An explicitly spatial version of the Lotka-Volterra model with interspecific competition

Annals of Applied Probability

Volumn 9, Issue 4, 1999, Pages 1226-1259

  Neuhauser, Claudia ^a   Pacala, Stephen W ^b  

^a UNIVERSITY OF MINNESOTA   (United States)

^b PRINCETON UNIVERSITY   (United States)

Author keywords

Competiton; Contact process; Interacting particle system; Voter model

Indexed keywords

EID: 0033274921     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/aoap/1029962871     Document Type: Article

References (34)

1. ANTONOVICS, J. and LEVIN, D. A. (1980). The ecological and genetic consequences of density dependent regulation in plants. Annual Review of Ecology and Systematics 11 411-452.
2. BRAMSON, M., WAN-DING, DING and DURRETT, R. (1991). Annihilating branching processes. Stochastic. Process Appl. 37 1-17.
3. BRAMSON, M. and NEUHAUSER, C. (1994). Survival of one-dimensional cellular automata under random perturbations. Ann. Probab. 22 244-263.
4. BRAMSON, M. and NEUHAUSER, C. (1997). Coexistence for a catalytic surface reaction model. Ann. Appl. Probab. 7 565-614.
5. CLIFFORD, P. and SUDBURY, A. (1973). A model for spatial conflict. Biometrika 60 581-588.
6. DURRETT, R. (1991). A new method for proving the existence of phase transitions. In Spatial Stochastic Processes (K. S. Alexander and J. S. Watkins, eds.) 141-170. Birkhäuser, Boston.
7. DURRETT, R. (1995). Ten lectures on particle systems. Lectures on Probability Theory, Ecole d'Eté de Probabilités de Saint-Flour XXIII-1993. Springer, New York.
8. DURRETT, R. (1998). Stochastic spatial models. Preprint.
9. DURRETT, R. and LEVIN, S. A. (1994). The importance of being discrete (and spatial). Theoret. Population Biol. 46 363-394.
10. DURRETT, R. and NEUHAUSER, C. (1994). Particle systems and reaction diffusion equations. Ann. Probab. 22 289-333.
11. DURRETT, R. and NEUHAUSER, C. (1997). Coexistence results for some competition models. Ann. Appl. Probab. 7 10-45.
12. GRIFFEATH, D. (1978). Limit theorems for nonergodic set-valued Markov processes. Ann. Probab. 6 379-387.
13. GRIFFEATH, D. (1979). Additive and Cancellative Interacting Particle Systems. Lecture Notes in Math. 724. Springer, Berlin.
14. HARRIS, T. E. (1972). Nearest neighbor Markov interaction processes. Adv. Math 9 66-89.
15. HOLLEY, R. and LIGGETT, T. M. (1975). Ergodic theorems for weakly interacting particle systems and the voter model. Ann. Probab. 3 643-663.
16. HOLLEY, R. and STROOCK, D. (1979). Dual processes and their applications to infinite interacting systems. Adv. Math. 32 149-174.
17. LEVIN, S. A. and PACALA, S. W. (1997). Theories of simplification and scaling of spatially distributed processes. In Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions (D. Tilman and P. Kareiva, eds.) 271-295. Princeton Univ. Press.
18. LlGGETT, T. M. (1985). Interacting Particle Systems. Springer, New York.
19. LOTKA, A. J. (1932). The growth of mixed populations: two species competing for a common food supply. Journal of the Washington Academy of Sciences 22 461-469.
20. MACARTHUR, R. H. (1972). Geographical Ecology. Patterns in the Distribution of Species. Harper and Row, New York.
21. NEUHAUSER, C. (1992). Ergodic theorems for the multitype contact process. Probab. Theory Related Fields 91 467-506.
22. NEUHAUSER, C. (1994). A long range sexual reproduction process. Stochastic Process Appl. 53 193-220.
23. NEUHAUSER, C. (1999). The ancestral graph and gene genealogy under frequency-dependent selection. Theoret. Population Biol. To appear.
24. NEUHAUSER, C. and SUDBURY, A. (1993). The biased annihilating branching process. Adv. Appl. Probab. 25 24-38.
25. PACALA, S. W. (1989). Plant population dynamic theory. In Perspectives in Ecological Theory (J. Roughgarden, R. M. May and S. A. Levin, eds.) 54-67. Princeton Univ. Press.
26. PACALA, S. W. (1997). Dynamics of plant communities. In Plant Ecology, 2nd ed. (M. C. Crawley, ed.) 532-555. Blackwell, Oxford.
27. PACALA, S. W., CANHAM, C. D., SAPONARA, J. SILANDER, J. A., KOBE, R. K. and RIBBENS, E. (1996). Forest models defined by field measurements II. Estimation, error analysis and dynamics. Ecological Monographs 66 1-44.
28. PACALA, S. W. and LEVIN, S. A. (1997). Biologically generated spatial pattern and the coexistence of competing species. In Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions (D. Tilman and P. Kareiva, eds.) 204-232. Princeton Univ. Press.
29. PACALA, S. W. and SILANDER, J. A. JR. (1990). Field tests of neighborhood population dynamic models of two annual weed species. Ecological Monographs 60 113-134.
30. REES, M. (1996). Evolutionary ecology of seed dormancy and seed size. Philos. Trans. Roy. Soc. London Ser. B 351 1299-1308.
31. SPITZER, F. (1970). Interaction of Markov processes. Adv. Math. 5 246-290.
32. THRALL, P. and ANTONOVICS, J. (1995). Theoretical and empirical studies of meta-populations: population and genetic dynamics of the Silene-Ustilago system. Canadian Journal of Botany 73 S1249-S1258.
33. TILMAN, D. (1982). Resource Competition and Community Structure. Princeton Univ. Press.
34. VOLTERRA, V. (1926). Variations and Fluctuations of the Number of Individuals in an Animal Species Living Together. In Animal Ecology (R. N. Chapman, ed.). (Reprinted in 1931. McGraw Hill, New York.)