Journal of Mathematical Biology

, Volume 54, Issue 2, pp 227–256

Stochastic stable population growth in integral projection models: theory and application


DOI: 10.1007/s00285-006-0044-8

Cite this article as:
Ellner, S.P. & Rees, M. J. Math. Biol. (2007) 54: 227. doi:10.1007/s00285-006-0044-8


Stochastic matrix projection models are widely used to model age- or stage-structured populations with vital rates that fluctuate randomly over time. Practical applications of these models rest on qualitative properties such as the existence of a long term population growth rate, asymptotic log-normality of total population size, and weak ergodicity of population structure. We show here that these properties are shared by a general stochastic integral projection model, by using results in (Eveson in D. Phil. Thesis, University of Sussex, 1991, Eveson in Proc. Lond. Math. Soc. 70, 411–440, 1993) to extend the approach in (Lange and Holmes in J. Appl. Prob. 18, 325–344, 1981). Integral projection models allow individuals to be cross-classified by multiple attributes, either discrete or continuous, and allow the classification to change during the life cycle. These features are present in plant populations with size and age as important predictors of individual fate, populations with a persistent bank of dormant seeds or eggs, and animal species with complex life cycles. We also present a case-study based on a 6-year field study of the Illyrian thistle, Onopordum illyricum, to demonstrate how easily a stochastic integral model can be parameterized from field data and then applied using familiar matrix software and methods. Thistle demography is affected by multiple traits (size, age and a latent “quality” variable), which would be difficult to accomodate in a classical matrix model. We use the model to explore the evolution of size- and age-dependent flowering using an evolutionarily stable strategy (ESS) approach. We find close agreement between the observed flowering behavior and the predicted ESS from the stochastic model, whereas the ESS predicted from a deterministic version of the model is very different from observed flowering behavior. These results strongly suggest that the flowering strategy in O. illyricum is an adaptation to random between-year variation in vital rates.


Stochastic demography Integral projection models Structured populations Hilbert’s projective metrix Onopordum illyricum 

Mathematics Subject Classification (2000)

92D25 60H25 37H15 47B65 

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Ecology and Evolutionary BiologyCornell UniversityIthacaUSA
  2. 2.Deparment of Animal and Plant SciencesUniversity of SheffieldSheffieldUK