Related Experiment Video
Updated: Jan 9, 2026
06:44
Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
Published on: September 23, 2025
480
Long-time behaviour and bifurcation analysis of a two-species aggregation-diffusion system on the torus
José A Carrillo1, Yurij Salmaniw1
1Mathematical Institute, University of Oxford, Oxford, England.
Calculus of Variations and Partial Differential Equations
|December 9, 2025
Summary
This study analyzes nonlocal aggregation-diffusion equations, revealing stable cell segregation patterns in a model of cell-cell adhesion. The findings advance understanding of pattern formation in biological systems.
Area of Science:
- Mathematical Biology
- Partial Differential Equations
- Nonlocal Analysis
Background:
- Nonlocal aggregation-diffusion equations model various phenomena, including biological pattern formation.
- Understanding the existence, stability, and bifurcation of stationary states is crucial for these models.
Purpose of the Study:
- To investigate stationary states in nonlocal aggregation-diffusion equations with linear diffusion and symmetric nonlocal interactions.
- To extend results for scalar equations under weaker hypotheses and perform a rigorous bifurcation analysis for two-species systems.
Main Methods:
- Bifurcation theory (Crandall & Rabinowitz framework) for the two-species system.
- Classification of solutions via fixed points of a nonlinear map.
- Derivation of Fréchet derivatives up to third order.
Main Results:
- Existence, regularity, bifurcation structure, and stability exchange confirmed for scalar equations under a bounded variation hypothesis.
- Classification of all solution branches from homogeneous states for the two-species system.
- Identification of stable segregation patterns relevant to cell-cell adhesion and cell sorting.
Conclusions:
- The study provides a comprehensive analysis of stationary states in nonlocal aggregation-diffusion equations.
- Rigorous mathematical framework confirms pattern formation in a cell-adhesion model.
- Findings offer insights into the onset of cell sorting driven by attractive interactions.
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