Bulletin of Mathematical Biology

, Volume 80, Issue 11, pp 2828–2855 | Cite as

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

  • Anudeep Surendran
  • Michael J. Plank
  • Matthew J. SimpsonEmail author
Original Article


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.


Collective cell migration Spatial moment dynamics Individual-based model Cell proliferation Cell migration 



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.

Supplementary material

11538_2018_488_MOESM1_ESM.pdf (1002 kb)
Supplementary material 1 (pdf 1002 KB)


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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesQueensland University of TechnologyBrisbaneAustralia
  2. 2.School of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand
  3. 3.Te Pūnaha Matatini, A New Zealand Centre of Research ExcellenceAucklandNew Zealand

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