Genotype selection model with two time-correlated white noises

Zhang, X.-M.; Ai, Bao-quan

doi:10.1140/epjb/e2010-00015-8

Cited by 10 publications

(6 citation statements)

References 20 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%

“…A variety of growth models are widely used in understanding the dynamic evolution of populations, not only in humans, ecology and biosystems, but also financial markets, environment, chemical and physical systems, etc. 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%

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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%

Section: Introductionmentioning

confidence: 99%

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

2017

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“…However, to the leading order in the correlation time, an approximate approach to the Markovian Fokker-Planck equation can be found [3,4]. The study of complex biological and physical systems driven by colored noise have been given much attention in the recent decades [5][6][7][8][9]. Moreover, Transitions in a logistic growth model induced by noise coupling and noise color shows some interesting behavior [10].…”

Section: Introductionmentioning

confidence: 99%

Effect of tumor microenvironmental factors on the steady state of tumor growth with non-zero correlation time

Idris¹,

Bakar²

2016

ams

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The effect of non-immunogenic tumor microenvironmental factors on the steady state tumor growth dynamics modeled by multiplicative colored noise is investigated. Using the Novikov theorem and Fox approach, an approximate Fokker-Planck equation for the non-Markovian stochastic model is obtained and analytic expression for the steady state distribution P st (x) is derived. We find that the strength of the tumor response to the microenvironmental factors effect θ reduces the value of the steady state distribution P st (x) of tumor growth at weak correlation time τ which is an inhibitive effect, and at strong correlation time the inhibitive effect is suppressed and instead the value of the steady state distribution is increased which corresponds to growth enhancement. The result indicated that the growth effect exerted by the non-immunogenic tumor microenvironmental factors on tumor growth depend on the correlation time strength τ of the tumor response.

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“…In application, the theoretical model equation is the Langevin stochastic equation and the corresponding Fokker Planck equation for the time evolution of probability density, and such theoretical descriptions have been used in different context to study the behavior of noise driven systems [1][2][3], where the driven noises are either Gaussian white or colored noises and in the form of additive or multiplicative. Research in [4] first discovered that physical systems driven by simultaneous noises with common origin leads to correlated effects, and the study of complex systems driven by correlated noises have been given much attention especially in bistable system [5][6][7], in mode laser system [8,9], in gene selection and genetic transcription regulatory models [10][11][12] and in all these systems, interesting properties and system behaviors were discovered at the influence of noise correlated effects. In the study of tumor cell growth, mathematical model equations that closely captures the general features of tumor growth are considered in literature, and logistic equation is the most widely used deterministic model for theoretical study [13].…”

Section: Introductionmentioning

confidence: 99%

Steady state analysis for the effect of tumor microenvironmental factors on tumor growth dynamics driven by correlated noises

Bakar¹,

Idris²

2016

ams

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The steady state analysis for the effect of non-immunogenic tumor microenvironmental factors modeled by correlated additive and multiplicative noises with zero correlation time is investigated. The underlying transition probability for the stochastic model equation satisfies the Markovian Fokker-Planck equation, and the steady state distribution ρ st (x) for the tumor growth system is derived. Based on the numerical computation, we find that the effect of non-immunogenic microenvironmental factors within the tumor site with strength θ have a diffusive effect on the steady state distribution ρ st (x), and the tumor response to the surrounding non-immunogenic microenvironmental factors effects with strength D inhibits growth on both the steady state distribution ρ st (x) and the mean x st of the tumor population. The result also indicates that the stronger the correlation strength φ, the more the tumor responded to the surrounding non-immunogenic microenvironmental factors effect.

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Genotype selection model with two time-correlated white noises

Cited by 10 publications

References 20 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

Effect of tumor microenvironmental factors on the steady state of tumor growth with non-zero correlation time

Steady state analysis for the effect of tumor microenvironmental factors on tumor growth dynamics driven by correlated noises

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