[1] Minus van Baalen. Pair approximations for different spatial geometries. In Dieckmann et al. [10], chapter 19, pages 359-387.
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[2] R. Law, D. W. Purves., D. J. Murrell, and U. Dieckmann. Causes and effects of small-scale spatial structure in plant populations. In J. Silvertown and J. Antonovics, editors, Integrating Ecology and Evolution in a Spatial Context, pages 21-44. Blackwell Science, Oxford, UK, 2001.
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This chapter is a slightly more gentle introduction to moment dynamics and the importance of considering local interactions for plant populations and communities

[3] M. van Baalen and D. A. Rand. The unit of selection in viscous populations and the evolution of altruism. Journal of Theoretical Biology, 193(4):631-648, 1998.
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[4] Benjamin M. Bolker. Analytic models for the patchy spread of plant disease. Bulletin of Mathematical Biology, 61:849-874, 1999.
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Basic application of spatial moment closure (power-1) to dynamics of a simple epidemic. Considers the covariance dynamics in a Poisson-distributed and aggregated host populations, and looks briefly at epidemics with removal/recovery.

[5] B. M. Bolker and S. W. Pacala. Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theoretical Population Biology, 52:179-197, 1997.
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Spatial moment equations (power-1 closure) for the spatial logistic model, one-species competition. Analytical methods for solving for equilibrium are presented (but these methods are somewhat clumsy, and are superseded by those in Bolker and Pacala 1999). Predicts when equilibrium population patterns will be even vs. aggregated.

[6] B. M. Bolker and S. W. Pacala. Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. American Naturalist, 153:575-602, 1999.
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Analyzes the spatial Lotka-Volterra model: two-species competition in a point process model. Decomposes spatial covariances into a series of terms that affect competitive invasion, attributing different terms to competition-colonization tradeoffs, successional niches, or phalanx growth. Uses power-1 closure and Bessel-function competition and dispersal kernels to get analytically tractable invasion criteria. (However, power-1 closure means that invasion criteria only act sensibly when the invader would lose in the non-spatial case.)

[7] Benjamin M. Bolker, Stephen W. Pacala, and Simon A. Levin. Moment methods for stochastic processes in continuous space and time. In U. Dieckmann, R. Law, and J. A. J. Metz, editors, The Geometry of Ecological Interactions: Simplifying Spatial Complexity, pages 388-411. Cambridge University Press, 2000.
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[8] T. Caraco, M. C. Duryea, S. Glavanakov, W. Maniatty, and B. K. Szymanski. Host spatial heterogeneity and the spread of vector-borne infection. Theoretical Population Biology, 59(3):185-206, 2001.
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[9] Ulf Dieckmann and Richard Law. Relaxation projections and the method of moments. In Dieckmann et al. [10], chapter 21, pages 412-455.
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[10] Ulf Dieckmann, Richard Law, and Johan A. J. Metz, editors. The Geometry of Ecological Interactions: Simplifying Spatial Complexity. Cambridge Studies in Adaptive Dynamics. Cambridge University Press, Cambridge, UK, 2000.
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[11] U. Dieckmann, B. O'Hara, and W. Weisser. The evolutionary ecology of dispersal. Trends in Ecology & Evolution, 14(3):88-90, 1999.
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[12] Jonathan Dushoff. Host heterogeneity and disease endemicity: A moment-based approach. Theoretical Population Biology, 56:325-335, 1999.
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non-spatial moment closure for the dynamics of the distribution of host susceptibility. Incorporates a particularly clever scaled closure that interpolates between invasion-phase and equilibrium-phase statistics

[13] S. P. Ellner. Pair approximation for lattice models with multiple interaction scales. Journal of Theoretical Biology, 210(4):435-447, jun 21 2001.
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[14] Stephen P. Ellner, Akira Sasaki, Yoshihiro Haraguchi, and Hirotsugu Matsuda. Speed of invasion in lattice population models: pair-edge approximation. Journal of Mathematical Biology, 36(5):469-484, 1998.
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[15] J. A. N. Filipe. Hybrid closure-approximation to epidemic models. Physica A, 266(1-4):238-241, 1999.
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A hybrid approximation method that elaborates on the common pair approximation is proposed and shown to give very accurate predictions for the order parameter (epidemic size) over the whole parameter-space, including the critical region.

[16] J. A. N. Filipe and G. J. Gibson. Comparing approximations to spatio-temporal models for epidemics with local spread. Bulletin of Mathematical Biology, 63(4):603-624, July 2001.
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Here we review and report recent progress on closure approximations applicable to lattice models with nearest-neighbour interactions, including cluster approximations and elaborations on the pair (or pairwise) approximation. (SIS model) A hybrid pairwise approximation is shown to provide the best predictions of transient and long-term, stationary behaviour over the whole parameter range of the model.

[17] J. A. N. Filipe and G. J. Gibson. Studying and approximating spatio-temporal models for epidemic spread and control. Philosophical Transactions of the Royal Society of London Series B-biological Sciences, 353(1378):2153-2162, dec 29 1998.
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A class of simple spatio-temporal stochastic models for the spread and control of plant disease is investigated. We consider a lattice-based susceptible-infected model in which the infection of a host occurs through two distinct processes: a background infective challenge representing primary infection from external sources, and a short-range :interaction representing the secondary infection of susceptibles by infectives within the population. Recent data-modelling studies have suggested that the above model may describe the spread of aphid-borne virus diseases in orchards. In addition, we extend the model to represent the effects of different control strategies involving replantation (or recovery). The Contact Process is a particular case of this model. The behaviour of the model has been studied using cellular-automata simulations. An alternative approach is to formulate a set of deterministic differential equations that captures the essential dynamics of the stochastic system. Approximate solutions to this set of equations, describing the time evolution over the whole parameter range, have been obtained using the pairwise approximation (PA) as well as the most commonly used mean-field approximation (MF). Comparison with simulation results shows that PA is significantly superior to MF, predicting accurately both transient and long-run, stationary behaviour over relevant parts of the parameter space. The conditions for the validity of the approximations to the present model and extensions thereof are discussed.

[18] A. Gandhi, S. Levin, and S. Orszag. Moment expansions in spatial ecological models and moment closure through gaussian approximation. Bulletin of Mathematical Biology, 62(4):595-632, July 2000.
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[19] Y. Harada, H. Ezoe, Y. Iwasa, H. Matsuda, and K. Sato. Population persistence and spatially limited social-interaction. Theoretical Population Biology, 48(1):65-91, August 1995.
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[20] Yuko Harada and Yoh Iwasa. Lattice population dynamics for plants with dispersing seeds and vegetative propagation. Researches on Population Ecology, 36(2):237-249, 1994.
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[21] Y. Harada and Y. Iwasa. Analyses of spatial patterns and population processes of clonal plants. Researches On Population Ecology, 38(2):153-164, December 1996.
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[22] Y. Haraguchi and A. Sasaki. The evolution of parasite virulence and transmission rate in a spatially structured population. Journal of Theoretical Biology, 203(2):85-96, mar 21 2000.
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[23] D. Hiebeler. Populations on fragmented landscapes with spatially structured heterogeneities: landscape generation and local dispersal. Ecology, 81(6):1629-1641, June 2000.
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[24] D. Hiebeler. Stochastic spatial models: from simulations to mean field and local structure approximations. Journal of Theoretical Biology, 187(3):307-319, aug 7 1997.
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[25] Yoh Iwasa. Lattice models and pair approximation in ecology. In Dieckmann et al. [10], chapter 13, pages 227-251.
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[26] Y. Iwasa, M. Nakamaru, and S. A. Levin. Allelopathy of bacteria in a lattice population: competition between colicin-sensitive and colicin-producing strains. Evolutionary Ecology, 12(7):785-802, October 1998.
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[27] Matthew J. Keeling. Evolutionary dynamics in spatial host-parasite systems. In Dieckmann et al. [10], chapter 15, pages 271-291.
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[28] M. J. Keeling. Metapopulation moments: coupling, stochasticity and persistence. Journal of Animal Ecology, 69(5):725-736, September 2000.
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[29] M. J. Keeling. Multiplicative moments and measures of persistence in ecology. Journal of Theoretical Biology, 205(2):269-281, jul 21 2000.
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[30] M. J. Keeling. Correlation equations for endemic diseases: externally imposed and internally generated heterogeneity. Proceedings of the Royal Society of London Series B, 266(1422):953-960, 1999.
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[31] M. J. Keeling, D. A. Rand, and A. J. Morris. Correlation models for childhood epidemics. Proceedings of the Royal Society of London B, 264:1149-1156, 1997.
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[32] T. Kubo, Y. Iwasa, and N. Furumoto. Forest spatial dynamics with gap expansion: total gap area and gap size distribution. Journal of Theoretical Biology, 180(3):18, 1996.
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[33] Richard Law and Ulf Dieckmann. Moment approximations of individual-based models. In Dieckmann et al. [10], chapter 14, pages 252-270.
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[34] R. Law and U. Dieckmann. A dynamical system for neighborhoods in plant communities. Ecology, 81(8):2137-2148, August 2000.
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[35] S.A. Levin and S.W. Pacala. Theories of simplification and scaling in ecological systems. In D. Tilman and P. Kareiva, editors, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions. Princeton University Press, Princeton, NJ, 1998.
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[36] M. A. Lewis. Spread rate for a nonlinear stochastic invasion. Journal of Mathematical Biology, 41(5):430-454, November 2000.
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[37] M. A. Lewis and S. Pacala. Modeling and analysis of stochastic invasion processes. Journal of Mathematical Biology, 41(5):387-429, November 2000.
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[38] Hirotsuga Matsuda, Naofumi Ogita, Akira Sasaki, and Kazunori Sato. Statistical mechanics of population: The lattice Lotka-Volterra model. Progress of Theoretical Physics, 88(6):1035-1049, 1992.
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[39] Ryan McAllister, Juan Lin, Benjamin Bolker, and Stephen W. Pacala. Spatial correlations in population models with competition and dispersal. In Proceedings of the Symposium on Biological Complexity, Montevideo, Uruguay, Dec. 12-14, 1995 to appear.
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[40] D. J. Murrell and R. Law. Beetles in fragmented woodlands: a formal framework for dynamics of movement in ecological landscapes. Journal of Animal Ecology, 69(3):471-483, May 2000.
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This paper uses a moment approximation to an individual-based model of animals moving through a heterogeneous landscape taken from a satellite image of the UK. Note that the moment closure used here works only when the first moments (population densities) have no dynamics; i.e. when only the spatial structure of the population changes.

[41] M. Boots and A. Sasaki. 'small worlds' and the evolution of virulence: infection occurs locally and at a distance. Proceedings of the Royal Society of London, Series B, 266:1933-1938, 1999.
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Uses the pairwise approach to look at the effects of long range (global) dispersal and short range dispersal on the evolutionary dynamics of infectious diseases

[42] M. Nakamaru, H. Matsuda, and Y. Iwasa. The evolution of cooperation in a lattice-structured population. Journal of Theoretical Biology, 184(1):65-81, jan 7 1997.
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[43] S.W. Pacala and S.A. Levin. Biologically generated spatial pattern and the coexistence of competing species. In D. Tilman and P. Kareiva, editors, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, chapter 9, pages 204-232. Princeton University Press, Princeton, NJ, 1998.
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[44] D. A. Rand. Correlation equations and pair approximations for spatial ecologies. In J. McGlade, editor, Theoretical Ecology 2. Blackwell, 1999.
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[45] Kazunori Sato and Yoh Iwasa. Pair approximations for lattice-based ecological models. In Dieckmann et al. [10], chapter 18, pages 341-358.
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[46] K. Sato, H. Matsuda, and A. Sasaki. Pathogen invasion and host extinction in lattice structured populations. Journal of Mathematical Biology, 32(3):251-268, February 1994.
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[47] Kazunori Sato, Hirotsugu Matsuda, and Akira Sasaki. Pathogen invasion and host extinction in lattice structured populations. Journal of Mathematical Biology, 32:251-268, 1994.
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[48] Robin E. Snyder and Roger M. Nisbet. Spatial structure and fluctuations in the contact process and related models. Bulletin of Mathematical Biology, 62(5):959-975, 2000.
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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.

[49] Kei-ichi Tainaka. Lattice model for the Lotka-Volterra system. Journal of the Physical Society of Japan, 57(88):2588-2590, 1988.
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[50] Kei ichi Tainaka. Vortices and strings in a model ecosystem. Physical Review E, 50(5):3401-3409, 1994.
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[51] Y. Takenaka, H. Matsuda, and Y. Iwasa. Competition and evolutionary stability of plants in a spatially structured habitat. Researches On Population Ecology, 39(1):67-75, June 1997.
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