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

The generalized logistic equation is proposed to model kinetics and statistics of natural processes such as earthquakes, forest fires, floods, landslides, and many others. This equation has the form dN(A)dA=s⋅(1−N(A))⋅N(A)q⋅A−α,q>0 and A>0 is the size of an element of a structure, and α≥0. The equation contains two exponents α and q taking into account two important properties of elements of a system: their fractal geometry, and their ability to interact either to enhance or to damp the process of aggregation. The function N(A) can be understood as an approximation to the number of elements the size of which is less than A. The function dN(A)/dA where N(A) is 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 relation with Tsallis non-extensive statistics is demonstrated by solving the generalized logistic equation for q>0. In the case 01 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 α. 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.