  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.
