Electronic Journal of Probability

Coexistence in a Two-Dimensional Lotka-Volterra Model

J. Theodore Cox, Mathieu Merle, and Edwin Perkins

Full-text: Open access

Abstract

We study the stochastic spatial model for competing species introduced by Neuhauser and Pacala in two spatial dimensions. In particular we confirm a conjecture of theirs by showing that there is coexistence of types when the competition parameters between types are equal and less than, and close to, the within types parameter. In fact coexistence is established on a thorn-shaped region in parameter space including the above piece of the diagonal. The result is delicate since coex- istence fails for the two-dimensional voter model which corresponds to the tip of the thorn. The proof uses a convergence theorem showing that a rescaled process converges to super-Brownian motion even when the parameters converge to those of the voter model at a very slow rate.

Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 38, 1190-1266.

Dates
Accepted: 9 August 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819823

Digital Object Identifier
doi:10.1214/EJP.v15-795

Mathematical Reviews number (MathSciNet)
MR2678390

Zentralblatt MATH identifier
1226.60131

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G57: Random measures 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Lotka-Volterra voter model super-Brownian motion spatial competition coalescing random walk coexistence survival

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Cox, J. Theodore; Merle, Mathieu; Perkins, Edwin. Coexistence in a Two-Dimensional Lotka-Volterra Model. Electron. J. Probab. 15 (2010), paper no. 38, 1190--1266. doi:10.1214/EJP.v15-795. https://projecteuclid.org/euclid.ejp/1464819823


Export citation

References

  • Billingsley, P. Convergence of probability measures. (1968), John Wiley & Sons.
  • Cox, J.T.; Durrett, R.; Perkins, E.A.; Rescaled voter models converge to super-Brownian motion. Ann. Probab. 28 (2000), no. 1, 185–234.
  • Cox, J.T.; Durrett, R.; Perkins, E.A.; Voter model perturbations and reaction diffusion equations. Preprint.
  • Cox, J.T.; Perkins, E.A. Rescaled Lotka-Volterra models converge to super-Brownian motion. Ann. Probab. 3 (2005), no. 3, 904–947.
  • Cox, J.T.; Perkins, E.A. Survival and coexistence in stochastic spatial Lotka-Volterra models. Probab. Theory Related Fields 139 (2007), no. 1-2, 89–142.
  • Cox, J.T.; Perkins, E.A. Renormalization of the two-dimensional Lotka-Volterra model. Ann. Appl. Probab. 18 (2008), no. 2, 747–812.
  • Durrett,R.; Remenik,D. Voter model perturbations in two dimensions. Preprint.
  • Lawler, G.F. Intersections of random walks. Probability and its Applications. Birkhauser. (1991)
  • Le Gall, J.-F.;Perkins, E.A. The Hausdorff measure of the support of two-dimensional super-Brownian motion. Ann. Probab. 23 (1995), no. 4, 1719–1747.
  • Liggett, T.M. Interacting particle systems. Springer-Verlag.
  • Neuhauser, C.; Pacala, S.W. An explicitly spatial version of the Lotka-Volterra model with interspecific competition. Ann. Appl. Probab. 9, (1999), no. 4, 1226–1259.
  • Perkins, E. Measure-valued processes and interactions. Ecole d'Ete de Probabilites de Saint Flour XXIX-1999, Lecture Notes in Mathematics 1781 (2002) 125–329.
  • Spitzer, F. Principles of random walks. 2nd ed. Springer-Verlag. 1976.