Tenkès, L-M, Hollerbach, R and Kim, E-J (2017) Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models. Journal of Statistical Mechanics: Theory and Experiment, 2017 (December). 123201. ISSN 1742-5468
Abstract
A probabilistic description is essential for understanding growth processes in non-stationary states. 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 multiplicative and additive noise sources 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 in non-stationary states, 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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2017, IOP Publishing Ltd and SISSA Medialab srl. This is an author-created, un-copyedited version of an article published in Journal of Statistical Mechanics: Theory and Experiment. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at: https://doi.org/10.1088/1742-5468/aa9a66 |
Keywords: | stochastic processes; population dynamics |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 13 Nov 2017 13:27 |
Last Modified: | 06 Nov 2018 01:39 |
Status: | Published |
Publisher: | IOP Publishing |
Identification Number: | 10.1088/1742-5468/aa9a66 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:123895 |