Global Dynamics of a Predator–Prey Model with General Holling Type Functional Responses

  • Wei DingEmail author
  • Wenzhang Huang


A predator–prey model with general Holling type functional response is considered. The aim of this paper is to investigate the global stability of the interior equilibrium of the model when it exists. By transforming the model to an equivalent system and with the detailed analysis of the crucial behavior of a corresponding function we are able to show that the local stability and the global stability of the interior equilibrium are equivalent.


Predator–prey model General Holling type functional response Global stability of interior equilibrium 



The first author of this paper would like to thank National Natural Science Foundation of China (No. 11431008) for the partial support of this research.


  1. 1.
    Castillo-Chavez, C., Feng, Z., Huang, W.: Global dynamics of a plant-herbivore model with toxin-determined functional response. SIAM J. Appl. Math. 72, 1002–1020 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ding, S.H.: On a kind of predator–prey system. SIAM J. Math. Anal. 20, 1426–1435 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gasull, A., Guillamon, A.: Non-existence of limit cycles for some predator–prey systems. In: Proceedings of Equadiff, vol 91, pp. 538–543. World Scientific, Singapore (1993)Google Scholar
  4. 4.
    Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 45, 1–60 (1965)Google Scholar
  5. 5.
    Hsu, S.B., Hwang, T.W.: Global stability for a class of predator–prey systems. SIAM J. Appl. Math. 55, 763–783 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Huang, X.C.: Uniqueness of limit cycles of generalized Liénard systems and predator–prey systems. J. Phys. A Math. Gen. 21, L685–L691 (1988)CrossRefzbMATHGoogle Scholar
  7. 7.
    Kuang, Y., Freedman, H.I.: Uniqueness of limit cycles in Gause-type models of predator–prey system. Math. Biosci. 88, 67–84 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ma, Z., Wang, S., Wang, T., Tang, H.: Stability analysis of prey–predator system with Holling type functional response and prey refuge. Adv. Differ. Equ. 2017, 243 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ma, Z., Li, W., Zhao, Y., Wang, W., Zhang, H., Li, Z.: Effects of prey refuges on a predator-prey model with a class of functional responses: the role of refuges. Math. Biosci. 218, 73–79 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    May, R.M.: Stability and Complexity in Ecosystems. Princeton University Press, Princeton (2001)zbMATHGoogle Scholar
  11. 11.
    Real, L.: The kinetics of functional response. Am. Nat. 111, 289–300 (1977)CrossRefGoogle Scholar
  12. 12.
    Sugie, J., Kohno, R., Miyazaki, R.: On a predator–prey system of Holling type. Proc. AMS 125, 2041–2050 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sugie, J., Miyamoto, K., Morino, K.: Absence of limit cycles of a predator–prey system with a sigmoid functional response. Appl. Math. Lett. 9, 85–90 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina
  2. 2.Department of Mathematical SciencesUniversity of Alabama in HuntsvilleHuntsvilleUSA

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