A general mathematical framework for the analysis of spatiotemporal point processes
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Author(s)
Ovaskainen, Otso
Finkelshtein, Dmitri
Kutoviy, Oleksandr
Cornell, Stephen
Bolker, Benjamin
Kondratiev, Yuri
Kutovyi, Oleksandr
Date Issued
October 2013
Journal
Theoretical Ecology
Publisher
Springer Netherlands
Citation
Ovaskainen, Otso, Dmitri Finkelshtein, Oleksandr Kutoviy, Stephen Cornell, Benjamin Bolker, and Yuri Kondratiev. “A General Mathematical Framework for the Analysis of Spatiotemporal Point Processes.” Theor Ecol 7, no. 1 (October 22, 2013): 101–113.
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Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s12080-013-0202-8
