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.
 Benton T.G., Grant A. (1996) How to keep fit in the real world: elasticity analyses and selection pressures on life histories in a variable environment. Am. Nat. 147, 115–139 CrossRef
 Birkhoff G. (1957) Extensions of Jentzch’s Theorem. Trans. Am. Math. Soc. 85, 219–227 CrossRef
 Caswell H. (2001) Matrix Population Models. Sinauer, Sunderland
 Childs D.Z., Rees M., Rose K.E., Grubb P.J., Ellner S.P. (2003) Evolution of complex flowering strategies: an age and sizestructured integral projection model. Proc. R. Soc. B 270, 1829–1839 CrossRef
 Childs D.Z., Rees M., Rose K.E., Grubb P.J., Ellner S.P. (2004) Evolution of size dependent flowering in a variable environment: construction and analysis of a stochastic integral projection model. Proc. R. Soc. B 271, 425–434 CrossRef
 Cohen J.E. (1976) Ergodicity of age structure in populations with Markovian vital rates. I. Countable states. J. Am. Stat. Assoc. 71, 335–339 CrossRef
 Cohen J.E. (1977) Ergodicity of age structure in populations with Markovian vital rates. 2. General states. Adv. Appl. Prob. 9, 18–37 CrossRef
 Crowder L.B., Crouse D.T., Heppell S.S., Martin T.H. (1994) Predicting the impact of turtle excluder devices on loggerhead seaturtle populations. Ecol. Appl. 4, 437–445
 Diekmann O., Gyllenberg M, Metz J.A.J., Thieme H.R. (1998) On the formulation and analysis of general deterministic structured population models I. Linear Theory. J. Math. Biol. 36, 349–388 CrossRef
 Diekmann O., Gyllenberg M., Huang H., Kirkilionis M., Metz J.A.J., Thieme H.R. (2001) On the formulation and analysis of general deterministic structured population models II. Nonlinear Theory. J. Math. Biol. 43, 157–189 CrossRef
 Easterling, M.R.: The integral projection model: theory, analysis and application. Doctoral thesis, North Carolina State University, Raleigh (1998)
 Easterling M.R., Ellner S.P., Dixon P.M. (2000) Sizespecific sensitivity: applying a new structured population model. Ecology 81, 694–708 CrossRef
 Ellner S. (1984) Asymptotic behavior of some stochastic difference equation population models. J. Math. Biol. 19, 169–200 CrossRef
 Ellner S.P., Guckenheimer J. (2006) Dynamics Models in Biology. Princeton University Press, Princeton
 Ellner S.P., Rees M. (2006) Integral projection models for species with complex demography. Am. Nat. 167, 410–428 CrossRef
 Eveson, S.P.: Theory and application of Hilbert’s projective metric to linear and nonlinear problems in positive operator theory. D. Phil. Thesis, University of Sussex (1991)
 Eveson S.P. (1993) Hilberts’ projective metric and the spectral properties of positive linear operators. Proc. Lond. Math. Soc. 70, 411–440
 Fieberg J., Ellner S.P. (2001) Stochastic matrix models for conservation and management: a comparative review of methods. Ecol. Lett. 4, 244–266 CrossRef
 Furstenburg H., Kesten H. (1960) Products of random matrices. Ann. Math. Stat. 31, 457–469
 Grafen A. (2006) A theory of Fisher’s reproductive value. J. Math. Biol. 53, 15–60 CrossRef
 Hall P., Heyde C.C. (1980) Martingale limit theory and its applications. Academic, New York
 Halley J.M. (1996) Ecology,evolution, and 1/fnoise. Trends Ecol. Evol. 11, 33–37 CrossRef
 Halley J.M., Inchausti P. (2004) The increasing importance of 1/fnoises as models of ecological variability. Fluct. Noise. Lett. 4, R1–R26 CrossRef
 Hardin D.P., Takáč P., Webb G.F. (1988) Asymptotic properties of a continuousspace discrete time population model in a random environment. J. Math. Biol. 26, 361–374
 Hardin D.P., Takáč P., Webb G.F. (1988) A comparison of dispersal strategies for survival of spatially heterogeneous populations. SIAM J. Appl. Math. 48, 1396–1423 CrossRef
 Hardin D.P., Takáč P., Webb G.F. (1990) Dispersion population models discrete in time and continuous in space. J. Math. Biol. 28, 406–409 CrossRef
 Heppell S.S., Crowder L.B., Crouse D.T. (1996) Models to evaluate headstarting as a management tool for longlived turtles Ecol. Appl. 6, 556–565
 Heppell S.S., Crouse D.R, Crowder L.B. (1998) Using matrix models to focus research and management efforts in conservation. In: Ferson S., Burgman M. (eds) Quantitative Methods for Conservation Biology. Springer, Berlin Heidelberg New York, pp. 148168
 Ishitani H. (1977) A Central Limit Theorem for the subadditive process and its application to products of random matrices. Publ Res Inst Math Sci Kyoto University 12, 565–575
 Kareiva P., Marvier M., McClure M. (2000) Recovery and management options for spring/summer Chinook salmon in the Columbia River basin. Science 290, 977–979 CrossRef
 Karlin S., Taylor H.M. (1975) A First Course in Stochastic Processes, 2nd ed. Academic, New York
 Kaye T.N., Pyke D.A. (1975) The effect of stochastic technique on estimates of population viability from transition matrix models. Ecology 84, 1464–1476
 Kifer Y. (1986) Ergodic Theory of Random Transformations. Birkhäuser, Boston
 Lange K, Holmes W. (1981) Stochastic stable population growth. J. Appl. Prob. 18, 325–344 CrossRef
 McEvoy P.B., Coombs E.M. (1999) Biological control of plant invaders: regional patterns, field experiments, and structured population models. Ecol. Appl. 9, 387–401
 Menges E.S. (2000) Population viability analyses in plants: challenges and opportunities. Trends Ecol. Evol. 15, 51–56 CrossRef
 Meyn S.P., Tweedie R.L. (1993) Markov Chains and Stochastic Stability. Springer, Berlin Heidelberg New York
 Morris W., Doak D. (2002) Quantitative Conservation Biology. Sinauer, Sunderland
 R Development Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. ISBN 3900051070, URL http://www.Rproject.org (2005)
 Ramula S., Kehtilä K. (2005) Importance of correlations among matrix entries in stochastic models in relation to number of transition matrices. Oikos 111, 9–18 CrossRef
 Rees M., Sheppard A., Briese D. Mangel M. (1999) Evolution of sizedependent flowering in Onopordum illyricim: a quantitative assessment of the role of stochastic selection pressures. Am. Nat. 154, 628–651 CrossRef
 Rees M., Childs D.Z., Rose K.E., Grubb P.J. (2004) Evolution of size dependent flowering in a variable environment: partitioning the effects of fluctuating selection. Proc. R. Soc. B 271, 471–475 CrossRef
 Rees M., Childs D.Z., Metcalf J.C., Rose K.E., Sheppard A.W., Grubb P.J. (2006) Seed dormancy and delayed flowering in monocarpic plants: selective interactions in a stochastic environment. Am. Nat. 168, E53–E71 CrossRef
 Rose K.E., Louda S., Rees M. (2005) Demographic and evolutionary impacts of native and invasive insect herbivores: a case study with Platte thistle, Cirsium canescens. Ecology 86, 453–465
 McCulloch C.E., Searle S.R. (2001) Generalized, Linear, and Mixed Models. Wiley, New York
 Shea K., Kelly D. (1998) Estimating biocontrol agent impact with matrix models: Carduus nutans in New Zealand. Ecol. Appl. 8, 824–832
 Shea K., Kelly D., Sheppard A.W., Woodburn T.L. (2005) Contextdependent biological control of an invasive thistle. Ecology 86, 3174–3181
 Tuljapurkar S. (1990) Population Dynamics in Variable Environments. Springer, Berlin Heidelberg New york
 Tuljapurkar S., Wiener P. (2000) Escape in time: stay young or age gracefully? Ecol. Model. 133, 143–159 CrossRef
 Tuljapurkar S., Haridas C.V. (2006) Temporal autocorrelation and stochastic population growth. Ecol. Lett. 9, 327–337 CrossRef
 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
 Industry Sectors
 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