Correlated noise in a logistic growth model

Ai, Bao-quan; Wang, Xianju; Liu, Guotao; Liang-Gang, Liu

doi:10.1103/physreve.67.022903

Cited by 174 publications

(107 citation statements)

References 11 publications

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“…Two of the most popular models are the logistic and Gompertz models [1][2][3][4][5]. Mathematically, the utility of these models stems from the simplicity in which the growth is saturated to a finite value by a simple nonlinear damping term.…”

Section: Introductionmentioning

confidence: 99%

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

2017

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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.

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Section: Introductionmentioning

confidence: 99%

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

2017

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“…[16] for t * (t) = t, which has been obtained from the method of characteristic. It is worth mentioning that the coefficients h(x, t) and g(x, t) given by h(x, t) ∼ x and g(x, t) ∼ x might be used to investigate the barrier crossing problem in heavy-ion fusion reaction [17], and they can also be used as a limiting case of a Langevin equation for describing the tumor cell growth system [18].…”

Section: From Eq (8) We Obtainmentioning

confidence: 99%

Solution of Fokker-Planck equation for a broad class of drift and diffusion coefficients

2011

Phys. Rev. E

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“…The logistic model has been used as a basic model of tumor cell growth. A more refined logistic model without immune response was presented in [24][25][26]. A similar model, including the presence of correlated noises for the case of correlation time, has also been considered [27].…”

Section: Introductionmentioning

confidence: 99%

Delay-induced state transition and resonance in periodically driven tumor model with immune surveillance

et al. 2014

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Abstract:The phenomenon of stochastic resonance (SR) in a tumor growth model under the presence of immune surveillance is investigated. Time delay and cross-correlation between multiplicative and additive noises are considered in the system. The signal-to-noise ratio (SNR) is calculated when periodic signal is introduced multiplicatively. Our results show that: (i) the time delay can accelerate the transition from the state of stable tumor to that of extinction, however the correlation between two noises can accelerate the transition from the state of extinction to that of stable tumor; (ii) the time delay and correlation between two noises can lead to a transition between SR and double SR in the curve of SNR as a function of additive noise intensity, however for the curve of SNR as a function of multiplicative noise intensity, the time delay can cause the SR phenomenon to disappear, and the cross-correlation between two noises can lead to a transition from SR to stochastic reverse-resonance. Finally, we compare the SR phenomenon for the multiplicative periodic signal with that for additive periodic signal in the tumor growth model with immune surveillance. PACS

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Correlated noise in a logistic growth model

Cited by 174 publications

References 11 publications

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

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

Solution of Fokker-Planck equation for a broad class of drift and diffusion coefficients

Delay-induced state transition and resonance in periodically driven tumor model with immune surveillance

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