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Spatial Moment Description of Birth–Death–Movement Processes Incorporating the Effects of Crowding and Obstacles

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Abstract

Birth–death–movement processes, modulated by interactions between individuals, are fundamental to many cell biology processes. A key feature of the movement of cells within in vivo environments is the interactions between motile cells and stationary obstacles. Here we propose a multi-species model of individual-level motility, proliferation and death. This model is a spatial birth–death–movement stochastic process, a class of individual-based model (IBM) that is amenable to mathematical analysis. We present the IBM in a general multi-species framework and then focus on the case of a population of motile, proliferative agents in an environment populated by stationary, non-proliferative obstacles. To analyse the IBM, we derive a system of spatial moment equations governing the evolution of the density of agents and the density of pairs of agents. This approach avoids making the usual mean-field assumption so that our models can be used to study the formation of spatial structure, such as clustering and aggregation, and to understand how spatial structure influences population-level outcomes. Overall the spatial moment model provides a reasonably accurate prediction of the system dynamics, including important effects such as how varying the properties of the obstacles leads to different spatial patterns in the population of agents.

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Acknowledgements

This work is supported by the Australian Research Council (DP170100474). MJP is partly supported by Te Pūnaha Matatini, a New Zealand Centre of Research Excellence. We thank the anonymous referee for their helpful suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, Australia

    Anudeep Surendran & Matthew J. Simpson

  2. School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand

    Michael J. Plank

  3. Te Pūnaha Matatini, A New Zealand Centre of Research Excellence, Auckland, New Zealand

    Michael J. Plank

Authors
  1. Anudeep Surendran
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  2. Michael J. Plank
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  3. Matthew J. Simpson
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Corresponding author

Correspondence to Matthew J. Simpson.

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Surendran, A., Plank, M.J. & Simpson, M.J. Spatial Moment Description of Birth–Death–Movement Processes Incorporating the Effects of Crowding and Obstacles. Bull Math Biol 80, 2828–2855 (2018). https://doi.org/10.1007/s11538-018-0488-1

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  • DOI: https://doi.org/10.1007/s11538-018-0488-1

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