Stochastic stable population growth in integral projection models: theory and application
 Stephen P. Ellner,
 Mark Rees
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Stochastic matrix projection models are widely used to model age or stagestructured 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 lognormality 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 crossclassified 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 casestudy based on a 6year 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 agedependent 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 betweenyear variation in vital rates.
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 Title
 Stochastic stable population growth in integral projection models: theory and application
 Journal

Journal of Mathematical Biology
Volume 54, Issue 2 , pp 227256
 Cover Date
 20070201
 DOI
 10.1007/s0028500600448
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Stochastic demography
 Integral projection models
 Structured populations
 Hilbert’s projective metrix
 Onopordum illyricum
 92D25
 60H25
 37H15
 47B65
 Authors

 Stephen P. Ellner ^{(1)}
 Mark Rees ^{(2)}
 Author Affiliations

 1. Department of Ecology and Evolutionary Biology, Cornell University, Ithaca, NY, USA
 2. Deparment of Animal and Plant Sciences, University of Sheffield, Sheffield, UK