### 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.

Original language | English |
---|---|

Article number | 123201 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2017 |

Issue number | 12 |

DOIs | |

Publication status | Published - 6 Dec 2017 |

Externally published | Yes |

### Fingerprint

### Keywords

- population dynamics
- stochastic processes

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Statistical Mechanics: Theory and Experiment*,

*2017*(12), [123201]. https://doi.org/10.1088/1742-5468/aa9a66

**Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models.** / Tenkes, Lucille Marie; Hollerbach, Rainer; Kim, Eun Jin.

Research output: Contribution to journal › Article

*Journal of Statistical Mechanics: Theory and Experiment*, vol. 2017, no. 12, 123201. https://doi.org/10.1088/1742-5468/aa9a66

}

TY - JOUR

T1 - Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models

AU - Tenkes, Lucille Marie

AU - Hollerbach, Rainer

AU - Kim, Eun Jin

PY - 2017/12/6

Y1 - 2017/12/6

N2 - 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.

AB - 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.

KW - population dynamics

KW - stochastic processes

UR - http://www.scopus.com/inward/record.url?scp=85039801840&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/aa9a66

DO - 10.1088/1742-5468/aa9a66

M3 - Article

VL - 2017

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 12

M1 - 123201

ER -