## The Annals of Probability

### Spatial models for species area curves

#### Abstract

The relationship between species number and area is an old problem in biology. We propose here an interacting particle system--the multitype voter model with mutation--as a mathematical model to study this problem. We analyze the species area curves of this model as the mutation rate $\alpha$ tends to zero. We obtain two basic types of behavior depending on the size of the spatial region under consideration. If the region is a square with area $\alpha^{-r}, r > 1$, then, for small $\alpha$, the number of species is of order $\alpha^{1-r}(\log \alpha)^2$, whereas if $r < 1$, the number of species is bounded.

#### Article information

Source
Ann. Probab. Volume 24, Number 4 (1996), 1727-1751.

Dates
First available in Project Euclid: 6 January 2003

Permanent link to this document
http://projecteuclid.org/euclid.aop/1041903204

Digital Object Identifier
doi:10.1214/aop/1041903204

Mathematical Reviews number (MathSciNet)
MR1415227

Zentralblatt MATH identifier
0939.92032

#### Citation

Bramson, Maury; Cox, J. Theodore; Durrett, Richard. Spatial models for species area curves. Ann. Probab. 24 (1996), no. 4, 1727--1751. doi:10.1214/aop/1041903204. http://projecteuclid.org/euclid.aop/1041903204.

#### References

• ARRATIA, R. 1981. Limiting point processes for rescalings of coalescing and annihilating random walks on Zd. Ann. Probab. 9 909 936. Z.
• ARRHENIUS, O. 1921. Species and area. Journal of Ecology 9 95 99. Z.
• BRAMSON, M., COX, J. T. and GRIFFEATH, D. 1986. Consolidation rates for two interacting sy stems in the plane. Probab. Theory Related Fields 73 613 625. Z.
• BRAMSON, M. and GRIFFEATH, D. 1979. Renormalizing the three-dimensional voter model. Ann. Probab. 7 418 432. Z. d
• BRAMSON, M. and GRIFFEATH, D. 1980. Asy mptotics for some interacting particle sy stems on Z. Z. Warsch. Verw. Gebiete 53 183 196. Z.
• CLIFFORD, P. and SUDBURY, A. 1973. A model for spatial conflict. Biometrika 60 681 588. Z.
• COLEMAN, B. D. 1982. On random placement and species-area-relations. Math. Biosci. 54 191 215. Z.
• CONNOR, E. F. and MCCOY, E. D. 1979. The statistics and biology of the species-area relationship. American Naturalist 113 791 833. Z.
• COX, J. T. and GRIFFEATH, D. 1986. Diffusive clustering in the two dimensional voter model. Ann. Probab. 14 347 370. Z.
• DONNELLY, P. and TAVARE, S. 1995. Coalescents and genealogical structure under neutrality. ´ Annual Review of Genetics 29 401 421. Z.
• DURRETT, R. 1988. Lecture Notes on Particle Sy stems and Percolation. Wadsworth, Belmont, CA. Z.
• DURRETT, R. and LEVIN, S. 1996. Spatial models for species area curves. J. Theoret. Biol. To appear. Z.
• EWENS, W. 1990. Population genetics the past and the future. In Mathematical and StatistiZ. cal Developments of Evolutionary Theory S. Lessard, ed. 177 277. Kluwer, Dordrecht. Z.
• FISHER, R. A., CORBET, A. S. and WILLIAMS, C. B. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology 12 42 58. Z.
• GRIFFEATH, D. 1979. Additive and Cancellative Interacting Particle Sy stems. Lecture Notes in Math. 724. Springer, Berlin. Z.
• HOLLEY, R. A. and LIGGETT, T. M. 1975. Ergodic theorems for weakly interacting sy stems and the voter model. Ann. Probab. 3 643 663. Z.
• HUBBELL, S. P. 1992. Speciation, dispersal, and extinction: an equilibrium theory of species-area relationships. Preprint.
• KARLIN, S. 1967. Central limit theorems for certain infinite urn schemes. Journal of Mathematical Mechanics 17 373 401. Z.
• KREITMAN, M. and AKASHI, H. 1995. Molecular evidence for natural selection. Annual Review of Ecological Sy stems 26 403 422. Z.
• LIGGETT, T. M. 1985. Interacting Particle Sy stems. Springer, New York. Z.
• MACARTHUR, R. H. and WILSON, E. O. 1967. The Theory of Island Biogeography. Princeton Univ. Press. Z.
• NEUHAUSER, C. 1992. Ergodic theorems for the multity pe contact process. Probab. Theory Related Fields 91 467 506. Z.
• PITMAN, J. 1996. Species sampling models. Unpublished manuscript. Z.
• PRESTON, F. W. 1962. The canonical distribution of commonness and rarity. I, II. Ecology 43 185 215; 410 432. Z.
• ROUAULT, A. 1978. Lois de Zipf et sources markoviennes. Ann. Inst. H. Poincare Probab. ´ Statist. 14 169 178. Z.
• WATSON, H. C. 1835. Remarks on the Geographical Distribution of British Plants. Longmans, London. Z.
• WILLIAMSON, M. 1988. Relationship of species number to area, distance and other variables. In Z. Analy tical Biogeography A. A. My ers and P. S. Giller, eds. Chap. 4. Chapman and Hall, London.
• MADISON, WISCONSIN 53706 Sy RACUSE, NEW YORK 13244 E-MAIL: bramson@math.wisc.edu E-MAIL: jtcox@gumby.sy r.edu
• ITHACA, NEW YORK 14853 E-MAIL: rtd1@cornell.edu