Bulletin of Mathematical Biology

, Volume 64, Issue 5, pp 913–958

# Extinction dynamics in mainland-island metapopulations: An N-patch stochastic model

• David Alonso
• Alan McKane
Article

## 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.

## Keywords

Master Equation Colonization Rate Extinction Rate Patch Area Extinction Dynamic
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Abramowitz, M. and I. A. Stegun (Eds) (1965). Handbook of Mathematical Functions, New York: Dover.Google Scholar
2. Ball, F. and I. Nasell (1994). The shape of the size distribution of an epidemic in a finite population. Math. Biosci. 123, 167–181.
3. Bascompte, J. and R. V. Solé (Eds) (1997). Modeling Spatiotemporal Dynamics in Ecology, Berlin: Springer and Landes Bioscience.Google Scholar
4. 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.Google Scholar
5. Durrett, R. (1999). Stochastic spatial models. SIAM Rev. 41, 677–718.
6. Durrett, R. and S. Levin (1994). The importance of being discrete (and spatial). Theor. Pop. Biol. 46, 363–394.
7. Gardiner, C. W. (1985). Handbook of Stochastic Processes, Berlin: Springer.Google Scholar
8. Gotelli, N. J. and W. G. Kelley (1993). A general model of metapopulation dynamics. OIKOS 68, 36–44.Google Scholar
9. Gurney, W. S. C. and R. M. Nisbet (1978). Single-species population fluctuations in patchy environments. Am. Nat. 112, 1075–1090.
10. Hanski, I. (1994a). A practical model of metapopulation dynamics. J. Anim. Ecol. 63, 151–162.Google Scholar
11. Hanski, I. (1994b). Spatial scale, patchiness and population dynamics on land. Phil. Trans. R. Soc. London B 343, 19–25.Google Scholar
12. Hanski, I. (1999). Metapopulation Ecology, Oxford: Oxford University Press.Google Scholar
13. Hanski, I. and O. Ovaskainen (2000). The metapopulation capacity of a fragmented landscape. Nature 404, 755–758.
14. Harrison, S. (1991). Local extinction in a metapopulation context: an empirical evaluation. Biol. J. Linnean Soc. 42, 73–88.Google Scholar
15. Hubbell, S. P. (2001). The Unified Neutral Theory of Biodiversity and Biogeography, Princeton: Princeton University Press.Google Scholar
16. Kendall, D. G. (1948). On some modes of population growth leading to R. A. Fisher’s logarithmic series distribution. Biometrika 35, 6–15.
17. Levins, R. (1969). Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entomol. Soc. Am. 15, 227–240.Google Scholar
18. Levins, R. (1970). Extinction. Lecture Notes Math. 2, 75–107.
19. MacArthur, R. H. and E. O. Wilson (1967). The Theory of Island Biogeography, Princeton: Princeton University Press.Google Scholar
20. Moilanen, A. (1999). Patch occupancy models of metapopulation dynamics: efficient parameter estimation using implicit statistical inference. Ecology 80, 1031–1043.
21. Mollison, D. (1977). Spatial contact models for ecological and epidemic spread. J. R. Stat. Soc. B 39, 283–326.
22. Naeem, S. et al. (1999). Biodiversity and ecosystem functioning: maintaining natural life support processes. Issues Ecol. 4, 2–11.Google Scholar
23. Nasell, I. (1996). The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Prob. 28, 895–932.
24. Nisbet, R. M. and W. S. C. Gurney (1982). Modelling Fluctuating Populations, New York: Wiley.Google Scholar
25. Press, W. H., S. A. Teukolsky, W. T. Vetterling and B. P. Flannery (1992). Numerical Recipes in C, Cambridge: Cambridge University Press.Google Scholar
26. Renshaw, E. (1991). Modelling Biological Populations in Space and Time, Cambridge Studies in Mathematical Biology 11, Cambridge: Cambridge University Press.Google Scholar
27. 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.
28. 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.Google Scholar
29. van Kampen, N. G. (1981). Stochastic Processes in Physics and Chemistry, Amsterdam: Elsevier.Google Scholar
30. Vitousek, P. M. (1994). Beyond global warming: ecology and global change. Ecology 75, 1861–1876.

© Society for Mathematical Biology 2002

## Authors and Affiliations

• David Alonso
• 1
• 2
• Alan McKane
• 3
1. 1.ICREA Complex Systems LaboratoryUniversitat Pompeu Fabra (GRIB)BarcelonaSpain
2. 2.Department of Ecology, Facultat de BiologiaUniversitat de BarcelonaBarcelonaSpain
3. 3.Department of Theoretical PhysicsUniversity of ManchesterManchesterUK