Journal of Mathematical Biology

, Volume 54, Issue 2, pp 227–256 | Cite as

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

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.

Keywords

Stochastic demography Integral projection models Structured populations Hilbert’s projective metrix Onopordum illyricum 

Mathematics Subject Classification (2000)

92D25 60H25 37H15 47B65 

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References

  1. 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, 115–139CrossRefGoogle Scholar
  2. 2.
    Birkhoff G. (1957) Extensions of Jentzch’s Theorem. Trans. Am. Math. Soc. 85, 219–227CrossRefMathSciNetGoogle Scholar
  3. 3.
    Caswell H. (2001) Matrix Population Models. Sinauer, SunderlandGoogle Scholar
  4. 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, 1829–1839CrossRefGoogle Scholar
  5. 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, 425–434CrossRefGoogle Scholar
  6. 6.
    Cohen J.E. (1976) Ergodicity of age structure in populations with Markovian vital rates. I. Countable states. J. Am. Stat. Assoc. 71, 335–339CrossRefGoogle Scholar
  7. 7.
    Cohen J.E. (1977) Ergodicity of age structure in populations with Markovian vital rates. 2. General states. Adv. Appl. Prob. 9, 18–37CrossRefGoogle Scholar
  8. 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, 437–445Google Scholar
  9. 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, 349–388CrossRefMathSciNetGoogle Scholar
  10. 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, 157–189CrossRefMathSciNetGoogle Scholar
  11. 11.
    Easterling, M.R.: The integral projection model: theory, analysis and application. Doctoral thesis, North Carolina State University, Raleigh (1998)Google Scholar
  12. 12.
    Easterling M.R., Ellner S.P., Dixon P.M. (2000) Size-specific sensitivity: applying a new structured population model. Ecology 81, 694–708CrossRefGoogle Scholar
  13. 13.
    Ellner S. (1984) Asymptotic behavior of some stochastic difference equation population models. J. Math. Biol. 19, 169–200CrossRefMathSciNetGoogle Scholar
  14. 14.
    Ellner S.P., Guckenheimer J. (2006) Dynamics Models in Biology. Princeton University Press, PrincetonGoogle Scholar
  15. 15.
    Ellner S.P., Rees M. (2006) Integral projection models for species with complex demography. Am. Nat. 167, 410–428CrossRefGoogle Scholar
  16. 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)Google Scholar
  17. 17.
    Eveson S.P. (1993) Hilberts’ projective metric and the spectral properties of positive linear operators. Proc. Lond. Math. Soc. 70, 411–440MathSciNetGoogle Scholar
  18. 18.
    Fieberg J., Ellner S.P. (2001) Stochastic matrix models for conservation and management: a comparative review of methods. Ecol. Lett. 4, 244–266CrossRefGoogle Scholar
  19. 19.
    Furstenburg H., Kesten H. (1960) Products of random matrices. Ann. Math. Stat. 31, 457–469Google Scholar
  20. 20.
    Grafen A. (2006) A theory of Fisher’s reproductive value. J. Math. Biol. 53, 15–60CrossRefMathSciNetGoogle Scholar
  21. 21.
    Hall P., Heyde C.C. (1980) Martingale limit theory and its applications. Academic, New YorkGoogle Scholar
  22. 22.
    Halley J.M. (1996) Ecology,evolution, and 1/f-noise. Trends Ecol. Evol. 11, 33–37CrossRefGoogle Scholar
  23. 23.
    Halley J.M., Inchausti P. (2004) The increasing importance of 1/f-noises as models of ecological variability. Fluct. Noise. Lett. 4, R1–R26CrossRefGoogle Scholar
  24. 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, 361–374MathSciNetGoogle Scholar
  25. 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, 1396–1423CrossRefMathSciNetGoogle Scholar
  26. 26.
    Hardin D.P., Takáč P., Webb G.F. (1990) Dispersion population models discrete in time and continuous in space. J. Math. Biol. 28, 406–409CrossRefGoogle Scholar
  27. 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, 556–565Google Scholar
  28. 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. 148-168Google Scholar
  29. 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, 565–575MathSciNetGoogle Scholar
  30. 30.
    Kareiva P., Marvier M., McClure M. (2000) Recovery and management options for spring/summer Chinook salmon in the Columbia River basin. Science 290, 977–979CrossRefGoogle Scholar
  31. 31.
    Karlin S., Taylor H.M. (1975) A First Course in Stochastic Processes, 2nd ed. Academic, New YorkMATHGoogle Scholar
  32. 32.
    Kaye T.N., Pyke D.A. (1975) The effect of stochastic technique on estimates of population viability from transition matrix models. Ecology 84, 1464–1476Google Scholar
  33. 33.
    Kifer Y. (1986) Ergodic Theory of Random Transformations. Birkhäuser, BostonMATHGoogle Scholar
  34. 34.
    Lange K, Holmes W. (1981) Stochastic stable population growth. J. Appl. Prob. 18, 325–344CrossRefMathSciNetGoogle Scholar
  35. 35.
    McEvoy P.B., Coombs E.M. (1999) Biological control of plant invaders: regional patterns, field experiments, and structured population models. Ecol. Appl. 9, 387–401Google Scholar
  36. 36.
    Menges E.S. (2000) Population viability analyses in plants: challenges and opportunities. Trends Ecol. Evol. 15, 51–56CrossRefGoogle Scholar
  37. 37.
    Meyn S.P., Tweedie R.L. (1993) Markov Chains and Stochastic Stability. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  38. 38.
    Morris W., Doak D. (2002) Quantitative Conservation Biology. Sinauer, SunderlandGoogle Scholar
  39. 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)Google Scholar
  40. 40.
    Ramula S., Kehtilä K. (2005) Importance of correlations among matrix entries in stochastic models in relation to number of transition matrices. Oikos 111, 9–18CrossRefGoogle Scholar
  41. 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, 628–651CrossRefGoogle Scholar
  42. 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, 471–475CrossRefGoogle Scholar
  43. 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–E71CrossRefGoogle Scholar
  44. 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, 453–465Google Scholar
  45. 45.
    McCulloch C.E., Searle S.R. (2001) Generalized, Linear, and Mixed Models. Wiley, New YorkMATHGoogle Scholar
  46. 46.
    Shea K., Kelly D. (1998) Estimating biocontrol agent impact with matrix models: Carduus nutans in New Zealand. Ecol. Appl. 8, 824–832Google Scholar
  47. 47.
    Shea K., Kelly D., Sheppard A.W., Woodburn T.L. (2005) Context-dependent biological control of an invasive thistle. Ecology 86, 3174–3181Google Scholar
  48. 48.
    Tuljapurkar S. (1990) Population Dynamics in Variable Environments. Springer, Berlin Heidelberg New yorkMATHGoogle Scholar
  49. 49.
    Tuljapurkar S., Wiener P. (2000) Escape in time: stay young or age gracefully? Ecol. Model. 133, 143–159CrossRefGoogle Scholar
  50. 50.
    Tuljapurkar S., Haridas C.V. (2006) Temporal autocorrelation and stochastic population growth. Ecol. Lett. 9, 327–337CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Ecology and Evolutionary BiologyCornell UniversityIthacaUSA
  2. 2.Deparment of Animal and Plant SciencesUniversity of SheffieldSheffieldUK

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