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On Discretely Structured Growth Models and Their Moments.
Benjamin J Walker1, Helen M Byrne2
1Department of Mathematics, University College London, Gordon Street, London, WC1H 0AY, UK. benjamin.walker@ucl.ac.uk.
Bulletin of Mathematical Biology
|May 12, 2025
Summary
We generalized the logistic growth model for structured populations, developing exact, low-dimensional moment equations. This approach aids in understanding complex population dynamics and aids model selection for biological systems.
Area of Science:
- Mathematical and Computational Biology
- Population Dynamics
- Applied Mathematics
Background:
- The logistic equation is a fundamental model for saturating growth.
- Real-world populations often exhibit discrete structures (e.g., age, health status).
- Existing models may not fully capture dynamics in structured populations.
Purpose of the Study:
- To generalize the logistic growth model for discretely structured populations.
- To derive exact, low-dimensional moment equations for these generalized models.
- To demonstrate the utility of moment information for analyzing structured population dynamics.
Main Methods:
- Analysis of polynomial kinetics.
- Derivation of conditions for closed moment equations.
- Exploration of concrete examples from biology (e.g., aging, immune cell exhaustion).
Main Results:
- Established necessary and sufficient conditions for exact moment equations in structured logistic models.
- Developed a framework for analyzing population dynamics using coarse-grained moment information.
- Demonstrated the potential for model selection and hypothesis testing.
Conclusions:
- Generalized logistic growth models for structured populations can yield exact moment equations.
- Moment equations provide valuable insights into complex population dynamics.
- This work offers a powerful tool for mathematical and computational biology research.
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