Abstract
The contact process is used as a simple spatial model in many disciplines, yet because of the buildup of spatial correlations, its dynamics remain difficult to capture analytically. We introduce an empirically based, approximate method of characterizing the spatial correlations with only a single adjustable parameter. This approximation allows us to recast the contact process in terms of a stochastic birth-death process, converting a spatiotemporal problem into a simpler temporal one. We obtain considerably more accurate predictions of equilibrium population than those given by pair approximations, as well as good predictions of population variance and first passage time distributions to a given (low) threshold. A similar approach is applicable to any model with a combination of global and nearest-neighbor interactions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allen, J. (1975). Mathematical models of species interactions in time and space. Am. Nat. 109, 319–342.
Barkham, J. P. and C. E. Hance (1982). Population dynamics of the wild daffodil (narcissus pseudonarcissus) III. Implications of a computer model of 1000 years of population change. J. Ecol. 70, 323–344.
Bascompte, J. (1999). Aggregate statistical measures and metapopulation dynamics. J. Theor. Biol. (Submitted).
Dickman, R. (1986). Kinetic phase transitions in a surface-reaction model: mean-field theory. Phys. Rev. A 34, 4246–4250.
Durrett, R. and S. A. Levin (1994). Stochastic spatial models: A user’s guide to ecological applications. Phil. Trans. R. Soc. Lond., Ser. B 343, 329–350.
Ellner, S. P., A. Sasaki, Y. Haraguchi and H. Matsuda (1998). Speed of invasion in lattice population models: Pair-edge approximation. J. Math. Biol. 36, 469–484.
Fisher, M. E. (1998). Renormalization group theory: Its basis and formulation in statistical physics. Rev. Mod. Phys. 70, 653–681.
Gurney, W. S. C. and R. M. Nisbet (1978). Single-species population fluctuations in patchy environments. Am. Nat. 112, 1075–1090.
Hanski, I. (1983). Coexistence of competitors in patchy environment. Ecology 64, 493–500.
Harada, Y., H. Ezoe, Y. Iwasa, H. Matsuda and K. Satō (1995). Population persistence and spatially limited social interaction. Theor. Popul. Biol. 48, 65–91.
Harada, Y. and Y. Iwasa (1994). Lattice population dynamics for plants with dispersing seeds and vegetative propagation. Res. Popul. Ecol. (Kyoto) 36, 237–249.
Kareiva, P. (1990). Population dynamics in spatially complex environments: Theory and data. Phil. Trans. R. Soc. Lond., Ser. B 330, 175–190.
Katori, M. and N. Konno (1990). Correlation inequalities and lower bounds for the critical value lambda/sub c/ of contact processes. J. Phys. Soc. Japan 59, 877–887.
Keeling, M. (1999). Spatial models of interacting populations, in Advanced Ecological Theory: Principles and Applications, J. McGlade (Ed.), Oxford: Blackwell Science, pp. 64–99.
Levins, R. (1969). Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entomol. Soc. Am. 15, 237–240.
Matsuda, H., N. Ogita, A. Sasaki and K. Satō (1992). Statistical mechanics of population-the lattice Lotka-Volterra model. Prog. Theor. Phys. 88, 1035–1049.
McCauley, E., W. G. Wilson and A. M. De Roos (1993). Dynamics of age-structured and spatially structured predator-prey interactions individual-based models and population-level formulations. Am. Nat. 142, 412–442.
Nachman, G. (1987). Systems analysis of acarine predator-prey interactions: II. The role of spatial processes in system stability spatial processes in system stability. J. Anim. Ecol. 56, 267–282.
Nisbet, R. M. and W. S. C. Gurney (1982). Modelling Fluctuating Populations, New York: John Wiley & Sons.
Nisbet, R. M., W. S. C. Gurney and M. A. Pettipher (1977). An evaluation of linear models of population fluctuations. J. Theor. Biol. 68, 143–160.
Ohtsuki, T. and T. Keyes (1986). Kinetic growth percolation: epidemic processes with immunization. Phys. Rev. A 33, 1223–1232.
Rand, D. A. (1999). Correlation equations and pair approximations for spatial ecologies, in Advanced Ecological Theory: Principles and Applications, J. McGlade (Ed.), Oxford: Blackwell Science, pp. 100–142.
Reeve, J. D. (1988). Environmental variability, migration, and persistence in host-parasitoid systems. Am. Nat. 132, 810–836.
Renfrew, E. (1991). Modelling Biological Populations in Space and Time, New York: Cambridge University Press.
Richards, S. A., W. G. Wilson and J. E. S. Socolar (1999). Selection for short-lived and offspring-limited individuals in a spatially-structured population which is subject to localized disturbances. Unpublished manuscript.
Satō, K., H. Matsuda and A. Sasaki (1994). Pathogen invasion and host extinction in lattice structured populations. J. Math. Biol. 32, 251–268.
Shigesada, N. and K. Kawasaki (1997). Biological Invasions: Theory and Practice, New York: Oxford University Press.
Slatkin, M. (1974). Competition and regional coexistence. Biometrika 55, 128–134.
Snyder, R. E. (2000). Effects of spatial correlations and fluctuations on population dynamics, PhD thesis, University of California, Santa Barbara (in preparation).
van Baalen, M. and D. A. Rand (1998). The unit of selection in viscous populations and the evolution of altruism. J. Theor. Biol. 193, 631–648.
van Kampen, N. G. (1981). Stochastic Processes in Physics and Chemistry, Amsterdam: North-Holland.
Wilson, K. G. (1983). The renormalization group and critical phenomena. Rev. Mod. Phys. 55, 583–600. Nobel Prize Lecture.
Wilson, W. G., R. M. Nisbet, A. H. Ross, C. Robles and R. A. Desharnais (1996). Abrupt population changes along smooth environmental gradients. Bull. Math. Biol. 58, 907–922.
Zeigler, B. P. (1977). Persistence and patchiness of predator-prey systems induced by discrete event population exchange mechanisms. J. Theor. Biol. 67, 687–713.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Snyder, R.E., Nisbet, R.M. Spatial structure and fluctuations in the contact process and related models. Bull. Math. Biol. 62, 959–975 (2000). https://doi.org/10.1006/bulm.2000.0191
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1006/bulm.2000.0191