Mathematics and Recurrent Population Outbreaks

  • Torsten LindströmEmail author
Living reference work entry


Despite that outbreaks had been observed for hundreds of years for many populations, it took until the 1920s before the first mechanisms that did not involve human interference were suggested. Just a few mechanisms were included in the first models and the question whether the inclusion of other, very plausible, mechanisms could alter the predictions remained.

In this chapter, we follow the development of models that have been proposed to explain oscillatory population dynamics from the early models suggested by Lotka (1925) and Volterra (1926) until global dynamical questions that are still open for models incorporating explicit resource dynamics, like the chemostat, cf Kuang (1989).


Global stability Limit cycle Lyapunov function Mechanistic population models Oscillatory dynamics Recurrent outbreaks 


  1. Ardito A, Ricciardi P (1995) Lyapunov functions for a generalized Gause-type model. J Math Biol 33:816–828MathSciNetCrossRefGoogle Scholar
  2. Brauer F, Castillo-Chávez C (2001) Mathematical models in population biology and epidemiology, volume 40 of Texts in applied mathematics. Springer, New YorkCrossRefGoogle Scholar
  3. Duff GFD (1953) Limit cycles and rotated vector fields. Ann Math 57(1):15–31MathSciNetCrossRefGoogle Scholar
  4. Elton CS (1930) Animal ecology and evolution. Clarendon Press, OxfordGoogle Scholar
  5. Elton CS (1942) Voles, mice and lemmings. Clarendon Press, OxfordGoogle Scholar
  6. Gause GF (1934) The struggle for existence. The Williams & Wilkins, BaltimoreCrossRefGoogle Scholar
  7. González-Olivares E, Rojas-Palma A (2011) Multiple limit cycles in a Gause type predator-prey model with Holling type III functional response and Allee effect on prey. Bull Math Biol 73:1378–1397MathSciNetCrossRefGoogle Scholar
  8. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, BerlinCrossRefGoogle Scholar
  9. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42(4):599–653MathSciNetCrossRefGoogle Scholar
  10. Hewitt CG (1921) The conservation of the wild life of Canada. Charles Scribner’s Sons, New YorkGoogle Scholar
  11. Hirsch MW, Smale S, Devaney RL (2013) Differential equations, dynamical systems, and an introduction to chaos. Academic, OxfordzbMATHGoogle Scholar
  12. Hofbauer J, Sigmund K (1988) The theory of evolution and dynamical systems. Cambridge University Press, CambridgezbMATHGoogle Scholar
  13. Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  14. Holling CS (1959) Some characteristics of simple types of predation and parasitism. Can Entomol 91(7):385–398CrossRefGoogle Scholar
  15. Kooi BW, Boer MP, Kooijman SALM (1998) On the use of the logistic equation in models of food chains. Bull Math Biol 60:231–246CrossRefGoogle Scholar
  16. Kuang Y (1988) Nonuniqueness of limit cycles of Gause-type predator-prey systems. Appl Anal 29:269–287MathSciNetCrossRefGoogle Scholar
  17. Kuang Y (1989) Limit cycles in a chemostat related model. SIAM J Appl Math 49(6):1759–1767MathSciNetCrossRefGoogle Scholar
  18. Kuang Y, Freedman HI (1988) Uniqueness of limit cycles in Gause-type models of predator–prey systems. Math Biosci 88:67–84MathSciNetCrossRefGoogle Scholar
  19. LaSalle JP (1960) Some extensions of Lyapunovs second method. IRE Trans Circuit Theory CT-7:520–527CrossRefGoogle Scholar
  20. Lindström T (1993) Qualitative analysis of a predator-prey system with limit cycles. J Math Biol 31:541–561MathSciNetCrossRefGoogle Scholar
  21. Lindström T, Cheng Y (2015) Uniqueness of limit cycles for a limiting case of the chemostat: does it justify the use of logistic growth rates. Electron J Qual Theory Differ Equ 47:1–14. Scholar
  22. Lindström T, Cheng Y (2016) A Rosenzweig–MacArthur (1963) criterion for the chemostat. Sci World J 2016:1–6. Scholar
  23. Lotka AJ (1925) Elements of physical biology. Williams and Wilkins, BaltimorezbMATHGoogle Scholar
  24. Metz JAJ, Diekmann O (1986) The dynamics of physiologically structured populations. Springer, BerlinCrossRefGoogle Scholar
  25. Nisbet RM, Gurney WSC (1982) Modelling fluctuating populations. The Blackburn Press, CaldwellzbMATHGoogle Scholar
  26. Rosenzweig ML (1973) Exploitation in three throphic levels. Am Nat 107(954):275–294CrossRefGoogle Scholar
  27. Rosenzweig ML, MacArthur RH (1963) Graphical representation and stability conditions of predator–prey interactions. Am Nat 97:209–223CrossRefGoogle Scholar
  28. Smith HL, Waltman P (1995) The theory of the chemostat: dynamics of microbial competition. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  29. Volterra V (1926) Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem R Accad Natl Lincei 6(2):31–113zbMATHGoogle Scholar
  30. Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  31. Ye Y-Q et al (1986) Theory of limit cycles, 2nd edn. American Mathematical Society, ProvidenceGoogle Scholar
  32. Zhang Z-f (1986) Proof of the uniqueness theorem of limit cycles of generalized Liénard equations. Appl Anal 29:63–76zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsLinnaeus UniversityVäxjöSweden

Section editors and affiliations

  • Torsten Lindström
    • 1
  • Bharath Sriraman
    • 2
  1. 1.Linneaeus UniversityVäxjöSweden
  2. 2.Department of Mathematical SciencesThe University of MontanaMissoulaUSA

Personalised recommendations