SPecial invited paper coexistence in stochastic spatial models

Annals of Applied Probability

Volumn 19, Issue 2, 2009, Pages 477-496

  Durrett, Rick ^a  

^a Department Of Mathematics ^*  (United States)

Author keywords

Block construction; Competitive exclusion principle.; Fast stirring limit; Interacting particle system

Indexed keywords

EID: 66149114872     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/08-AAP590     Document Type: Article

References (32)

1. Bramson, M. and Durrett, R. (1988). A simple proof of the stability criterion of Gray and Griffeath. Probab. Theory Related Fields 80 293-298. MR968822
2. Bramson, M. and Griffeath, D. (1989). Flux and fixation in cyclic particle systems. Ann. Probab. 17 26-45. MR972768
3. Bramson, M. and Neuahuser, C. (1992). A catalytic surface reaction model. J. Comput. Appl. Math. 40 157-161.
4. Chan, B. and Durrett, R. (2006). A new coexistence result for competing contact processes. Ann. Appl. Probab. 16 1155-1165. MR2260060
5. Durrett, R. (1992). Multicolor particle systems with large threshold and range. J. Theoret. Probab. 5 127-152. MR1144730
6. 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. MR1383122
7. Durrett, R. (2002). Mutual invadability implies coexistence in spatial models. Mem. Amer. Math. Soc. 156 viii-118. MR1879853
8. Durrett, R. and Griffeath, D. (1993). Asymptotic behavior of excitable cellular automata. Experiment. Math. 2 183-208. MR1273408
9. Durrett, R. and Lanchier, N. (2008). Coexistence in host-pathogen systems. Stochastic Process. Appl. 118 1004-1021. MR2418255
10. Durrett, R. and Levin, S. (1994). The importance of being discrete (and spatial). Theor. Pop. Biol. 46 363-394.
11. Durrett, R. and Levin, S. (1997). Allelopathy in spatially distributed populations. J. Theoret. Biol. 185 165-172.
12. Durrett, R. and Levin, S. (1998). Spatial aspects of interspecific competition. Theor. Pop. Biol. 53 30-43.
13. Durrett, R. and Møller, A. M. (1991). Complete convergence theorem for a competition model. Probab. Theory Related Fields 88 121-136. MR1094080
14. Durrett, R. and Neuhauser, C. (1994). Particle systems and reaction-diffusion equations. Ann. Probab. 22 289-333. MR1258879
15. Durrett, R. and Neuhauser, C. (1997). Coexistence results for some competition models. Ann. Appl. Probab. 7 10-45. MR1428748
16. Durrett, R. and Schinazi, R. (1993). Asymptotic critical value for a competition model. Ann. Appl. Probab. 3 1047-1066. MR1241034
17. Durrett, R. and Swindle, G. (1991). Are there bushes in a forest? Stochastic Process. Appl. 37 19-31. MR1091691
18. Durrett, R. and Swindle, G. (1994). Coexistence results for catalysts. Probab. Theory Related Fields 98 489-515. MR1271107
19. 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. MR1144096
20. Grannan, E. R. and Swindle, G. (1990). Rigorous results on mathematical models of catalytic surfaces. J. Statist. Phys. 61 1085-1103. MR1083897
21. Harris, T. E. (1974). Contact interactions on a lattice. Ann. Probab. 2 969-988. MR0356292
22. 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.
23. Lanchier, N. and Neuhauser, C. (2006). Stochastic spatial models of host-pathogen and hostmutualist interactions. I. Ann. Appl. Probab. 16 448-474. MR2209349
24. Levin, S. A. (1970). Community equilibria and stability, and an extension of the competitive exclusion. principle. Am. Naturalist. 104 413-423.
25. Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York. MR776231
26. Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 324. Springer, Berlin. MR1717346
27. 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. MR1170090
28. Neuhauser, C. (1992). Ergodic theorems for the multitype contact process. Probab. Theory Related Fields 91 467-506. MR1151806
29. 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.
30. Sinvervo, B. and Lively, C. M. (1996). The rock-paper-scissors game and the evolution of alternative male strategies. Nature 380 240-243.
31. 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. MR936502
32. Ziff, R. M., Gulari, E. and Barshad, Y. (1986). Kinetic phase transitions in an irreversible surface-reaction model. Phys. Rev. Lett. 56 2553-2556.