Keywords
- Patch Size
- Spatial Model
- Particle System
- Vacant Site
- Voter Model
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References
Bezuidenhout, C. and Gray, L. (1994) Critical attractive spin systems. Ann. Probbab. 22, 1160–1194
Bramson, M., Cox, J. T., and Durrett, R. (1996) Spatial models for species area curves. Ann. Prob. 24, 1727–1751
Bramson, M., Cox, J. T., and Durrett, R. (1997) A spatial model for the abundance of species. Ann. Prob., to appear
Bramson, M., Durrett, R. and Swindle, G. (1989) Statistical mechanics of Crabgrass. Ann. Prob. 17, 444–481.
Bramson, M. and Griffeath, D. (1980). Asymptotics for some interacting particle systems on Z d. Z. Warsch. verw. Gebiete 53, 183–196.
Bramson, M. and Griffeath, D. (1989). Flux and fixation in cyclic particle systems. Ann. Probab. 17, 26–45.
Bramson, M., Durrett, R. and Swindle, S. (1989) Statistical mechanics of crabgrass. Ann. Probab. 17, 444–481
Bramson, M. and Gray, L. (1991) A useful renormalization argument. Random Walks, Brownian Motion and Intracting Brownian Motion. Edited by R. Durett and H. Kesten. Birkhauser, Boston.
Bramson, M. and Griffeath, D. (1980) On the Williams-Bjerknes tumor growth model, II. Math. Proc. Camb. Phil. Soc. 88, 339–357
Bramson, M. and Griffeath, D. (1981) On the Williams-Bjerknes tumor growth model, I. Ann. Probab. 9, 173–185
Brower, R. C., Furman, M. A., and Moshe, M. (1978) Critical exponents for the Reggeon quantum spin model. Phys. Lett. B. 76, 213–219
Brown, D. B. and Hansel, R. I. C. (1987) Convergence to an evolutionary stable strategy in the two-policy game. Am. Nat. 130, 929–940
Chao, L. (1979) The population of colicinogenic bacteria: a model for the evolution of allelopathy. Ph.D. dissertation, U. of Massachusetts.
Chao, L. and Levin, B. R. (1981) Structured habitats and the evolution of anti-competitor toxins in bacteria. Proc. Nat. Acad. Sci. 78, 6324–6328
Connor, E. F. and McCoy, E. D. (1979) The statistics and biology of the species-area realtionship. Amer. Nat. 113, 791–833.
Cox, J. T. and Durrett, R. (1995) Hybrid zones and voter model interfaces. Bernoulli 1, 343–370
Crawley, M. J. and R. M. May (1987) Population dynamics and plant community structure: competition between annuals and perennials. J. Theor. Biol. 125, 475–489
DeMasi, A., Ferrari, P. and Lebowitz, J. (1986) Reaction diffusion equations for interacting particle systems. J. Stat. Phys. 44, 589–644
Dunford, N. and Schwarz, J. T. (1957) Linear Operators, Vol. 1. Interscience Publishers, John Wiley and Sons, New York
Durrett, R. (1988) Lecture Notes on Particle Systems and Percolation. Wadsworth Pub. Co. Belmont, CA
Durrett, R. (1992) A new method for proving the existence of phase transitions. Pages 141–170 in Spatial Stochastic Processes, edited by K. S. Alexander and J. C. Watkins, Birkhauser, Boston
Durrett, R. (1993) Predator-prey systems. Pages 37–58 in Asymptotic problems in probability theory: stochastic models and diffusions on fractals. Edited by K. D. Elworthy and N. Ikeda, Pitman Research Notes 83, Longman Scientific, Essex, England
Durrett, R. (1995a) Probability: Theory and Examples. Duxbury Press, Belmont, CA.
Durrett, R. (1995b) Ten Lectures on Particle Systems. Ecole d'Eté de Probabilités de Saint Flour, 1993. Lecture Notes in Math 1608, Springer, New York
Durrett, R. (1995c) Spatial epidemic models. Pages 187–201 in Epidemic Models: Their Structure and Relation to Data. Edited by D. Mollison. Cambridge U. Press
Durrett, R. (1996) Stochastic Calculus. CRC Press, Boca Raton, FL
Durrett, R. and Levin, S. A. (1994a) Stochastic spatial models: A user's guide for ecological applications. Phil. Trans. Roy. Soc. B, 343, 1047–1066
Durrett, R. and Levin, S. (1994b) The importance of being discrete (and spatial). Theoret. Pop. Biol. 46, 363–394
Durrett, R. and Levin, S. (1996) Spatial models for species area curves. J. Theor. Biol., 179, 119–127
Durrett, R. and Levin, S. A. (1997) Spatial aspects of interspecific competition. Preprint.
Durrett, R. and Levin, S. A. (1998) Pattern formation on planet WATOR. In preparation.
Durrett, R. and Neuhauser, C. (1991) Epidemics with recovery in d=2. Ann. Applied Prob. 1, 189–206
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. Prob. 7, 10–45
Durrett, R. and Schinazi, R. (1993) Asymptotic critical value for a competition model. Ann. Applied. Prob. 3, 1047–1066
Durrett, R. and Swindle, G. (1991) Are there bushes in a forest? Stoch. Proc. Appl. 37, 19–31
Durrett, R., and Swindle, G. (1994) Coexistence results for catalysts. Prob. Th. Rel. Fields 98, 489–515
Engen, S. and R. Lande (1996) Population dynamic models generating the lognormal species abundance distribution. Math. Biosci. 132, 169–183
Fife, P. C. and McLeod, J. B. (1977) The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rat. Mech. Anal. 65, 335–361
Fisch, R., Gravner, J. and Griffeath, D. (1991a). Cyclic cellular automata in two dimensions. Pages 171–185 in Spatial Stochastic Processes. edited by K. Alexander and J. Watkins. Birkhauser, Boston.
Fisch, R., Gravner, J. and Griffeath, D. (1991b). Threshold-range scaling of excitable cellular automata. Statistics and Computing. 1, 23–39.
Fisher, R. A., Corbet, A. S., and Williams, C. B. (1943). The relation between the number of species and the number of individuals in a random sample of an animal population. J. Animal Ecol. 12, 42–58.
Frank, S. A. (1994) Spatial polymorphism of bacteriocins and other allelopathic traits. Evolutionary Ecology
Gilpin, M. E. (1975). Limit cycles in competition communities. Am. Nat. 109, 51–60.
Grassberger, P. (1982) On phase transitions in Schlogl's second model. Z. Phys. B. 47, 365–376
Grassberger, P. and de la Torre, A. (1979) Reggeon field theory (Schlogl's first model) on a lattice: Monte Carlo calculation of critical behavior. Ann. Phys. 122, 373–396
Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Springer Lecture Notes in Mathematics, 724.
Griffeath, D. (1988). Cyclic random competition: a case history in experimental mathematics. Notices of the Amer. Math. Soc. 1472–1480
Harada, Y., Ezoe, H., Iwasa, Y., Matsuda, H., and Sato, K. Population persistence and spatially limited local interaction. Theor. Pop. Biol. 48, 65–91
Harris, T. E. (1974) Contact interactions on a lattice. Ann. Prob. 2, 969–988
Harris, T. E. (1977) A correlation inequality for Markov processes in partially ordered state spaces. Ann. Probab. 6, 355–378
Holley, R. A. and Liggett, T. M. (1975). Ergodic theorems for weakly interacting systems and the voter model. Ann. Prob. 3, 643–663.
Hirsch, M. W. and S. Smale (1974) Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York
Hubbell, S. P. (1992). Speciation, dispersal, and extinction: An equilibrium theory of species-area relationships. Preprint.
Hubbell, S. P. (1995) Towards a theory of biodiversity and biogeograph on continuous landscapes. Pages 173–201 in Preparing for Global Change: A Midwestern Perspective. Edited by G. R. Carmichael, G. E. Folk, and J. L. Schnoor. SPB Academic Publishing, Amsterdam.
Huberman, B.A. and Glance, N.S. (1993) Evolutionary games and computer simulations. Proc. Nat. Acad. Sci., USA. 90, 7712–7715
Keeling, M.J., Mezic, I., Hendry, R.J., McGlade, J., and Rand, D.A. (1997) Characteristic length scales of spatial models in ecology. Phil. Trans. R. Soc. London B 352, 1589–1601
Levin, B.R. (1988) Frequency dependent selection in bacterial populations. Phil. Trans. R. Soc. London B 319, 459–472
Liggett, T.M. (1985) Interacting Particle Systems. Springer-Verlag, New York
Longuet-Higgins, M.S. (1971) On the Shannon-Weaver index of diversity, in relation to the distribution of species in bird censuses. Theor. Pop. Biol. 2, 271–289
Luce, R.D., and Raiffa, H. (1968) Games and Decisions. John Wiley and Sons, New York
MacArthur, R.H. (1957) On the relative abundance of bird species. Proc. Nat. Acad. Sci. USA. 43, 293–295
MacArthur, R.H. (1960) On the relative abundance of species. Am. Nat. 94, 25–36
MacArthur, R.H. and Wilson, E.O. (1967). The Theory of Island Biogeography. Princeton Monographs in Population Biology.
MacCauley, E., Wilson, W.G., and de Roos, A.M. (1993) Dynamics of agestructured and spatially structured predator-prey interactions: Individual bassed models and population-level formulations. Amer, Natur. 142, 412–442
Matsuda, H., Ogita, N., Sasaki, A., and Sato, K. (1992) Statistical mechanics of population: the lattice Lotka-Volterra model. Prog. Theor. Phys. 88, 1035–1049
Matsuda, H., Tamachi, N., Sasaki, A., and Ogita, N. (1987) A lattice model for population biology. Pages 154–161 in Mathematical Topics in Biology edited by E. Teramoto and M. Yamaguchi, Springer Lecture Notes in Biomathematics.
May, R.M. (1975) Patterns of species abundance and diversity. Pages 81–120 in Ecology and Evolution of Communities. Edited by M.L. Coday and J.M. Diamond, Belknap Press, Cambridge, MA
May, R.M. (1994) Spatial chaos and its role in ecology and evolution. Pages 326–344 in Frontiers in Mathematical Biology. Lecture Notes in Biomathematics 100, Springer, New York
May, R.M. (1995) Necessity and chance: Deterministic chaos in ecology and evolution. Bulletin of the AMS., New Series, 32, 291–308
May, R.M. and Leonard, W.J. (1975). Nonlinear aspects of competition between species. SIAM J. of Applied Math 29, 243–253.
Maynard Smith, J. (1982) Evolution and the Theory of Games. Cambridge U. Press, Cambridge, England
Mollison, D. (1995) Epidemic Models: Their Structure and Relation to Data. Cambridge U. Press
Neubert, M.G., Kot, M., and Lewis, M.A. (1995) Dispersal and pattern formation in a discrete-time predator-prey model. Theoret. Pop. Biol. 48, 7–43
Neuhauser, C. (1992) Ergodic theorems for the multi-type contact process. Prob. Theor. Rel. Fields. 91, 467–506
Noble, C. (1992) Equilibrium behavior of the sexual reproduction process with rapid diffusion. Ann. Probab. 20, 724–745
Nowak, M.A., and May, R.M. (1993). Evolutionary games and spatial chaos. Nature 359, 826–829
Nowak, M.A., and May, R.M. (1993). The spatial dilemmas of evolution. Int. J. Bifurcation and Chaos. 3, 35–78
Nowak, M.A., Bonhoeffer, S., and May, R.M. (1994). More spatial games. Int. J. Bifurcation and Chaos. 4, 33–56
Owen, G. (1968) Game Theory. W.B. Saunders Co., Philadelphia
Pacala, S.W. and Levin, S.A. (1996). Biologically generated spatial pattern and the coexistence of competing species. In: (D. Tilman and P. Kareiva, eds.) Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions. Princeton University Press, Princeton, NJ. To appear.
Pascual, M. and Levin, S.A. (1998) From individuals to population densities: searching for the intermediate scale of nontrivial determinism. Preprint
Preston, F.W. (1948) The commoness, and rarity, of species. Ecology 29, 254–283
Preston, F.W. (1962). The canonical distribution of commonness and rarity. Ecology 43, I. 185–215, II. 410–432.
Rand, D.A., Keeling, M., and Wilson, H.B. (1995). Invasion, stability, and evolution to criticality in spatially extended, artificial host-pathogen ecologies. Proc. Roy. Soc. London B. 259, 55–63
Rand, D.A. and Wilson, H.B. (1995) Using spatio-temporal chaos and intermediate scale determinism in artificial ecologies to quantify spatially-extended systems. Proc. Roy. Soc. London. 259, 111–117
Redheffer, R., Redlinger, R. and Walter, W. (1988) A theorem of La Salle-Lyapunov type for parabolic systems. SIAM J. Math. Anal. 19, 121–132
Sawyer, S. (1979) A limit theorem for patch sizes in a selectively-neutral migration model. J Appl. Prob. 16, 482–495
Schlogl, F. (1972) Chemical reaction models for non-equilibrium phase transitions. Z. Physik 253, 147–161
Shah, N. (1997) Predator-mediated coexistence. Ph.D. Thesis Cornell U.
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.
Tainaka, K. (1993). Paradoxical effect in a three candidate voter model. Physics Letters A 176, 303–306.
Tainaka, K. (1995). Indirect effects in cyclic voter models. Physics Letters A 207, 53–57
Tramer, E.J. (1969) Bird species diversity; components of Shannon's formula. Ecology, 50, 927–929
Thoday, J.M. et al (1959–64) Effects of disruptive selection. I–IX. Heredity. 13, 187–203, 205–218; 14, 35–49; 15, 119–217; 16, 219–223; 17, 1–27; 18, 513–524; 19, 125–130
Tilman, D. (1994) Competition and bio-diversity in spatially structured habits. Ecology. 75, 2–16
Watson, H. (1835) Remarks on the Geographical Distribution of British Plants. Longman, London.
Webb, D.J. (1974) The statistics of relative abundance and diversity. J. Theor. Biol. 43, 277–292
Whittaker, R.H. (1970) Communities and Ecosystems, MacMillan, New York.
Williams, T. and Bjerknes, R. (1972) Stochastic model for abnormal clone spread through epithelial basal layer. Nature. 236, 19–21
Williamson, M. (1988). Relationship of species number to area, distance and other variables. Chapter 4 in Analytical Biogeography edited by A.A. Myers and P.S. Giller, Chapman and Hall, London.
Wilson, W.G. (1996) Lotka's game in predator-prey theory: linking populations to individuals. Theoret. Pop. Biol. 50, 368–393
Wilson, W.G., de Roos, A.M., and MacCauley, E. (1993) Spatial instabilities with the diffusive Lotka-Volterra system: Individual-based simulation results. Theoret. Pop. Biol. 43, 91–127
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Durrett, R. (1999). Stochastic spatial models. In: Capasso, V. (eds) Mathematics Inspired by Biology. Lecture Notes in Mathematics, vol 1714. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092375
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