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A probabilistic description is essential for understanding growth processesfar from equilibrium. In this paper, we compute time-dependent ProbabilityDensity Functions (PDFs) in order to investigate stochastic logistic andGompertz models, which are two of the most popular growth models. We considerdifferent types of short-correlated internal (multiplicative) and external(additive) stochastic noises and compare the time-dependent PDFs in the twomodels, elucidating the effects of the additive and multiplicative noises onthe form of PDFs. We demonstrate an interesting transition from a unimodal to abimodal PDF as the multiplicative noise increases for a fixed value of theadditive noise. A much weaker (leaky) attractor in the Gompertz model leads toa significant (singular) growth of the population of a very small size. Wepoint out the limitation of using stationary PDFs, mean value and variance inunderstanding statistical properties of the growth far from equilibrium,highlighting the importance of time-dependent PDFs. We further compare thesetwo models from the perspective of information change that occurs during thegrowth process. Specifically, we define an infinitesimal distance at any timeby comparing two PDFs at times infinitesimally apart and sum these distances intime. The total distance along the trajectory quantifies the total number ofdifferent states that the system undergoes in time, and is called theinformation length. We show that the time-evolution of the two models becomemore similar when measured in units of the information length and point out themerit of using the information length in unifying and understanding the dynamicevolution of different growth processes.

Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models