Modeling statistics of the natural aggregation structures and processes with the solution of generalized logistic equation

TL;DR

This paper introduces a generalized logistic equation to model natural aggregation processes, linking it with non-extensive statistics and applying it to earthquake data for improved distribution modeling.

Contribution

It derives a new solution to the generalized logistic equation that captures natural process statistics and demonstrates its application to earthquake data fitting.

Findings

The model fits earthquake foreshock and aftershock distributions well.

The solution links with Tsallis non-extensive statistics.

It explains the power-law and stretched exponential behaviors.

Abstract

The generalized logistic equation is derived to model kinetics and statistics of natural processes such as earthquakes, forest fires, floods, landslides, and many others. The general solution of this equation for q=1 is a product of an increasing bounded function and power-law function with stretched exponential cut-off; the power-law distribution is asymptotically nested in the stretched exponential distribution. The relation with Tsallis non-extensive statistics is demonstrated by solving the generalized logistic equation for q>0. In the case 0<q<1 the equation models super-additive, and the case q>1 it models sub-additive structures. The Gutenberg-Richter G-R) formula results from interpretation of empirical data as a straight line in the area of stretched exponent with small {\alpha}. The solution is applied for modeling distribution of foreshocks and aftershocks in the regions of Napa Valley 2014, and Sumatra 2004 earthquakes fitting the observed data well, both qualitatively and quantitatively.