Abstract
Spatial and stochastic models are often straightforward to simulate but difficult to analyze mathematically. Most of the mathematical methods available for nonlinear stochastic and spatial models are based on heuristic rather than mathematically justified assumptions, so that, e.g., the choice of the moment closure can be considered more of an art than a science. In this paper, we build on recent developments in specific branch of probability theory, Markov evolutions in the space of locally finite configurations, to develop a mathematically rigorous and practical framework that we expect to be widely applicable for theoretical ecology. In particular, we show how spatial moment equations of all orders can be systematically derived from the underlying individual-based assumptions. Further, as a new mathematical development, we go beyond mean-field theory by discussing how spatial moment equations can be perturbatively expanded around the mean-field model. While we have suggested such a perturbation expansion in our previous research, the present paper gives a rigorous mathematical justification. In addition to bringing mathematical rigor, the application of the mathematically well-established framework of Markov evolutions allows one to derive perturbation expansions in a transparent and systematic manner, which we hope will facilitate the application of the methods in theoretical ecology.
Similar content being viewed by others
References
Albeverio S, Kondratiev YG, Röckner M (1998) Analysis and geometry on configuration spaces. J Funct Anal 154(2):444–500
Baddeley A (2010) Multivariate and marked point processes. Handbook of spatial statistics
Baddeley A, Turner R (2005) spatstat: an R package for analyzing spatial point patterns. J Stat Softw 12(6):1–42
Barraquand F, Murrell DJ (2013) Scaling up predator-prey dynamics using spatial moment equations. Methods Ecol Evol 4(3):276–289
Berec L (2002) Techniques of spatially explicit individual-based models: construction, simulation, and mean-field analysis. Ecol Model 150(1–2):55–81
Bolker BM (2004) Continuous-space models for population dynamics. In: Hanski I, Gaggiotti O (eds) Ecology, genetics, and evolution in metapopulations. Academic, New York, pp 45–69
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
Cantrell S, Cosner C (2003) Spatial ecology via reaction-diffusion equations. Wiley, New York
Cantrell S, Cosner C, Ruan S (2010) Spatial ecology. Mathematical and computational biology series. Chapman and Hall/CRC, New York
Chesson P (2012) Scale transition theory: its aims, motivations and predictions. Ecol Complex 10:52–68
Cornell SJ, Ovaskainen O (2008) Exact asymptotic analysis for metapopulation dynamics on correlated dynamic landscapes. Theor Popul Biol 74(3):209–225
Dieckmann U, Law R, Metz JAJ (2000) The geometry of ecological interactions: simplifying spatial complexity. Cambridge University Press, Cambridge
Dodd PJ, Ferguson NM (2009) A many-body field theory approach to stochastic models in population biology. Plos One 4(9). doi:10.1371/journal.pone.0006855
Durrett R, Levin S (1994) The importance of being discrete (and spatial). Theor Popul Biol 46(3):363–394
Ellner SP (2001) Pair approximation for lattice models with multiple interaction scales. J Theor Biol 210(4):435–447
Filipe JAN, Gibson GJ (2001) Comparing approximations to spatio-temporal models for epidemics with local spread. Bull Math Biol 63(4):603–624
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
Finkelshtein D, Kondratiev Y, Kutoviy O (2010) Vlasov scaling for stochastic dynamics of continuous systems. J Stat Phys 141(1):158–178
Finkelshtein D, Kondratiev Y, Kutoviy O (2011) Vlasov scaling for the Glauber dynamics in continuum. Infinite Dimensional Analysis Quantum Probability and Related Topics 14(4):537– 569
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, Kozitsky Y (2013) Glauber dynamics in continuum: a constructive approach to evolution of states. Discrete and Continuous Dynamical Systems 33(4):1431– 1450
Gillespie DT (1977) Exact stochastic simulation of coupled chemical-reactions. J Phys Chem 81(25):2340–2361
Grimm V, Railsback SF (2005) Individual-based modelling and ecology. Princeton University Press, Princeton
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, Jorgensen C, Mooij WM, Mueller B, Pe’er G, Piou C, Railsback SF, Robbins AM, Robbins MM, Rossmanith E, Rueger N, Strand E, Souissi S, Stillman RA, Vabo R, Visser U, DeAngelis DL (2006) A standard protocol for describing individual-based and agent-based models. Ecol Model 198(1–2):115–126
Gurarie E, Ovaskainen O (2013) Towards a general formalization of encounter rates in ecology. Theor Ecol 6:189–202
Haase P (1995) Spatial pattern-analysis in ecology based on Ripley K-function—introduction and methods of edge correction. J Veg Sci 6(4):575–582
Hanski I, Gaggiotti O (2004) Ecology, genetics, and evolution in metapopulations. Academic, New York
Hanski I, Ovaskainen O (2000) The metapopulation capacity of a fragmented landscape. Nature 404(6779):755–758
Hiebeler D (2000) Populations on fragmented landscapes with spatially structured heterogeneities: landscape generation and local dispersal. Ecology 81(6):1629–1641
Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns. Statistics in practice. Wiley, Chichester
Illian JB, Sorbye SH, Rue H (2012) A toolbox for fitting complex spatial point process models using integrated nested Laplace approximation (INLA). Ann Appl Stat 6(4):1499–1530
Iwasa Y, Andreasen V, Levin S (1987) Aggregation in model-ecosystems. 1. Perfect aggregation. Ecol Model 37(3–4):287–302
Keeling MJ (2000) Multiplicative moments and measures of persistence in ecology. J Theor Biol 205(2):269–281
Keeling MJ, Rand DA, Morris AJ (1997) Correlation models for childhood epidemics. Proc R Soc Lond Ser B Biol Sci 264(1385):1149–1156
Kondratiev YG, Kuna T (2002) Harmonic analysis on configuration space - I. General theory. Infinite Dimensional Analysis Quantum Probability and Related Topics 5(2):201–233
Kondratiev Y, Skorokhod A (2006) On contact processes in continuum. Infinite Dimensional Analysis Quantum Probability and Related Topics 9(2):187–198
Kondratiev YG, Kuna T, Oliveira MJ (2006a) Holomorphic Bogoliubov functionals for interacting particle systems in continuum. J Funct Anal 238(2):375–404
Kondratiev YG, Kutoviy OV, Zhizhina E (2006b) Nonequilibrium Glauber-type dynamics in continuum. J Math Phys 47(11)
Kondratiev Y, Kutoviy O, Minlos R (2008a) On non-equilibrium stochastic dynamics for interacting particle systems in continuum. J Funct Anal 255(1):200–227
Kondratiev Y, Kutoviy O, Pirogov S (2008b) Correlation functions and invariant measures in continuous contact model. Infinite Dimensional Analysis Quantum Probability and Related Topics 11(2):231–258
Kondratiev Y, Kutoviy O, Minlos R (2010) Ergodicity of non-equilibrium Glauber dynamics in continuum. J Funct Anal 258(9):3097–3116
Kot M, Lewis MA, van den Driessche P (1996) Dispersal data and the spread of invading organisms. Ecology 77(7):2027–2042
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
Levermore CD (1996) Moment closure hierarchies for kinetic theories. J Stat Phys 83(5–6):1021–1065
Marion G, Mao XR, Renshaw E, Liu JL (2002) Spatial heterogeneity and the stability of reaction states in autocatalysis. Phys Rev E 66(5):051915
Matsuda H, Ogita N, Sasaki A, Sato K (1992) Statistical-mechanics of population—the lattice Lotka-Volterra model. Prog Theor Phys 88(6):1035–1049
Morozov A, Poggiale J-C (2012) From spatially explicit ecological models to mean-field dynamics: the state of the art and perspectives. Ecol Complex 10:1–11
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
North A, Ovaskainen O (2007) Interactions between dispersal, competition, and landscape heterogeneity. Oikos 116:1106–1119
North A, Cornell S, Ovaskainen O (2011a) Evolutionary responses of dispersal distance to landscape structure and habitat loss. Evolution 65(6):1739–1751
North A, Pennanen J, Ovaskainen O, Laine A-L (2011b) Local adaptation in a changing world: the roles of gene-flow, mutation, and sexual reproduction. Evolution 65:79–89
O’Dwyer JP, Green JL (2010) Field theory for biogeography: a spatially explicit model for predicting patterns of biodiversity. Ecol Lett 13(1):87–95
Okubo A, Levin SA (2001) Diffusion and ecological problems: modern perspectives. Interdisciplinary applied mathematics. Springer-Verlag, New York
Ovaskainen O, Cornell SJ (2006a) Asymptotically exact analysis of stochastic metapopulation dynamics with explicit spatial structure. Theor Popul Biol 69:13–33
Ovaskainen O, Cornell SJ (2006b) Space and stochasticity in population dynamics. PNAS 103:12781–12786
Penttinen A, Stoyan D, Henttonen HM (1992) Marked point-processes in forest statistics. For Sci 38(4):806–824
Presutti E (2009) Scaling limits in statistical mechanics and microstructures in continuum mechanics. Theoretical and mathematical physics. Springer, Berlin
Ruelle D (1964) Cluster property of the correlation functions of classical gases. Rev Mod Phys 35:580–584
Ruelle D (1969) Statistical mechanics. Rigorous results. Benjamins, New York
Shimatani K (2002) Point processes for fine-scale spatial genetics and molecular ecology. Biom J 44(3):325–352
Thompson HR (1955) Spatial point processes, with applications to ecology. Biometrika 42(1–2):102–115
Acknowledgments
The authors thank the Center for Interdisciplinary Research (ZIF) in Bielefeld, Germany for the possibility for organizing an International Research Programme in Mathematical Biology in 2012–2013. This work greatly benefited from the interactions during the Research Programme. Two anonymous reviewers are acknowledged for providing helpful comments. The study was supported financially by the Academy of Finland (grant no. 250444 to O.O) and the European Research Council (ERC starting grant no. 205905 to O.O.).
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Ovaskainen, O., Finkelshtein, D., Kutoviy, O. et al. A general mathematical framework for the analysis of spatiotemporal point processes. Theor Ecol 7, 101–113 (2014). https://doi.org/10.1007/s12080-013-0202-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12080-013-0202-8