Title: Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models,
Abstract: A probabilistic description is essential for understanding growth processes
far from equilibrium. In this paper, we compute time-dependent Probability
Density Functions (PDFs) in order to investigate stochastic logistic and
Gompertz models, which are two of the most popular growth models. We consider
different types of short-correlated internal (multiplicative) and external
(additive) stochastic noises and compare the time-dependent PDFs in the two
models, elucidating the effects of the additive and multiplicative noises on
the form of PDFs. We demonstrate an interesting transition from a unimodal to a
bimodal PDF as the multiplicative noise increases for a fixed value of the
additive noise. A much weaker (leaky) attractor in the Gompertz model leads to
a significant (singular) growth of the population of a very small size. We
point out the limitation of using stationary PDFs, mean value and variance in
understanding statistical properties of the growth far from equilibrium,
highlighting the importance of time-dependent PDFs. We further compare these
two models from the perspective of information change that occurs during the
growth process. Specifically, we define an infinitesimal distance at any time
by comparing two PDFs at times infinitesimally apart and sum these distances in
time. The total distance along the trajectory quantifies the total number of
different states that the system undergoes in time, and is called the
information length. We show that the time-evolution of the two models become
more similar when measured in units of the information length and point out the
merit of using the information length in unifying and understanding the dynamic
evolution of different growth processes.