Abstract
A generalization of the well-known Levins’ model of metapopulations is studied. The generalization consists of (i) the introduction of immigration from a mainland, and (ii) assuming the dynamics is stochastic, rather than deterministic. A master equation, for the probability that n of the patches are occupied, is derived and the stationary probability P s (n), together with the mean and higher moments in the stationary state, determined. The time-dependence of the probability distribution is also studied: through a Gaussian approximation for general n when the boundary at n = 0 has little effect, and by calculating P(0, t), the probability that no patches are occupied at time t, by using a linearization procedure. These analytic calculations are supplemented by carrying out numerical solutions of the master equation and simulations of the stochastic process. The various approaches are in very good agreement with each other. This allows us to use the forms for P s 0) and P(0, t) in the linearization approximation as a basis for calculating the mean time for a metapopulation to become extinct. We give an analytical expression for the mean time to extinction derived within a mean field approach. We devise a simple method to apply our mean field approach even to complex patch networks in realistic model metapopulations. After studying two spatially extended versions of this nonspatial metapopulation model—a lattice metapopulation model and a spatially realistic model—we conclude that our analytical formula for the mean extinction time is generally applicable to those metapopulations which are really endangered, where extinction dynamics dominates over local colonization processes. The time evolution and, in particular, the scope of our analytical results, are studied by comparing these different models with the analytical approach for various values of the parameters: the rates of immigration from the mainland, the rates of colonization and extinction, and the number of patches making up the metapopulation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M. and I. A. Stegun (Eds) (1965). Handbook of Mathematical Functions, New York: Dover.
Ball, F. and I. Nasell (1994). The shape of the size distribution of an epidemic in a finite population. Math. Biosci. 123, 167–181.
Bascompte, J. and R. V. Solé (Eds) (1997). Modeling Spatiotemporal Dynamics in Ecology, Berlin: Springer and Landes Bioscience.
Dieckmann, U., R. Law and J. A. J. Metz (Eds) (2000). The Geometry of Ecological Interactions: Simplifying Spatial Complexity, Cambridge Studies in Adaptive Dynamics 1, Cambridge: Cambridge University Press.
Durrett, R. (1999). Stochastic spatial models. SIAM Rev. 41, 677–718.
Durrett, R. and S. Levin (1994). The importance of being discrete (and spatial). Theor. Pop. Biol. 46, 363–394.
Gardiner, C. W. (1985). Handbook of Stochastic Processes, Berlin: Springer.
Gotelli, N. J. and W. G. Kelley (1993). A general model of metapopulation dynamics. OIKOS 68, 36–44.
Gurney, W. S. C. and R. M. Nisbet (1978). Single-species population fluctuations in patchy environments. Am. Nat. 112, 1075–1090.
Hanski, I. (1994a). A practical model of metapopulation dynamics. J. Anim. Ecol. 63, 151–162.
Hanski, I. (1994b). Spatial scale, patchiness and population dynamics on land. Phil. Trans. R. Soc. London B 343, 19–25.
Hanski, I. (1999). Metapopulation Ecology, Oxford: Oxford University Press.
Hanski, I. and O. Ovaskainen (2000). The metapopulation capacity of a fragmented landscape. Nature 404, 755–758.
Harrison, S. (1991). Local extinction in a metapopulation context: an empirical evaluation. Biol. J. Linnean Soc. 42, 73–88.
Hubbell, S. P. (2001). The Unified Neutral Theory of Biodiversity and Biogeography, Princeton: Princeton University Press.
Kendall, D. G. (1948). On some modes of population growth leading to R. A. Fisher’s logarithmic series distribution. Biometrika 35, 6–15.
Levins, R. (1969). Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entomol. Soc. Am. 15, 227–240.
Levins, R. (1970). Extinction. Lecture Notes Math. 2, 75–107.
MacArthur, R. H. and E. O. Wilson (1967). The Theory of Island Biogeography, Princeton: Princeton University Press.
Moilanen, A. (1999). Patch occupancy models of metapopulation dynamics: efficient parameter estimation using implicit statistical inference. Ecology 80, 1031–1043.
Mollison, D. (1977). Spatial contact models for ecological and epidemic spread. J. R. Stat. Soc. B 39, 283–326.
Naeem, S. et al. (1999). Biodiversity and ecosystem functioning: maintaining natural life support processes. Issues Ecol. 4, 2–11.
Nasell, I. (1996). The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Prob. 28, 895–932.
Nisbet, R. M. and W. S. C. Gurney (1982). Modelling Fluctuating Populations, New York: Wiley.
Press, W. H., S. A. Teukolsky, W. T. Vetterling and B. P. Flannery (1992). Numerical Recipes in C, Cambridge: Cambridge University Press.
Renshaw, E. (1991). Modelling Biological Populations in Space and Time, Cambridge Studies in Mathematical Biology 11, Cambridge: Cambridge University Press.
Snyder, R. E. and R. M. Nisbet (2000). Spatial structure and fluctuations in the contact process and related models. Bull. Math. Biol. 62, 959–975.
Tilman, D. and P. Kareiva (Eds) (1997). Spatial Ecology. The Role of Space in Population Dynamics and Interspecific Interactions, Monographs in Population Biology 30, Princeton University Press.
van Kampen, N. G. (1981). Stochastic Processes in Physics and Chemistry, Amsterdam: Elsevier.
Vitousek, P. M. (1994). Beyond global warming: ecology and global change. Ecology 75, 1861–1876.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Alonso, D., McKane, A. Extinction dynamics in mainland-island metapopulations: An N-patch stochastic model. Bull. Math. Biol. 64, 913–958 (2002). https://doi.org/10.1006/bulm.2002.0307
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1006/bulm.2002.0307