Probability Surveys

Limit theorems for discrete-time metapopulation models

F.M. Buckley and P.K. Pollett

Full-text: Open access

Abstract

We describe a class of one-dimensional chain binomial models of use in studying metapopulations (population networks). Limit theorems are established for time-inhomogeneous Markov chains that share the salient features of these models. We prove a law of large numbers, which can be used to identify an approximating deterministic trajectory, and a central limit theorem, which establishes that the scaled fluctuations about this trajectory have an approximating autoregressive structure.

Article information

Source
Probab. Surveys, Volume 7 (2010), 53-83.

Dates
First available in Project Euclid: 12 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.ps/1273670365

Digital Object Identifier
doi:10.1214/10-PS158

Mathematical Reviews number (MathSciNet)
MR2645217

Zentralblatt MATH identifier
1194.60024

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 92B05: General biology and biomathematics
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Citation

Buckley, F.M.; Pollett, P.K. Limit theorems for discrete-time metapopulation models. Probab. Surveys 7 (2010), 53--83. doi:10.1214/10-PS158. https://projecteuclid.org/euclid.ps/1273670365


Export citation

References

  • [1] Akçakaya, H. and Ginzburg, L. (1991). Ecological risk analysis for single and multiple populations. In Species Conservation: A Population Biological Approach, A. Seitz and V. Loescheke, Eds., Birkhauser, Basel, 78–87.
  • [2] Alonso, D. and McKane, A. (2002). Extinction dynamics in mainland-island metapopulations: an N-patch stochastic model. Bull. Math. Biol. 64, 913–958.
  • [3] Anderson, W. (1991). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer-Verlag, New York.
  • [4] Andrewartha, H. and Birch, L. (1954). The Distribution and Abundance of Animals. University of Chicago Press, Chicago, USA.
  • [5] Arrigoni, F. (2001). Approssimazione deterministica di modelli stocastici di metapopulazione. Bollettino dell’Unione Matematica Italiana, Serie VIII IV-A, 387–390.
  • [6] Arrigoni, F. (2003). A deterministic approximation of a stochastic metapopulation model. Adv. Appl. Probab. 3, 691–720.
  • [7] Arrigoni, F. and Pugliese, A. (2002). Limits of a multi-patch SIS epidemic model. J. Math. Biol. 45, 419–440.
  • [8] Asmussen, S. and Hering, H. (1983). Branching Processes. Birkhauser, Boston.
  • [9] Athreya, K. and Jagers, P. (1997). Classical and Modern Branching Processes. The IMA Volumes in Mathematics and its Applications, Vol. 84. Springer-Verlag, New York. Papers from the IMA Workshop held at the University of Minnesota, Minneapolis, MN, June 13–17, 1994.
  • [10] Athreya, K. and Ney, P. (1972). Branching Processes. Springer-Verlag, Berlin.
  • [11] Barbour, A. (1974). On a functional central limit theorem for Markov population processes. Adv. Appl. Probab. 6, 21–39.
  • [12] Barbour, A. (1976). Quasi-stationary distributions in Markov population processes. Adv. Appl. Probab. 8, 296–314.
  • [13] Barbour, A. (1980a). Density dependent Markov population processes. In Biological growth and spread: mathematical theories and applications (Proc. Conf., Heidelberg, 1979), Lecture Notes in Biomathematics, W. Jager, H. Rost, and P. Tautu, Eds. Vol. 38. Springer, Berlin, 36–49.
  • [14] Barbour, A. (1980b). Equilibrium distributions for Markov population processes. Adv. Appl. Probab. 12, 591–614.
  • [15] Barbour, A. and Pugliese, A. (2005). Asymptotic behaviour of a metapopulation model. Ann. Appl. Probab. 15, 1306–1338.
  • [16] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York.
  • [17] Brown, J. and Kodric-Brown, A. (1977). Turnover rates in insular biogeography: effect of immigration on extinction. Ecology 58, 445–449.
  • [18] Buckley, F. and Pollett, P. (2009). Analytical methods for a stochastic mainland-island metapopulation model. In Proceedings of the 18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation, R. Anderssen, R. Braddock, and L. Newham, Eds. Modelling and Simulation Society of Australia and New Zealand and International Association for Mathematics and Computers in Simulation, Canberra, Australia, 1767–1773.
  • [19] Daley, D. and Gani, J. (1999). Epidemic Modelling: an Introduction. Cambridge Studies in Mathematical Biology, Vol. 15. Cambridge University Press, Cambridge.
  • [20] Darling, R. and Norris, J. (2008). Differential equation approximations for Markov chains. Probab. Surv. 5, 37–79.
  • [21] Darroch, J. and Seneta, E. (1965). On quasi-stationary distributions in absorbing discrete-time Markov chains. J. Appl. Probab. 2, 88–100.
  • [22] Day, J. and Possingham, H. (1995). A stochastic metapopulation model with variable patch size and position. Theoret. Pop. Biol. 48, 333–360.
  • [23] Feller, W. (1939). Die grundlagen der volterraschen theorie des kampfes ums dasein in wahrscheinlichkeitsteoretischer behandlung. Acta Biotheoretica 5, 11–40.
  • [24] Grimmett, G. and Stirzaker, D. (2001). Probability and Random Processes, 3rd ed. Oxford University Press, New York.
  • [25] Hanski, I. (1994). A practical model for metapopulation dynamics. J. Animal Ecol. 63, 151–162.
  • [26] Hanski, I. and Gilpin, M. (1991). Metapopulation dynamics: brief history and conceptual domain. Biol. J. Linnean Soc. 42, 3–16.
  • [27] Hanski, I. and Simberloff, D. (1997). The metapopulation approach, its history, conceptual domain, and application to conservation. In Metapopulation Biology, I. Hanski and M. Gilpin, Eds. Academic Press, San Diego, CA, USA, 5–26.
  • [28] Harris, T. (1963). The Theory of Branching Processes. Springer-Verlag, Berlin.
  • [29] Hill, M. and Caswell, H. (2001). The effects of habitat destruction in finite landscapes: a chain-binomial metapopulation model. OIKOS 93, 321–331.
  • [30] Karr, A. (1975). Weak convergence of a sequence of Markov chains. Probability Theory and Related Fields 33, 41–48.
  • [31] Kelly, F. (1979). Reversibility and Stochastic Networks. Wiley, Chichester.
  • [32] Klebaner, F. (1993). Population-dependent branching processes with a threshold. Stochastic Process. Appl. 46, 115–127.
  • [33] Klebaner, F. and Nerman, O. (1994). Autoregressive approximation in branching processes with a threshold. Stochastic Process. Appl. 51, 1–7.
  • [34] Klok, C. and De Roos, A. (1998). Effects of habitat size and quality on equilibrium density and extinction time of Sorex araneus populations. J. Animal Ecol. 67, 195–209.
  • [35] Kurtz, T. (1970). Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 7, 49–58.
  • [36] Kurtz, T. (1971). Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab. 8, 344–356.
  • [37] Kurtz, T. (1972). The relationship between stochastic and deterministic models in chemical reactions. J. Chem. Phys. 57, 2976–2978.
  • [38] Kurtz, T. (1976). Limit theorems and diffusion approximations for density dependent Markov chains. Math. Prog. Study 5, 67–78.
  • [39] Kurtz, T. (1978). Strong approximation theorems for density dependent Markov chains. Stochastic Process. Appl. 6, 223–240.
  • [40] Levins, R. (1969). Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entom. Soc. Amer. 15, 237–240.
  • [41] Levins, R. (1970). Extinction. In Some Mathematical Questions in Biology, M. Gerstenhaber, Ed. American Mathematical Society, Providence, RI, USA, 75–107.
  • [42] MacArthur, R. and Wilson, E. (1967). The Theory of Island Biogeography. Princeton University Press, Princeton, NJ, USA.
  • [43] McKenzie, E. (1985). Some simple models for discrete variate time series. Water Resources Bull. 21, 645–650.
  • [44] McKenzie, E. (2003). Discrete variate time series. In Handbook of Statistics, C. Rao and D. Shanbhag, Eds., Elsevier Science, Amsterdam, 573–606.
  • [45] McVinish, R. and Pollett, P. (2009). Limits of large metapopulations with patch dependent extinction probabilities. Submitted for publication.
  • [46] Moilanen, A. (1999). Patch occupancy models of metapopulation dynamics: efficient parameter estimation using implicit statistical inference. Ecology 80, 1031–1043.
  • [47] Nåsell, I. (1996). The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Probab. 28, 895–932.
  • [48] Nåsell, I. (1999). On the quasi-stationary distribution of the stochastic logistic epidemic. Math. Biosci. 156, 21–40.
  • [49] Norman, M. (1972). Markov Processes and Learning Models. Academic Press, New York.
  • [50] Ovaskainen, O. and Hanski, I. (2004). Metapopulation dynamics in highly fragmented landscapes. In Ecology, Genetics, and Evolution in Metapopulations, I. Hanski and O. Gaggiotti, Eds. Academic Press, San Diego, CA, USA, 73–103.
  • [51] Pearl, R. (1925). The Biology of Population Growth. Alfred A. Knopf, New York.
  • [52] Pearl, R. (1927). The growth of populations. Quart. Rev. Biol. 2, 532–548.
  • [53] Pearl, R. and Reed, L. (1920). On the rate of growth of population of the united states since 1790 and its mathematical representation. Proc. Nat. Academy Sci. 6, 275–288.
  • [54] Pollett, P. (2001). Diffusion approximations for ecological models. In Proceedings of the International Congress on Modelling and Simulation, F. Ghassemi, Ed. Vol. 2. Modelling and Simulation Society of Australia and New Zealand, Canberra, Australia, 843–848.
  • [55] Renshaw, E. (1991). Modelling Biological Populations in Space and Time. Cambridge Studies in Mathematical Biology, Vol. 11. Cambridge University Press, Cambridge.
  • [56] Ross, J. (2006a). A stochastic metapopulation model accounting for habitat dynamics. J. Math. Biol. 52, 788–806.
  • [57] Ross, J. (2006b). Stochastic models for mainland-island metapopulations in static and dynamic landscapes. Bull. Math. Biol. 68, 417–449.
  • [58] Ross, J., Pagendam, D., and Pollett, P. (2009). On parameter estimation in population models II: multi-dimensional processes and transient dynamics. Theoret. Pop. Biol. 75, 123–132.
  • [59] Ross, J., Taimre, T., and Pollett, P. (2006). On parameter estimation in population models. Theoret. Pop. Biol. 70, 498–510.
  • [60] Rout, T., Hauser, C., and Possingham, H. (2007). Minimise long-term loss or maximise short-term gain? Optimal translocation strategies for threatened species. Ecological Modelling 201, 67–74.
  • [61] Schach, S. (1971). Weak convergence results for a class of multivariate Markov processes. Ann. Math. Statist. 42, 451–465.
  • [62] Tenhumberg, B., Tyre, A., Shea, K., and Possingham, H. (2004). Linking wild and captive populations to maximise species persistence: optimal translocation strategies. Conservation Biology 18, 1304–1314.
  • [63] Verhulst, P. (1838). Notice sur la loi que la population suit dans son accroisement. Corr. Math. et Phys. X, 113–121.
  • [64] Weiss, G. and Dishon, M. (1971). On the asymptotic behaviour of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261–265.
  • [65] Weiss, C. (2009a) A new class of autoregressive models for time series of binomial counts. Comm. Statist. Theory Meth. 38, 447–460.
  • [66] Weiss, C. (2009b) Monitoring correlated processes with binomial margin–als. J. Appl. Statist. 36, 399–414.
  • [67] Wright, S. (1940). Breeding structure of populations in relation to speciation. American Naturalist 74, 232–248.
  • [68] Zhang, Y., Liu, L., and Xu, R. (2009). The effect of migration on the viability, dynamics and structure of two coexisting metapopulations. Ecological Modelling 220, 272–282.