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A mathematical biologist's guide to absolute and convective instability.
Jonathan A Sherratt1, Ayawoa S Dagbovie, Frank M Hilker
1Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK, j.a.sherratt@hw.ac.uk.
Bulletin of Mathematical Biology
|November 26, 2013
Summary
This study reviews absolute and convective instability theory to improve mathematical understanding of complex biological spatiotemporal phenomena. The absolute spectrum approach offers new methods for distinguishing instabilities and predicting solution behavior in models.
Area of Science:
- Mathematical Biology
- Theoretical Ecology
- Nonlinear Dynamics
Background:
- Mathematical models excel at reproducing biological spatiotemporal phenomena.
- Numerical simulation capabilities exceed current mathematical understanding.
- Instability theory is crucial for understanding complex system dynamics.
Purpose of the Study:
- To review the theory of absolute and convective instability.
- To explain concepts of remnant and transient instability.
- To demonstrate the application of instability theory in biological models.
Main Methods:
- Explanation of absolute and convective instability concepts.
- Introduction to the absolute spectrum for distinguishing instability types.
- Numerical simulations of a river-based predator-prey model.
Main Results:
- Convectively unstable states can persist in solutions, unlike absolutely unstable states.
- Instability theory can explain qualitative transitions in solution behavior.
- The absolute spectrum provides a method for quantitative prediction of parameter-dependent changes.
Conclusions:
- Instability theory, particularly the absolute spectrum, enhances mathematical understanding of spatiotemporal biological systems.
- This framework aids in distinguishing and predicting system behaviors.
- Applications include understanding pattern formation and persistent features in ecological models.
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