The Annals of Applied Probability

Coexistence in stochastic spatial models

Rick Durrett

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In this paper I will review twenty years of work on the question: When is there coexistence in stochastic spatial models? The answer, announced in Durrett and Levin [Theor. Pop. Biol. 46 (1994) 363–394], and that we explain in this paper is that this can be determined by examining the mean-field ODE. There are a number of rigorous results in support of this picture, but we will state nine challenging and important open problems, most of which date from the 1990’s.

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Ann. Appl. Probab. Volume 19, Number 2 (2009), 477-496.

First available in Project Euclid: 7 May 2009

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Interacting particle system block construction fast stirring limit competitive exclusion principle


Durrett, Rick. Coexistence in stochastic spatial models. Ann. Appl. Probab. 19 (2009), no. 2, 477--496. doi:10.1214/08-AAP590.

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  • Bramson, M. and Durrett, R. (1988). A simple proof of the stability criterion of Gray and Griffeath. Probab. Theory Related Fields 80 293–298.
  • Bramson, M. and Griffeath, D. (1989). Flux and fixation in cyclic particle systems. Ann. Probab. 17 26–45.
  • Bramson, M. and Neuahuser, C. (1992). A catalytic surface reaction model. J. Comput. Appl. Math. 40 157–161.
  • Chan, B. and Durrett, R. (2006). A new coexistence result for competing contact processes. Ann. Appl. Probab. 16 1155–1165.
  • Durrett, R. (1992). Multicolor particle systems with large threshold and range. J. Theoret. Probab. 5 127–152.
  • Durrett, R. (1995). Ten lectures on particle systems. In Lectures on Probability Theory (Saint-Flour, 1993). Lecture Notes in Mathematics 1608 97–201. Springer, Berlin.
  • Durrett, R. (2002). Mutual invadability implies coexistence in spatial models. Mem. Amer. Math. Soc. 156 viii–118.
  • Durrett, R. and Griffeath, D. (1993). Asymptotic behavior of excitable cellular automata. Experiment. Math. 2 183–208.
  • Durrett, R. and Lanchier, N. (2008). Coexistence in host-pathogen systems. Stochastic Process. Appl. 118 1004–1021.
  • Durrett, R. and Levin, S. (1994). The importance of being discrete (and spatial). Theor. Pop. Biol. 46 363–394.
  • Durrett, R. and Levin, S. (1997). Allelopathy in spatially distributed populations. J. Theoret. Biol. 185 165–172.
  • Durrett, R. and Levin, S. (1998). Spatial aspects of interspecific competition. Theor. Pop. Biol. 53 30–43.
  • Durrett, R. and Møller, A. M. (1991). Complete convergence theorem for a competition model. Probab. Theory Related Fields 88 121–136.
  • Durrett, R. and Neuhauser, C. (1994). Particle systems and reaction-diffusion equations. Ann. Probab. 22 289–333.
  • Durrett, R. and Neuhauser, C. (1997). Coexistence results for some competition models. Ann. Appl. Probab. 7 10–45.
  • Durrett, R. and Schinazi, R. (1993). Asymptotic critical value for a competition model. Ann. Appl. Probab. 3 1047–1066.
  • Durrett, R. and Swindle, G. (1991). Are there bushes in a forest? Stochastic Process. Appl. 37 19–31.
  • Durrett, R. and Swindle, G. (1994). Coexistence results for catalysts. Probab. Theory Related Fields 98 489–515.
  • Fisch, R., Gravner, J. and Griffeath, D. (1991). Cyclic cellular automata in two dimensions. In Spatial Stochastic Processes. Progress in Probability 19 171–185. Birkhäuser Boston, Boston, MA.
  • Grannan, E. R. and Swindle, G. (1990). Rigorous results on mathematical models of catalytic surfaces. J. Statist. Phys. 61 1085–1103.
  • Harris, T. E. (1974). Contact interactions on a lattice. Ann. Probab. 2 969–988.
  • Kirkup, B. C. and Riley, M. A. (2004). Antibiotic-meidated antagonism leads to a bacterial game of rock-paper-scissor in vivo. Nature 428 412–414.
  • Lanchier, N. and Neuhauser, C. (2006). Stochastic spatial models of host-pathogen and host-mutualist interactions. I. Ann. Appl. Probab. 16 448–474.
  • Levin, S. A. (1970). Community equilibria and stability, and an extension of the competitive exclusion. principle. Am. Naturalist. 104 413–423.
  • Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 324. Springer, Berlin.
  • Mountford, T. S. and Sudbury, A. (1992). An extension of a result of Grannan and Swindle on the poisoning of catalytic surfaces. J. Statist. Phys. 67 1219–1222.
  • Neuhauser, C. (1992). Ergodic theorems for the multitype contact process. Probab. Theory Related Fields 91 467–506.
  • Silvertown, J., Holtier, S., Johnson, J. and Dale, P. (1992). Cellular automaton models of interspecific competition for space—the effect of pattern on process. J. Ecol. 80 527–534.
  • Sinvervo, B. and Lively, C. M. (1996). The rock-paper-scissors game and the evolution of alternative male strategies. Nature 380 240–243.
  • Volpert, V. A. and Volpert, A. I. (1988). Application of the Leray–Schauder method to the proof of the existence of wave solutions of parabolic systems. Dokl. Akad. Nauk SSSR 298 784–787.
  • Ziff, R. M., Gulari, E. and Barshad, Y. (1986). Kinetic phase transitions in an irreversible surface-reaction model. Phys. Rev. Lett. 56 2553–2556.