The Annals of Applied Probability

Survival and extinction in a locally regulated population

A. M. Etheridge

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Abstract

Bolker and Pacala recently introduced a model of an evolving population in which an individual's fecundity is reduced in proportion to the "local population density." We consider two versions of this model and prove complementary extinction/persistence results, one for each version. Roughly, if individuals in the population disperse sufficiently quickly relative to the range of the interaction induced by the density dependent regulation, then the population has positive chance of survival, whereas, if they do not, then the population will die out.

Article information

Source
Ann. Appl. Probab. Volume 14, Number 1 (2004), 188-214.

Dates
First available in Project Euclid: 3 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1075828051

Digital Object Identifier
doi:10.1214/aoap/1075828051

Mathematical Reviews number (MathSciNet)
MR2023020

Zentralblatt MATH identifier
02072698

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Interacting superprocess regulated population extinction persistence

Citation

Etheridge, A. M. Survival and extinction in a locally regulated population. Ann. Appl. Probab. 14 (2004), no. 1, 188--214. doi:10.1214/aoap/1075828051. http://projecteuclid.org/euclid.aoap/1075828051.


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References

  • Barton, N. H., Depaulis, F. and Etheridge, A. M. (2002). Neutral evolution in spatially continuous populations. Theoretical Population Biology 61 31--48.
  • Bolker, B. M. and Pacala, S. W. (1997). Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theoretical Population Biology 52 179--197.
  • Bolker, B. M. and Pacala, S. W. (1999). Spatial moment equations for plant competition: Understanding spatial strategies and the advantages of short dispersal. American Naturalist 153 575--602.
  • Dawson, D. A. (1993). Measure-valued Markov processes. Ecole d'été de probabilités de Saint Flour XXI. Lecture Notes in Math. 1541 1--260. Springer, New York.
  • Dawson, D. A., Iscoe, I. and Perkins, E. A. (1989). Super-Brownian motion: Path properties and hitting probabilities. Probab. Theory Related Fields 83 135--205.
  • Durrett, R. (1995). Ten lectures on particle systems. Ecole d'été de probabilités de Saint Flour XXIII. Lecture Notes in Math. 1608 97--201. Springer, New York.
  • Durrett, R. and Perkins, E. A. (1999). Rescaled contact processes converge to super-Brownian motion in two or more dimensions. Probab. Theory Related Fields 114 309--399.
  • Etheridge, A. M. (2000). An Introduction to Superprocesses 20. AMS, Providence, RI.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Evans, S. N. and Perkins, E. A. (1994). Measure-valued branching diffusions with singular interactions. Canad. J. Math. 46 120--168.
  • Evans, S. N. and Perkins, E. A. (1998). Collision local times, historical stochastic calculus and competing species. Electron. J. Probab. 3 1--120.
  • Feller, W. (1951). Diffusion processes in genetics. Proc. Second Berkeley Symp. Math. Statist. Probab. 227--246. Univ. California Press, Berkeley.
  • Felsenstein, J. (1975). A pain in the torus: Some difficulties with the model of isolation by distance. American Naturalist 109 359--368.
  • Iscoe, I. (1986). A weighted occupation time for a class of measure-valued critical branching Brownian motions. Probab. Theory Related Fields 71 85--116.
  • Kimura, M. (1953). Stepping stone model of population. Ann. Rep. Nat. Inst. Genetics Japan 3 62--63.
  • Knight, F. B. (1981). Essentials of Brownian Motion and Diffusion. Amer. Math. Soc., Providence, RI.
  • Law, R., Murrell, D. J. and Dieckmann, U. (2003). On population growth in space and time: Spatial logistic equations. Ecology 84 252--262.
  • Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
  • Mueller, C. and Tribe, R. (1994). A phase-transition for a stochastic pde related to the contact process. Probab. Theory Related Fields 100 131--156.
  • Perkins, E. A. (2002). Dawson--Watanabe superprocesses and measure-valued diffusions. Ecole d'été de probabilités de Saint Flour. Lecture Notes in Math. 1781 125--329. Springer, New York.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion. Springer, New York.
  • Walsh, J. B. (1986). An introduction to stochastic partial differential equations. Ecole d'été de probabilités de Saint Flour. Lecture Notes in Math. 1180 265--439. Springer, New York.