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).

References 1.

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 2.

Birkhoff G. (1957) Extensions of Jentzch’s Theorem. Trans. Am. Math. Soc. 85, 219–227

CrossRef MathSciNet 3.

Caswell H. (2001) Matrix Population Models. Sinauer, Sunderland

4.

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

CrossRef 5.

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 6.

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 7.

Cohen J.E. (1977) Ergodicity of age structure in populations with Markovian vital rates. 2. General states. Adv. Appl. Prob. 9, 18–37

CrossRef 8.

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

9.

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 MathSciNet 10.

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 MathSciNet 11.

Easterling, M.R.: The integral projection model: theory, analysis and application. Doctoral thesis, North Carolina State University, Raleigh (1998)

12.

Easterling M.R., Ellner S.P., Dixon P.M. (2000) Size-specific sensitivity: applying a new structured population model. Ecology 81, 694–708

CrossRef 13.

Ellner S. (1984) Asymptotic behavior of some stochastic difference equation population models. J. Math. Biol. 19, 169–200

CrossRef MathSciNet 14.

Ellner S.P., Guckenheimer J. (2006) Dynamics Models in Biology. Princeton University Press, Princeton

15.

Ellner S.P., Rees M. (2006) Integral projection models for species with complex demography. Am. Nat. 167, 410–428

CrossRef 16.

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)

17.

Eveson S.P. (1993) Hilberts’ projective metric and the spectral properties of positive linear operators. Proc. Lond. Math. Soc. 70, 411–440

MathSciNet 18.

Fieberg J., Ellner S.P. (2001) Stochastic matrix models for conservation and management: a comparative review of methods. Ecol. Lett. 4, 244–266

CrossRef 19.

Furstenburg H., Kesten H. (1960) Products of random matrices. Ann. Math. Stat. 31, 457–469

20.

Grafen A. (2006) A theory of Fisher’s reproductive value. J. Math. Biol. 53, 15–60

CrossRef MathSciNet 21.

Hall P., Heyde C.C. (1980) Martingale limit theory and its applications. Academic, New York

22.

Halley J.M. (1996) Ecology,evolution, and 1/f-noise. Trends Ecol. Evol. 11, 33–37

CrossRef 23.

Halley J.M., Inchausti P. (2004) The increasing importance of 1/f-noises as models of ecological variability. Fluct. Noise. Lett. 4, R1–R26

CrossRef 24.

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

MathSciNet 25.

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 MathSciNet 26.

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 27.

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

28.

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

29.

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

MathSciNet 30.

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 31.

Karlin S., Taylor H.M. (1975) A First Course in Stochastic Processes, 2nd ed. Academic, New York

MATH 32.

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

33.

Kifer Y. (1986) Ergodic Theory of Random Transformations. Birkhäuser, Boston

MATH 34.

Lange K, Holmes W. (1981) Stochastic stable population growth. J. Appl. Prob. 18, 325–344

CrossRef MathSciNet 35.

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

36.

Menges E.S. (2000) Population viability analyses in plants: challenges and opportunities. Trends Ecol. Evol. 15, 51–56

CrossRef 37.

Meyn S.P., Tweedie R.L. (1993) Markov Chains and Stochastic Stability. Springer, Berlin Heidelberg New York

MATH 38.

Morris W., Doak D. (2002) Quantitative Conservation Biology. Sinauer, Sunderland

39.

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)

40.

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 41.

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

CrossRef 42.

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 43.

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 44.

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

45.

McCulloch C.E., Searle S.R. (2001) Generalized, Linear, and Mixed Models. Wiley, New York

MATH 46.

Shea K., Kelly D. (1998) Estimating biocontrol agent impact with matrix models: Carduus nutans in New Zealand. Ecol. Appl. 8, 824–832

47.

Shea K., Kelly D., Sheppard A.W., Woodburn T.L. (2005) Context-dependent biological control of an invasive thistle. Ecology 86, 3174–3181

48.

Tuljapurkar S. (1990) Population Dynamics in Variable Environments. Springer, Berlin Heidelberg New york

MATH 49.

Tuljapurkar S., Wiener P. (2000) Escape in time: stay young or age gracefully? Ecol. Model. 133, 143–159

CrossRef 50.

Tuljapurkar S., Haridas C.V. (2006) Temporal autocorrelation and stochastic population growth. Ecol. Lett. 9, 327–337

CrossRef