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Distinguishing Between Long-Transient and Asymptotic States in a Biological Aggregation Model.

Jonathan R Potts1, Kevin J Painter2

  • 1School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, Sheffield, S3 7RH, UK. j.potts@sheffield.ac.uk.

Bulletin of Mathematical Biology
|February 10, 2024
PubMed
Summary

Mathematical models of biological aggregation, like the aggregation-diffusion equation, often show multi-peaked solutions. New energy minimization techniques reveal most multi-peaked solutions are transient, not stable states.

Keywords:
Aggregation–diffusion equationAsymptoticsBiological aggregationLong transientsMetastabilityNonlocal advection

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Area of Science:

  • Mathematical Biology
  • Nonlinear Dynamics
  • Computational Science

Background:

  • Biological systems exhibit emergent aggregations, necessitating mathematical models for understanding.
  • The aggregation-diffusion equation is a key model for studying aggregation phenomena.
  • Multi-peaked solutions in this model are often suspected to be long-lived transients rather than stable states.

Purpose of the Study:

  • To develop a novel analytical technique for distinguishing between transient and asymptotic states in the aggregation-diffusion equation.
  • To investigate the stability of multi-peaked solutions in a one-dimensional aggregation-diffusion model with linear diffusion.

Main Methods:

  • Approximation of the aggregation-diffusion equation using a limiting process and moment closure.
  • Analysis of local energy minima in the approximate system to predict asymptotic patterns.
  • Numerical verification of the analytical predictions regarding solution stability.

Main Results:

  • A novel energy minimization technique effectively distinguishes between transient and asymptotic states.
  • Almost all twin-peaked and multi-peaked solutions are found to be transient.
  • Transient solutions can exhibit arbitrarily long lifetimes, dependent on system parameters.

Conclusions:

  • The developed analytical technique provides accurate predictions for the stability of aggregation patterns.
  • Multi-peaked solutions in the studied model are predominantly transient, challenging previous assumptions.
  • Understanding the transient nature of these solutions is crucial for accurately modeling biological aggregation dynamics.