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A general mathematical framework for the analysis of spatiotemporal point processes

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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.

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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.).

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Authors and Affiliations

  1. Department of Biosciences, University of Helsinki, Viikinkaari 1, Helsinki, 00014, Finland

    Otso Ovaskainen

  2. Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK

    Dmitri Finkelshtein

  3. Fakultät für Mathematik, Universität Bielefeld, 33615, Bielefeld, Germany

    Oleksandr Kutoviy & Yuri Kondratiev

  4. Department of Mathematics, MIT, 77 Massachusetts Avenue2-155, Cambridge, MA, USA

    Oleksandr Kutoviy

  5. Institute of Integrative Biology, University of Liverpool, Liverpool, L69 7ZB, UK

    Stephen Cornell

  6. Department of Mathematics and Statistics, McMaster Universit, Hamilton, Ontario, L8S 4K1, Canada

    Benjamin Bolker

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  1. Otso Ovaskainen
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  2. Dmitri Finkelshtein
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  3. Oleksandr Kutoviy
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  6. Yuri Kondratiev
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Correspondence to Otso Ovaskainen.

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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

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