Abstract
Mathematical models of dynamical systems in the life sciences typically assume that biological systems are spatially well mixed (the mean-field assumption). Even spatially explicit differential equation models typically make a local mean-field assumption. In effect, the assumption is that diffusive movement is strong enough to destroy spatial structure or that interactions between individuals are sufficiently long-range that the effects of spatial structure are weak. However, many important biophysical processes, such as chemical reactions of biomolecules within cells, disease transmission among humans, and dispersal of plants, have characteristic spatial scales that can generate strong spatial structure at the scale of individuals, with important effects on the behaviour of biological systems. This calls for mathematical methods that incorporate spatial structure. Here, we focus on one method, spatial-moment dynamics, which is based on the idea that important information about a spatial point process is held in its low-order spatial moments. The method goes beyond the dynamics of the first moment, i.e. the mean density or concentration of agents in space, in which no information about spatial structure is retained. By including the dynamics of at least the second moment, the method retains some information about spatial structure. Whereas mean-field models effectively use a closure assumption for the second moment, spatial-moment models use a closure assumption for the third (or a higher-order) moment. The aim of the paper was to provide a parsimonious and intuitive derivation of spatial-moment dynamic equations that is accessible to non-specialists. The derivation builds naturally from the first moment to the second, and we show how it can be extended to higher-order moments. Rather than tying the model to a specific biological example, we formulate a general model of movement, birth, and death of multiple types of interacting agents. This model can be applied to problems from a range of disciplines, some of which we discuss. The derivation is performed in a spatially non-homogeneous setting, to facilitate future investigations of biological scenarios, such as invasions, in which the spatial patterns are non-stationary over space.
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References
Adams T, Holland EP, Law R, Plank MJ, Raghib M (2013) On the growth of locally interacting plants: differential equations for the dynamics of spatial moments. Ecology 94:2732–2743
Baker RE, Simpson MJ (2010) Correcting mean-field approximations for birth-death-movement processes. Phys. Rev. E 82(4):041905
Barraquand F, Murrell DJ (2012) Intense or spatially heterogeneous predation can select against prey dispersal. PLoS One 7:28924
Barraquand F, Murrell DJ (2013) Scaling up predator-prey dynamics using spatial moment equations. Methods Ecol Evol 4:276–289
Barraquand F, Murrell DJ (2012) Evolutionarily stable consumer home range size in relation to resource demography and consumer spatial organization. Theor Ecol 5(4):567–589
Blath J, Etheridge A, Meredith M (2007) Coexistence in locally regulated competing populations and survival of branching annihilating random walk. Ann Appl Probab 17(5–6): 1474–1507
Bolker B, Pacala SW (1997) Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor Popul Biol 52(3):179–197
Bolker BM (1999) Analytic models for the patchy spread of plant disease. Bull Math Biol 61(5):849–874
Bolker BM (2003) Combining endogenous and exogenous spatial variability in analytical population models. Theor Popul Biol 64(3):255–270
Bolker BM, Pacala SW (1999) Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. Am Nat 153(6):575–602
Bolker BM, Pacala SW, Neuhauser C (2003) Spatial dynamics in model plant communities: what do we really know? Am Nat 162:135–148
Bronstein JL, Wilson WG, Morris WF (2003) Ecological dynamics of mutualist/antagonist communities. Am Nat 162:S24–S39
Brown DH, Bolker BM (2004) The effects of disease dispersal and host clustering on the epidemic threshold in plants. Bull Math Biol 66(2):341–371
Bruna M, Chapman SJ (2012a) Diffusion of multiple species with excluded-volume effects. J Chem Phys 137(20):204116
Bruna M, Chapman SJ (2012b) Excluded-volume effects in the diffusion of hard spheres. Phys Rev E 85(1):011103
Bruna M, Chapman SJ (2014) Diffusion of finite-size particles in confined geometries. Bull Math Biol 76(4):947–982
Cantrell RS, Cosner C (2004) Deriving reaction-diffusion models in ecology from interacting particle systems. J Math Biol 48(2):187–217
Champagnat N, Méléard S (2007) Invasion and adaptive evolution for individual-based spatially structured populations. J Math Biol 55(2):147–188
Champagnat N, Ferrière R, Méléard S (2006) Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theor Popul Biol 69(3):297–321
Codling EA, Plank MJ, Benhamou S (2008) Random walk models in biology. J R Soc Interface 5(25):813–834
Cornell SJ, Ovaskainen O (2008) Exact asymptotic analysis for metapopulation dynamics on correlated dynamic landscapes. Theor Popul Biol 74:209–225
Deroulers C, Aubert M, Badoual M, Grammaticos B (2009) Modeling tumor cell migration: from microscopic to macroscopic models. Phys Rev E 79(3):031917
Dieckmann U, Law R (2000) Relaxation projections and the method of moments. In: Dieckmann U, Law R, Metz JAJ (eds) The Geometry of ecological interactions: simplifying spatial complexity. Cambridge University Press, Cambridge, pp 412–455
Ellner SP (2001) Pair approximation for lattice models with multiple interaction scales. J Theor Biol 210:435–447
Fernando AE, Landman KA, Simpson MJ (2010) Nonlinear diffusion and exclusion processes with contact interactions. Phys Rev E 81(1):011903
Finkelshtein D, Kondratiev Y, Kutoviy O (2009) Individual based model with competition in spatial ecology. SIAM J Math Anal 41(1):297–317
Finkelshtein D, Kondratiev Y, Kutoviy O (2012) Semigroup approach to birth-and-death stochastic dynamics in continuum. J Funct Anal 262(3):1274–1308
Finkelshtein D, Kondratiev Y, Kutoviy O (2013) Establishment and fecundity in spatial ecological models: statistical approach and kinetic equations. Infinit Dimens Anal Quantum Probab Relat Top 16(02)
Finkelshtein DL, Kondratiev YG, Oliveira MJ (2009) Markov evolutions and hierarchical equations in the continuum. I: one-component systems. J Evol Equ 9(2):197–233
Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7(4):355–369
Grey D (2000) personal communication
Grimm V, Berger U, Bastiansen F, Eliassen S, Ginot V, Giske J, Goss-Custard J, Grand T, Heinz SK, Huse G, Huth A, Jepsen JU, Jørgensen C, Mooij WM, Müller B, Pe’er G, Piou C, Railsback SF, Robbins AM, Robbins MM, Rossmanith E, Rüger N, Strand E, Souissi S, Stillman RA, Vabø R, Visser U, DeAngelis DL (2006) A standard protocol for describing individual-based and agent-based models. Ecol Model 198(1):115–126
Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns. Wiley, Chichester
Johnston ST, Simpson MJ, Baker RE (2012) Mean-field descriptions of collective migration with strong adhesion. Phys Rev E 85(5):051922
Keeling MJ, Rand DA, Morris AJ (1997) Correlation models for childhood epidemics. Proc R Soc Lond B 264(1385):1149–1156
Keeling MJ (2000) Multiplicative moments and measures of persistence in ecology. J Theor Biol 205(2):269–281
Keeling MJ (1999) The effects of local spatial structure on epidemiological invasions. Proc R Soc Lond B 266(1421):859–867
Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond A 115(772):700–721
Khain E, Sander LM, Schneider-Mizell CM (2007) The role of cell-cell adhesion in wound healing. J Stat Phys 128(1–2):209–218
Kirkwood JG (1935) Statistical mechanics of fluid mixtures. J Chem Phys 3:300–313
Kiss IZ, Green DM, Kao RR (2005) Disease contact tracing in random and clustered networks. Proc R Soc Lond B 272(1570):1407–1414
Kolmogorov AN, Petrovsky IG, Piskunov NS (1937) Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Moscow Univ Bull Math 1:1–25
Kondratiev YG, Kuna T (2002) Harmonic analysis on configuration space I: general theory. Infinit Dimens Anal Quantum Probab Relat Top 5(02):201–233
Law R, Dieckmann U (2000) A dynamical system for neighborhoods in plant communities. Ecology 81:2137–2148
Law R, Murrell DJ, Dieckmann U (2003) Population growth in space and time: spatial logistic equations. Ecology 84(1):252–262
Law R, Illian J, Burslem DFRP, Gratzer G, Gunatilleke CVS, Gunatilleke IAUN (2009) Ecological information from spatial patterns of plants: insights from point process theory. J Ecol 97(4):616–628
Lewis MA (2000) Spread rate for a nonlinear stochastic invasion. J Math Biol 41(5):430–454
Lewis MA, Pacala S (2000) Modeling and analysis of stochastic invasion processes. J Math Biol 41(5):387–429
Liggett TM (1999) Stochastic interacting systems: contact, voter and exclusion processes. Springer, Berlin
Llambi LD, Law R, Hodge A (2004) Temporal changes in local spatial structure of late-successional species: establishment of an Andean caulescent rosette plant. J Ecol 92:122–131
Lotka AJ (1920) Undamped oscillations derives from the law of mass action. J Am Chem Soc 42:1595–1599
Matsuda H, Ogita N, Sasaki A, Sato K (1992) Statistical mechanics of population—the lattice Lotka–Volterra model. Prog Theor Phys 88:1035–1049
Medlock J, Kot M (2003) Spreading disease: integro-differential equations old and new. Math Biosci 184(2):201–222
Murray JD (1989) Mathematical biology. Springer, New York
Murrell DJ (2005) Local spatial structure and predator-prey dynamics: counterintuitive effects of prey enrichment. Am Nat 166:354–367
Murrell DJ (2009) On the emergent spatial structure of size-structured populations: when does self-thinning lead to a reduction in clustering? J Ecol 97:256–266
Murrell DJ, Law R (2000) Beetles in fragmented woodlands: a formal framework for dynamics of movement in ecological landscapes. J Anim Ecol 69(3):471–483
Murrell DJ, Law R (2003) Heteromyopia and the spatial coexistence of similar competitors. Ecol Lett 6(1):48–59
Murrell DJ, Dieckmann U, Law R (2004) On moment closures for population dynamics in continuous space. J Theor Biol 229(3):421–432
Niazi M, Hussain A (2011) Agent-based computing from multi-agent systems to agent-based models: a visual survey. Scientometrics 89:479–499
Noble JV (1974) Geographic and temporal development of plagues. Nature 250:726–729
North A, Ovaskainen O (2007) Interactions between dispersal, competition, and landscape heterogeneity. Oikos 116:1106–1119
Okubo A, Maini PK, Williamson MH, Murray JD (1989) On the spatial spread of the grey squirrel in Britain. Proc R Soc Lond B 238(1291):113–125
Ovaskainen O, Cornell SJ (2006) Asymptotically exact analysis of stochastic metapopulation dynamics with explicit spatial structure. Theor Popul Biol 69(1):13–33
Ovaskainen O, Finkelshtein D, Kutoviy O, Cornell S, Bolker B, Kondratiev Y (2014) A general mathematical framework for the analysis of spatiotemporal point processes. Theor Ecol 7:101–113
Petermann T, De Los P (2004) Rios, Cluster approximations for epidemic processes: a systematic description of correlations beyond the pair level. Theor Ecol 229:1–11
Plank MJ, Simpson MJ (2012) Models of collective cell behaviour with crowding effects: comparing lattice-based and lattice-free approaches. J R Soc Interface 9(76):2983–2996
Purves DW, Law R (2002) Fine-scale spatial structure in a grassland community: quantifying the plant’s-eye view. J Ecol 90:121–129
Raghib M, Hill NA, Dieckmann U (2011) A multiscale maximum entropy moment closure for locally regulated space–time point process models of population dynamics. J Math Biol 62(5):605–653
Shigesada N, Kawasaki K (1997) Biological invasions: theory and practice. Oxford University Press, Oxford
Simpson MJ, Baker RE (2011) Corrected mean-field models for spatially dependent advection–diffusion–reaction phenomena. Phys Rev E 83(5):051922
Simpson MJ, Landman KA, Hughes BD (2009) Multi-species simple exclusion processes. Phys A 388(4):399–406
Simpson MJ, Landman KA, Hughes BD (2010) Cell invasion with proliferation mechanisms motivated by time-lapse data. Phys A 389(18):3779–3790
Simpson MJ, Merrifield A, Landman KA, Hughes BD (2007) Simulating invasion with cellular automata: connecting cell-scale and population-scale properties. Phys Rev E 76(2):021918
Simpson MJ, Treloar KK, Binder BJ, Haridas P, Manton KJ, Leavesley DI, McElwain DLS, Baker RE (2013) Quantifying the roles of cell motility and cell proliferation in a circular barrier assay. J R Soc Interface 10(82):20130007. doi:10.1098/rsif.2013.0007
Singer A (2004) Maximum entropy formulation of the Kirkwood superposition approximation. J Chem Phys 121:3657–3666
Stoyan D, Penttinen A (2000) Recent applications of point process methods in forestry statistics. Stat Sci 15(1):61–78
Van Baalen M (2000) Pair approximations for different spatial geometries. In: Dieckmann U, Law R, Metz JAJ (eds) The Geometry of ecological interactions: simplifying spatial complexity. Cambridge University Press, Cambridge, pp 359–387
Verhulst PF (1836) Notice sur la loi que la population suit dans son accroissement. Corresp Math Phys 10:113–121
Volterra V (1927) Variazioni e fluttuazioni del numero d’individui in specie animali conviventi
Young HM, Bergner AJ, Anderson RB, Enomoto H, Milbrandt J, Newgreen DF, Whitington PM (2004) Dynamics of neural crest-derived cell migration in the embryonic mouse gut. Dev Biol 270(2):455–473
Acknowledgments
This work owes its origin to unpublished ideas of Dr. David Grey (University of Sheffield), and we are very grateful to him for his interest in the research area and for sharing his ideas. The research was supported by the RSNZ Marsden Fund, Grant Number 11-UOC-005.
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Plank, M.J., Law, R. Spatial Point Processes and Moment Dynamics in the Life Sciences: A Parsimonious Derivation and Some Extensions. Bull Math Biol 77, 586–613 (2015). https://doi.org/10.1007/s11538-014-0018-8
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DOI: https://doi.org/10.1007/s11538-014-0018-8