[1] 
Minus van Baalen.
Pair approximations for different spatial geometries.
In Dieckmann et al. [10], chapter 19, pages 359387. [ bib ] 
[2] 
R. Law, D. W. Purves., D. J. Murrell, and U. Dieckmann.
Causes and effects of smallscale spatial structure in plant
populations.
In J. Silvertown and J. Antonovics, editors, Integrating Ecology
and Evolution in a Spatial Context, pages 2144. Blackwell Science, Oxford,
UK, 2001. [ bib ]
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):631648, 1998. [ bib ] 
[4] 
Benjamin M. Bolker.
Analytic models for the patchy spread of plant disease.
Bulletin of Mathematical Biology, 61:849874, 1999. [ bib ]
Basic application of spatial moment closure (power1) to dynamics of a simple epidemic. Considers the covariance dynamics in a Poissondistributed 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, 52:179197, 1997. [ bib ]
Spatial moment equations (power1 closure) for the spatial logistic model, onespecies 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 Analytical methods for solving for equilibrium are presented (but these methods are somewhat clumsy, and are superseded by those in Bolker and Pacala 1999).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, 153:575602, 1999. [ bib ]
Analyzes the spatial LotkaVolterra model: twospecies competition in a point process model. Decomposes spatial covariances into a series of terms that affect competitive invasion, attributing different terms to competitioncolonization tradeoffs, successional niches, or phalanx growth. Uses power1 closure and Besselfunction competition and dispersal kernels to get analytically tractable invasion criteria. (However, power1 closure means that invasion criteria only act sensibly when the invader would lose in the nonspatial 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
388411. Cambridge University Press, 2000. [ bib ] 
[8] 
T. Caraco, M. C. Duryea, S. Glavanakov, W. Maniatty, and B. K. Szymanski.
Host spatial heterogeneity and the spread of vectorborne infection.
Theoretical Population Biology, 59(3):185206, 2001. [ bib ] 
[9] 
Ulf Dieckmann and Richard Law.
Relaxation projections and the method of moments.
In Dieckmann et al. [10], chapter 21, pages 412455. [ bib ] 
[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. [ bib ] 
[11] 
U. Dieckmann, B. O'Hara, and W. Weisser.
The evolutionary ecology of dispersal.
Trends in Ecology & Evolution, 14(3):8890, 1999. [ bib ] 
[12] 
Jonathan Dushoff.
Host heterogeneity and disease endemicity: A momentbased approach.
Theoretical Population Biology, 56:325335, 1999. [ bib ]
nonspatial moment closure for the dynamics of the distribution of host susceptibility. Incorporates a particularly clever scaled closure that interpolates between invasionphase and equilibriumphase statistics

[13] 
S. P. Ellner.
Pair approximation for lattice models with multiple interaction
scales.
Journal of Theoretical Biology, 210(4):435447, jun 21 2001. [ bib ] 
[14] 
Stephen P. Ellner, Akira Sasaki, Yoshihiro Haraguchi, and Hirotsugu Matsuda.
Speed of invasion in lattice population models: pairedge
approximation.
Journal of Mathematical Biology, 36(5):469484, 1998. [ bib ] 
[15] 
J. A. N. Filipe.
Hybrid closureapproximation to epidemic models.
Physica A, 266(14):238241, 1999. [ bib ]
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 parameterspace, including the critical region.

[16] 
J. A. N. Filipe and G. J. Gibson.
Comparing approximations to spatiotemporal models for epidemics with
local spread.
Bulletin of Mathematical Biology, 63(4):603624, July 2001. [ bib ]
Here we review and report recent progress on closure approximations applicable to lattice models with nearestneighbour 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 longterm, stationary behaviour over the whole parameter range of the model.

[17] 
J. A. N. Filipe and G. J. Gibson.
Studying and approximating spatiotemporal models for epidemic spread
and control.
Philosophical Transactions of the Royal Society of London Series
Bbiological Sciences, 353(1378):21532162, dec 29 1998. [ bib ]
A class of simple spatiotemporal stochastic models for the spread and control of plant disease is investigated. We consider a latticebased susceptibleinfected 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 shortrange :interaction representing the secondary infection of susceptibles by infectives within the population. Recent datamodelling studies have suggested that the above model may describe the spread of aphidborne 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 cellularautomata 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 meanfield approximation (MF). Comparison with simulation results shows that PA is significantly superior to MF, predicting accurately both transient and longrun, 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):595632, July 2000. [ bib ] 
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Y. Harada, H. Ezoe, Y. Iwasa, H. Matsuda, and K. Sato.
Population persistence and spatially limited socialinteraction.
Theoretical Population Biology, 48(1):6591, August 1995. [ bib ] 
[20] 
Yuko Harada and Yoh Iwasa.
Lattice population dynamics for plants with dispersing seeds and
vegetative propagation.
Researches on Population Ecology, 36(2):237249, 1994. [ bib ] 
[21] 
Y. Harada and Y. Iwasa.
Analyses of spatial patterns and population processes of clonal
plants.
Researches On Population Ecology, 38(2):153164, December
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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):8596, mar 21 2000. [ bib ] 
[23] 
D. Hiebeler.
Populations on fragmented landscapes with spatially structured
heterogeneities: landscape generation and local dispersal.
Ecology, 81(6):16291641, June 2000. [ bib ] 
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D. Hiebeler.
Stochastic spatial models: from simulations to mean field and local
structure approximations.
Journal of Theoretical Biology, 187(3):307319, aug 7 1997. [ bib ] 
[25] 
Yoh Iwasa.
Lattice models and pair approximation in ecology.
In Dieckmann et al. [10], chapter 13, pages 227251. [ bib ] 
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Y. Iwasa, M. Nakamaru, and S. A. Levin.
Allelopathy of bacteria in a lattice population: competition between
colicinsensitive and colicinproducing strains.
Evolutionary Ecology, 12(7):785802, October 1998. [ bib ] 
[27] 
Matthew J. Keeling.
Evolutionary dynamics in spatial hostparasite systems.
In Dieckmann et al. [10], chapter 15, pages 271291. [ bib ] 
[28] 
M. J. Keeling.
Metapopulation moments: coupling, stochasticity and persistence.
Journal of Animal Ecology, 69(5):725736, September 2000. [ bib ] 
[29] 
M. J. Keeling.
Multiplicative moments and measures of persistence in ecology.
Journal of Theoretical Biology, 205(2):269281, jul 21 2000. [ bib ] 
[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):953960, 1999. [ bib ] 
[31] 
M. J. Keeling, D. A. Rand, and A. J. Morris.
Correlation models for childhood epidemics.
Proceedings of the Royal Society of London B, 264:11491156,
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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. [ bib ] 
[33] 
Richard Law and Ulf Dieckmann.
Moment approximations of individualbased models.
In Dieckmann et al. [10], chapter 14, pages 252270. [ bib ] 
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R. Law and U. Dieckmann.
A dynamical system for neighborhoods in plant communities.
Ecology, 81(8):21372148, August 2000. [ bib ] 
[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
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[36] 
M. A. Lewis.
Spread rate for a nonlinear stochastic invasion.
Journal of Mathematical Biology, 41(5):430454, November 2000. [ bib ] 
[37] 
M. A. Lewis and S. Pacala.
Modeling and analysis of stochastic invasion processes.
Journal of Mathematical Biology, 41(5):387429, November 2000. [ bib ] 
[38] 
Hirotsuga Matsuda, Naofumi Ogita, Akira Sasaki, and Kazunori Sato.
Statistical mechanics of population: The lattice LotkaVolterra
model.
Progress of Theoretical Physics, 88(6):10351049, 1992. [ bib ] 
[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. 1214, 1995 to appear. [ bib ] 
[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):471483, May 2000. [ bib ]
This paper uses a moment approximation to an individualbased 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,
266:19331938, 1999. [ bib ]
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 latticestructured population.
Journal of Theoretical Biology, 184(1):6581, jan 7 1997. [ bib ] 
[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 204232. Princeton University Press, Princeton, NJ, 1998. [ bib ] 
[44] 
D. A. Rand.
Correlation equations and pair approximations for spatial ecologies.
In J. McGlade, editor, Theoretical Ecology 2. Blackwell, 1999. [ bib ] 
[45] 
Kazunori Sato and Yoh Iwasa.
Pair approximations for latticebased ecological models.
In Dieckmann et al. [10], chapter 18, pages 341358. [ bib ] 
[46] 
K. Sato, H. Matsuda, and A. Sasaki.
Pathogen invasion and host extinction in lattice structured
populations.
Journal of Mathematical Biology, 32(3):251268, February 1994. [ bib ] 
[47] 
Kazunori Sato, Hirotsugu Matsuda, and Akira Sasaki.
Pathogen invasion and host extinction in lattice structured
populations.
Journal of Mathematical Biology, 32:251268, 1994. [ bib ] 
[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):959975, 2000. [ bib ]
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 birthdeath 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 nearestneighbor interactions.

[49] 
Keiichi Tainaka.
Lattice model for the LotkaVolterra system.
Journal of the Physical Society of Japan, 57(88):25882590,
1988. [ bib ] 
[50] 
Kei ichi Tainaka.
Vortices and strings in a model ecosystem.
Physical Review E, 50(5):34013409, 1994. [ bib ] 
[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):6775, June 1997. [ bib ] 