Global Stability of a Delayed Eco-Epidemiological Model with Holling Type III Functional Response

  • Hongfang Bai
  • Rui Xu
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 225)


In this paper, we consider an eco-epidemiological model with Holling type III functional response and a time delay representing the gestation period of the predator. In the model, it is assumed that the predator population suffers a transmissible disease. By means of Lyapunov functionals and Laselle’s invariance principle, sufficient conditions are obtained for the global stability of the endemic coexistence of the system.


Eco-epidemiological model Delay Laselle’s invariance principle Global stability 


  1. 1.
    Beretta, E., Hara, T., Ma, W., Takeuchi, Y.: Global asymptotic stability of an SIR epidemic model with distributed time delay. Nonlinear Anal. 47, 4017–4115 (2001)Google Scholar
  2. 2.
    Gakkhar, S., Negi, K.: Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate. Chaos Solitions Fractals 35, 626–638 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Xu, R., Ma, Z.E., Wang, Z.P.: Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity. Comput. Math. Appl. 59, 3211–3221 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Xu, R.: Global dynamics of an SEIRI epidemioligical model with time delay. Appl. Math. Comput. 232, 436–444 (2014)MathSciNetGoogle Scholar
  5. 5.
    Kermack, W.Q., Mckendrick, A.G.: Contributions to the mathematical theory of epidemics (Part I). Proc. R. Soc. A 115, 700–721 (1927)CrossRefzbMATHGoogle Scholar
  6. 6.
    Anderson, R.M., May, R.M.: Regulation stability of host-parasite population interactions: I. Regulatory processes. J. Anim. Ecol. 47, 219–267 (1978)CrossRefGoogle Scholar
  7. 7.
    Zhang, J., Li, W., Yan, X.: Hopf bifurcation and stability of periodic solutions in a delayed eco-epidemiological system. Appl. Math. Comput. 198, 865–876 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Debasis, M.: Hopf bifurcation in an eco-epidemic model. Appl. Math. Comput. 217, 2118–2124 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bairagi, N.: Direction and stability of bifurcating periodic solution in a delay-induced eco-epidemiological system. Int. J. Differ. Equ. 1–25 (2011)Google Scholar
  10. 10.
    Xu, R., Tian, X.H.: Global dynamics of a delayed eco-epidemiological model with Holling type-III functional response. Math. Method Appl. Sci. 37, 2120–2134 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sahoo, B.: Role of additional food in eco-epidemiological system with disease in the prey. Appl. Math. Comput. 259, 61–79 (2015)MathSciNetGoogle Scholar
  12. 12.
    Holling, C.S.: The components of predation as revealed by a study of small mammal predation of the European pine sawfly. Can. Entomol. 91, 293–320 (1959)CrossRefGoogle Scholar
  13. 13.
    Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385–398 (1959)CrossRefGoogle Scholar
  14. 14.
    Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 45, 3–60 (1965)Google Scholar
  15. 15.
    Hale, J., Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20, 383–395 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Haddock, J.R., Terjéki, J.: Liapunov-Razumikhin functions and an invariance principle for functional-differential equations. J. Differ. equ. 48, 95–122 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsShijiazhuang Mechanical Engineering CollegeShijiazhuangPeople’s Republic of China

Personalised recommendations