Free energy of a chemotactic model with nonlinear diffusion

Area of Science:

• Mathematical Biology
• Biophysics
• Chemical Ecology

Background:

• The Patlak-Keller-Segel equation models chemotaxis and self-organized aggregation.
• Chemotaxis describes how organisms move towards or away from chemical signals.
• Understanding aggregation dynamics is crucial in ecology and developmental biology.

Purpose of the Study:

• To investigate a variant of the Patlak-Keller-Segel model with organism-exerted pressure.
• To identify conditions favoring self-organized aggregation in this modified model.
• To analyze the transition dynamics from a homogeneous state to aggregation.

Main Methods:

• Derivation of a Lyapunov functional (free energy) from a modified Patlak-Keller-Segel system.
• Minimization of the free energy using Monte Carlo simulations.
• Analysis of radially symmetric solutions on a two-dimensional disc.

Main Results:

• A threshold for relative interaction strength was identified for aggregation to occur.
• Chemical interaction and diffusion were found to compete, influencing aggregation.
• The transition to aggregation was characterized as abrupt yet continuous.

Conclusions:

• The modified model provides insights into self-organized aggregation driven by chemical signals and organism pressure.
• The free-energy landscape analysis reveals the nature of the aggregation transition.
• This work contributes to understanding collective behaviors in biological systems.