Proc. Am. Math. Soc. 96(1), 75–78 (1986)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Li, M.Y., Lin, X.H., Wang, H.: Global Hopf branches in a delayed model for immune response to HTLV-1 infections: coexistence of multiple limit cycles. Can. Appl. Math. Q. 20(1), 39–50 (2012)MathSciNetGoogle Scholar
  17. 17.
    Li, M.Y., Shu, H.Y.: Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection. Bull. Math. Biol. 73(8), 1774–1793 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Li, X.G., Ruan, S.G., Wei, J.J.: Stability and bifurcation in delay-differential equations with two delays. J. Math. Anal. Appl. 236(2), 254–280 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Liao, K.L., Lou, Y.: The effect of time delay in a two-patch model with random dispersal. Bull. Math. Biol. 76(2), 335–376 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mahaffy, J.M., Joiner, K.M., Zak, P.J.: A geometric analysis of stability regions for a linear differential equation with two delays. Int. J. Bifur. Chaos Appl. Sci. Eng. 5(3), 779–796 (1995)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Miller, R.K.: On Volterra’s population equation. SIAM J. Appl. Math. 14, 446–452 (1966)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Niculescu, S.I., Kim, P.S., Gu, K., Lee, P.P., Levy, D.: Stability crossing boundaries of delay systems modeling immune dynamics in leukemia. Discret. Contin. Dyn. Syst. Ser. B 13(1), 129–156 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Piotrowska, M.J.: A remark on the ODE with two discrete delays. J. Math. Anal. Appl. 329(1), 664–676 (2007)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Reynolds, J.J.H., Sherratt, J.A., White, A.: Stability switches in a host-pathogen model as the length of a time delay increases. J. Nonlinear Sci. 23(6), 1073–1087 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ruan, S.G.: Delay differential equations in single species dynamics. Delay differential equations and applications. NATO Sci. Ser. II Math. Phys. Chem., vol. 205, pp. 477–517. Springer, Berlin (2006)CrossRefGoogle Scholar
  26. 26.
    Ruan, S.G.: On nonlinear dynamics of predator-prey models with discrete delay. Math. Model. Nat. Phenom. 4(02), 140–188 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Ruan, S.G., Wei, J.J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn. Contin. Discret. Impuls. Syst. Ser. A 10(6), 863–874 (2003)MathSciNetMATHGoogle Scholar
  28. 28.
    Schoen, G.M., Geering, H.P.: Stability condition for a delay differential system. Int. J. Control 58(1), 247–252 (1993)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Seifert, G.: On a delay-differential equation for single specie population variations. Nonlinear Anal. 11(9), 1051–1059 (1987)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Smith, H.: An introduction to delay differential equations with applications to the life sciences. Texts in Applied Mathematics, vol. 57. Springer, New York (2011)Google Scholar
  31. 31.
    Song, Y.L., Wei, J.J.: Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system. J. Math. Anal. Appl. 301(1), 1–21 (2005)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Song, Z.G., Xu, J.: Stability switches and multistability coexistence in a delay-coupled neural oscillators system. J. Theoret. Biol. 313, 98–114 (2012)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Su, Y., Wei, J.J., Shi, J.P.: Hopf bifurcation in a diffusive logistic equation with mixed delayed and instantaneous density dependence. J. Dynam. Differ. Equ. 24(4), 897–925 (2012)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Wei, J.J., Ruan, S.G.: Stability and bifurcation in a neural network model with two delays. Phys. D 130(3–4), 255–272 (1999)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsLanzhou Jiaotong UniversityLanzhouChina
  2. 2.Department of MathematicsCollege of William and MaryWilliamsburgUSA

Personalised recommendations