Random death process for the regularization of subdiffusive anomalous equations

TL;DR

This paper introduces a random death process into subdiffusive fractional equations to improve their structural stability against spatial variations, resulting in a new advection-diffusion model applicable to morphogen gradient formation.

Contribution

The paper proposes a novel regularization method for fractional equations using a random death process, enhancing their stability and applicability to biological modeling.

Findings

The modified equation is stable under spatial variations of the anomalous exponent.

The asymptotic behavior is characterized analytically and via Monte Carlo simulations.

The resulting model describes morphogen gradient formation with novel advection-diffusion dynamics.

Abstract

Subdiffusive fractional equations are not structurally stable with respect to spatial perturbations to the anomalous exponent (Phys. Rev. E 85, 031132 (2012)). The question arises of applicability of these fractional equations to model real world phenomena. To rectify this problem we propose the inclusion of the random death process into the random walk scheme from which we arrive at the modified fractional master equation. We analyze the asymptotic behavior of this equation, both analytically and by Monte Carlo simulation, and show that this equation is structurally stable against spatial variations of anomalous exponent. Additionally, in the continuous and long time limit we arrived at an unusual advection-diffusion equation, where advection and diffusion coefficients depend on both the death rate and anomalous exponent. We apply the regularized fractional master equation to the problem of morphogen gradient formation.