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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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
Birkhoff G. (1957) Extensions of Jentzch’s Theorem. Trans. Am. Math. Soc. 85, 219–227
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 size-structured integral projection model. Proc. R. Soc. B 270, 1829–1839
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
Cohen J.E. (1976) Ergodicity of age structure in populations with Markovian vital rates. I. Countable states. J. Am. Stat. Assoc. 71, 335–339
Cohen J.E. (1977) Ergodicity of age structure in populations with Markovian vital rates. 2. General states. Adv. Appl. Prob. 9, 18–37
Crowder L.B., Crouse D.T., Heppell S.S., Martin T.H. (1994) Predicting the impact of turtle excluder devices on loggerhead sea-turtle 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
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
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) Size-specific sensitivity: applying a new structured population model. Ecology 81, 694–708
Ellner S. (1984) Asymptotic behavior of some stochastic difference equation population models. J. Math. Biol. 19, 169–200
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
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
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
Hall P., Heyde C.C. (1980) Martingale limit theory and its applications. Academic, New York
Halley J.M. (1996) Ecology,evolution, and 1/f-noise. Trends Ecol. Evol. 11, 33–37
Halley J.M., Inchausti P. (2004) The increasing importance of 1/f-noises as models of ecological variability. Fluct. Noise. Lett. 4, R1–R26
Hardin D.P., Takáč P., Webb G.F. (1988) Asymptotic properties of a continuous-space 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
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
Heppell S.S., Crowder L.B., Crouse D.T. (1996) Models to evaluate headstarting as a management tool for long-lived 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. 148-168
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
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
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
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 3-900051-07-0, URL http://www.R-project.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
Rees M., Sheppard A., Briese D. Mangel M. (1999) Evolution of size-dependent flowering in Onopordum illyricim: a quantitative assessment of the role of stochastic selection pressures. Am. Nat. 154, 628–651
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
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
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) Context-dependent 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
Tuljapurkar S., Haridas C.V. (2006) Temporal autocorrelation and stochastic population growth. Ecol. Lett. 9, 327–337
Author information
Authors and Affiliations
Corresponding author
Additional information
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).
Rights and permissions
About this article
Cite this article
Ellner, S.P., Rees, M. Stochastic stable population growth in integral projection models: theory and application. J. Math. Biol. 54, 227–256 (2007). https://doi.org/10.1007/s00285-006-0044-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-006-0044-8
Keywords
- Stochastic demography
- Integral projection models
- Structured populations
- Hilbert’s projective metrix
- Onopordum illyricum