Stochastic stable population growth in integral projection models: theory and application Authors Stephen P. Ellner Department of Ecology and Evolutionary Biology Cornell University Mark Rees Deparment of Animal and Plant Sciences University of Sheffield Article

First Online: 23 November 2006 Received: 08 March 2006 Revised: 11 October 2006 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 Abstract 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.

Keywords Stochastic demography Integral projection models Structured populations Hilbert’s projective metrix Onopordum illyricum Research supported by NSF grant OCE 0326705 in the NSF/NIH Ecology of Infectious Diseases program and the Cornell College of Arts and Sciences (SPE), and NERC grant NER/A/S/2002/00940 (MR).

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