Skip to main content

Stochastic stable population growth in integral projection models: theory and application

Journal of Mathematical Biology volume 54, pages 227–256 (2007)Cite this article

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

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

    Article  Google Scholar 

  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

    Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  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  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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

    Article  Google Scholar 

  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

    Google Scholar 

  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

    Google Scholar 

  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  Google Scholar 

  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

    Article  Google Scholar 

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

    MATH  Google Scholar 

  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

    Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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

    Google Scholar 

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

    Article  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Ecology and Evolutionary Biology, Cornell University, Ithaca, NY, USA

    Stephen P. Ellner

  2. Deparment of Animal and Plant Sciences, University of Sheffield, Sheffield, UK

    Mark Rees

Authors
  1. Stephen P. Ellner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark Rees
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephen P. Ellner.

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

Reprints 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

Download citation

  • 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

Mathematics Subject Classification (2000)

  • 92D25
  • 60H25
  • 37H15
  • 47B65