The Annals of Applied Probability

A microscopic probabilistic description of a locally regulated population and macroscopic approximations

Nicolas Fournier and Sylvie Méléard

Full-text: Open access


We consider a discrete model that describes a locally regulated spatial population with mortality selection. This model was studied in parallel by Bolker and Pacala and Dieckmann, Law and Murrell. We first generalize this model by adding spatial dependence. Then we give a pathwise description in terms of Poisson point measures. We show that different normalizations may lead to different macroscopic approximations of this model. The first approximation is deterministic and gives a rigorous sense to the number density. The second approximation is a superprocess previously studied by Etheridge. Finally, we study in specific cases the long time behavior of the system and of its deterministic approximation.

Article information

Ann. Appl. Probab., Volume 14, Number 4 (2004), 1880-1919.

First available in Project Euclid: 5 November 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Interacting measure-valued processes regulated population deterministic macroscopic approximation nonlinear superprocess equilibrium


Fournier, Nicolas; Méléard, Sylvie. A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 (2004), no. 4, 1880--1919. doi:10.1214/105051604000000882.

Export citation


  • Aldous, D. (1978). Stopping times and tightness. Ann. Probab. 6 335–340.
  • Bolker, B. and Pacala, S. (1997). Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theoret. Population Biol. 52 179–197.
  • Bolker, B. and Pacala, S. (1999). Spatial moment equations for plant competition: Understanding spatial strategies and the advantages of short dispersal. The American Naturalist 153 575–602.
  • Dieckmann, U. and Law, R. (2000). Relaxation projections and the method of moments. In The Geometry of Ecological Interactions (U. Dieckmann, R. Law and J. A. J. Metz, eds.) 412–455. Cambridge Univ. Press.
  • Evans, S. N. and Perkins, E. A. (1994). Measure-valued branching diffusions with singular interactions. Canad. J. Math. 46 120–168.
  • Etheridge, A. (2001). Survival and extinction in a locally regulated population. Preprint.
  • Joffe, A. and Métivier, M. (1986). Weak convergence of sequences of semimartingales with applications to multi-type branching processes. Adv. in Appl. Probab. 18 20–65.
  • Kallenberg, O. (1975). Random Measures. Akademie Verlag, Berlin.
  • Law, R. and Dieckmann, U. (2002). Moment approximations of individual-based models. In The Geometry of Ecological Interactions (U. Dieckmann, R. Law and J. A. J. Metz, eds.) 252–270. Cambridge Univ. Press.
  • Law, R., Murrell, D. J. and Dieckmann, U. (2003). Population growth in space and time: Spatial logistic equations. Ecology 84 252–262.
  • Liggett, T. (1985). Interacting Particle Systems. Springer, Berlin.
  • Méléard, S. and Roelly, S. (1993). Sur les convergences étroite ou vague de processus à valeurs mesures. C. R. Acad. Sci. Paris Sér. I Math. 317 785–788.
  • Moller, J. (1994). Lectures on Random Voronoi Tesselations. Lecture Notes in Statist. 87. Springer, Berlin.
  • Olivares-Rieumont, P. and Rouault, A. (1991). Unscaled spatial branching processes with interaction: Macroscopic equation and local equilibrium. Stochastic Anal. Appl. 8 445–461.
  • Roelly, S. (1986). A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics Stochastics Rep. 17 43–65.
  • Roelly, S. and Rouault, A. (1990). Construction et propriétés de martingales des branchements spatiaux interactifs. Internat. Statist. Rev. 58 173–189.