Free energy of a chemotactic model with nonlinear diffusion

Seung Ki Baek,Beom Jun Kim

doi:10.1038/s41598-017-09369-w

Seung Ki Baek, Beom Jun Kim

Open Access

https://doi.org/10.1038/s41598-017-09369-w

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Abstract

The Patlak-Keller-Segel equation is a canonical model of chemotaxis to describe self-organized aggregation of organisms interacting with chemical signals. We investigate a variant of this model, assuming that the organisms exert effective pressure proportional to the number density. From the resulting set of partial differential equations, we derive a Lyapunov functional that can also be regarded as the free energy of this model, and minimize it with a Monte Carlo method to detect the condition for self-organized aggregation. Focusing on radially symmetric solutions on a two-dimensional disc, we find that the chemical interaction competes with diffusion so that aggregation occurs when the relative interaction strength exceeds a certain threshold. Based on the analysis of the free-energy landscape, we argue that the transition from a homogeneous state to aggregation is abrupt yet continuous.

Highlights

We show that the system described by Eqs (1) and (2) has a Lyapunov functional whose time derivative is smaller than or equal to zero all the time
We have derived its Lyapunov functional W in Eq (18), which may be called the free energy of this system
The linear stability analysis of the homogeneous solution predicts a jump in the amplitude of aggregation as a parameter K, defined in Eq (24), exceeds j11/l

Summary

Introduction

From the resulting set of partial differential equations, we derive a Lyapunov functional that can be regarded as the free energy of this model, and minimize it with a Monte Carlo method to detect the condition for self-organized aggregation. After examining two stationary states, of which one is homogeneous and the other is not, we investigate the Lyapunov functional in the normal-mode coordinates to examine the transition between the homogeneous and inhomogeneous states, restricting ourselves to radially symmetric solutions.

Results

Conclusion