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Random walk models in the life sciences: including births, deaths and local interactions
Michael J Plank1, Matthew J Simpson2,3, Ruth E Baker4
1School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand.
Journal of the Royal Society, Interface
|January 14, 2025
Summary
Spatial stochastic models, or random walks, are enhanced by including individual interactions. This review covers models with interactions, focusing on continuum-limit descriptions and resulting spatial patterns for various applications.
Area of Science:
- Mathematical Biology
- Ecology
- Epidemiology
- Developmental Biology
- Oncology
Background:
- Classical random walk models assume independent individuals, simplifying mathematical analysis.
- Ignoring interactions can limit the accuracy of spatial stochastic models in real-world applications.
- Interactions like crowding, adhesion, and competition significantly influence population dynamics.
Purpose of the Study:
- To review progress in spatial stochastic models that incorporate interactions between individuals.
- To provide an accessible overview for both application researchers and specialist modelers.
- To focus on continuum-limit descriptions and mean-field models of spatial patterns.
Main Methods:
- Reviewing mathematical models of interacting individuals in spatial stochastic processes.
- Deriving asymptotically exact or approximate continuum-limit descriptions.
- Developing simplified deterministic models for mean-field behavior and spatial patterns.
Main Results:
- Models with interactions offer more realistic population dynamics compared to classical independent models.
- Continuum-limit descriptions and mean-field models capture emergent spatial patterns.
- Worked examples illustrate the behavior of selected interacting random walk models.
Conclusions:
- Models incorporating individual interactions are crucial for understanding complex population behaviors.
- Further research is needed to address current challenges and explore new frontiers in interacting spatial models.
- These models have broad applicability across ecology, epidemiology, and developmental biology.
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