A spatially explicit, stochastic Lotka–Volterra model was introduced by Neuhauser and Pacala in Neuhauser and Pacala (Ann. Appl. Probab. 9, 1226–1259, 1999). A low density limit theorem for this process was proved by the authors in Cox and Perkins (Ann. Probab. 33, 904–947, 2005), showing that certain generalized rescaled Lotka–Volterra models converge to super-Brownian motion with drift. Here we use this convergence result to extend what is known about the parameter regions for the Lotka–Volterra process where (i) survival of one type holds, and (ii) coexistence holds.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Blath J., Etheridge, A.M., Meredith, M.E. Coexistence in locally regulated competing populations. Preprint (2004)
Bramson M., Griffeath D. (1981) On the Williams–Bjerknes tumour growth model. I. Ann. Probab. 9, 173–185
Bramson M., Neuhauser C. (1997) Coexistence for a catalytic surface reaction model. Ann. Appl. Probab. 7, 565–614
Cox J.T., Durrett R., Perkins E.A. (2000) Rescaled voter models converge to super-Brownian motion. Ann. Probab. 28, 185–234
Cox J.T., Perkins E.A. (2005) Rescaled Lotka–Volterra models converge to super-Brownian motion. Ann. Probab. 33, 904–947
Cox, J.T., Perkins, E.A. Renormalization of the two-dimensional Lotka–Volterra Model. Preprint (2006)
Durrett, R. Ten lectures on particle systems. Ecole d’été de Probabilités de St. Flour, XXIII. Lecture Notes in Math 1608, pp. 97–201. Springer, Berlin Heidelberg New York (1995)
Durrett R., Perkins E. (1998) Rescaled contact processes converge to super-Brownian motion for d ≥ 2. Probab. Theory Rel. Fields 114, 309–399
Liggett T.L. (1985) Interacting particle systems. Springer, Berlin Heidelberbg New York
Liggett T.L. (1999) Interacting stochastic systems: contact, voter and exclusion processes. Springer, Berlin Heidelberg New York
Neuhauser C., Pacala S.W. (1999) An explicitly spatial version of the Lotka–Volterra model with interspecific competition. Ann. Appl. Probab. 9, 1226–1259
Perkins, E. Measure-valued processes and interactions. Pages 125-324, in École d’Été de Probabilités de Saint Flour XXIX-1999, Lecture Notes in Mathematics, 1781, pp. 97–201. Springer, Berlin Heidelberg New York (2002)
Supported in part by NSF grants DMS-024422/DMS-0505439. Part of the research was done while the author was visiting The University of British Columbia.
Supported in part by an NSERC Research grant.
About this article
Cite this article
Cox, J.T., Perkins, E.A. Survival and coexistence in stochastic spatial Lotka–Volterra models. Probab. Theory Relat. Fields 139, 89–142 (2007). https://doi.org/10.1007/s00440-006-0040-3
- Voter model
- Super–Brownian motion
Mathematics Subject Classification (2000)
- Primary 60K35
- Primary 60G57
- Secondary 60F05
- Secondary 60J80