scholarly journals Dynamic Analysis of a Tumor-Immune System under Allee Effect

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Chunmei Zeng ◽  
Shaojuan Ma
Keyword(s):  
Immune System ◽  
Equilibrium Point ◽  
Allee Effect ◽  
Deterministic Model ◽  
First Passage Time ◽  
Stochastic Averaging ◽  
Threshold Value ◽  
Passage Time ◽  
Steady State Probability ◽  
Coexistence Equilibrium

In this paper, we develop a definite tumor-immune model considering Allee effect. The deterministic model is studied qualitatively by mathematical analysis method, including the positivity, boundness, and local stability of the solution. In addition, we explore the effect of random factors on the transition of the tumor-immune system from a stable coexistence equilibrium point to a stable tumor-free equilibrium point. Based on the method of stochastic averaging, we obtain the expressions of the steady-state probability density and the mean first-passage time. And we find that the Allee effect has the greatest impact on the number of cells in the system when the Allee threshold value is within a certain range; the intensity of random factors could affect the likelihood of the system crossing from the coexistence equilibrium to the tumor-free equilibrium.

scholarly journals Determinants of Brain Rhythm Burst Statistics

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Arthur S. Powanwe ◽  
André Longtin
Keyword(s):  
Optimal Parameter ◽  
First Passage Time ◽  
Stochastic Averaging ◽  
Gamma Oscillations ◽  
Passage Time ◽  
Recurrent Network ◽  
Stochastic Averaging Method ◽  
Model Parameters ◽  
Brain Rhythm

AbstractBrain rhythms recorded in vivo, such as gamma oscillations, are notoriously variable both in amplitude and frequency. They are characterized by transient epochs of higher amplitude known as bursts. It has been suggested that, despite their short-life and random occurrence, bursts in gamma and other rhythms can efficiently contribute to working memory or communication tasks. Abnormalities in bursts have also been associated with e.g. motor and psychiatric disorders. It is thus crucial to understand how single cell and connectivity parameters influence burst statistics and the corresponding brain states. To address this problem, we consider a generic stochastic recurrent network of Pyramidal Interneuron Network Gamma (PING) type. Using the stochastic averaging method, we derive dynamics for the phase and envelope of the amplitude process, and find that they depend on only two meta-parameters that combine all the model parameters. This allows us to identify an optimal parameter regime of healthy variability with similar statistics to those seen in vivo; in this regime, oscillations and bursts are supported by synaptic noise. The probability density for the rhythm’s envelope as well as the mean burst duration are then derived using first passage time analysis. Our analysis enables us to link burst attributes, such as duration and frequency content, to system parameters. Our general approach can be extended to different frequency bands, network topologies and extra populations. It provides the much needed insight into the biophysical determinants of rhythm burst statistics, and into what needs to be changed to correct rhythms with pathological statistics.

scholarly journals The Protein Hourglass: Exact First Passage Time Distributions for Protein Thresholds

2020 ◽  
Author(s):  
Krishna Rijal ◽  
Ashok Prasad ◽  
Dibyendu Das
Keyword(s):  
First Passage Time ◽  
Self Regulation ◽  
Threshold Value ◽  
Passage Time ◽  
First Passage Times ◽  
Lambda Phage ◽  
First Passage ◽  
Analytical Expression ◽  
Protein Levels ◽  
Passage Times

Protein thresholds have been shown to act as an ancient timekeeping device, such as in the time to lysis of E. coli infected with bacteriophage lambda. The time taken for protein levels to reach a particular threshold for the first time is defined as the first passage time of the protein synthesis system, which is a stochastic quantity. The first few moments of the distribution of first passage times were known earlier, but an analytical expression for the full distribution was not available. In this work, we derive an analytical expression for the first passage times for a long-lived protein. This expression allows us to calculate the full distribution not only for cases of no self-regulation, but also for both positive and negative self-regulation of the threshold protein. We show that the shape of the distribution matches previous experimental data on lambda-phage lysis time distributions. We also provide analytical expressions for the FPT distribution with non-zero degradation in Laplace space. Furthermore, we study the noise in the precision of the first passage times described by coefficient of variation (CV) of the distribution as a function of the protein threshold value. We show that under conditions of positive self-regulation, the CV declines monotonically with increasing protein threshold, while under conditions of linear negative self-regulation, there is an optimal protein threshold that minimizes the noise in the first passage times.

Population system with coupling between non-Gaussian and Gaussian colored noise under Allee effect

2018 ◽  
Vol 32 (24) ◽  
pp. 1850279 ◽  
Author(s):  
Yachao Yang ◽  
Dongxi Li
Keyword(s):  
Allee Effect ◽  
Colored Noise ◽  
First Passage Time ◽  
Planck Equation ◽  
Single Species ◽  
Passage Time ◽  
Population System ◽  
Stationary Probability Distribution ◽  
Gaussian Colored Noise ◽  
Non Gaussian

We investigate a stochastic model for single species population growth with strong and weak Allee effects subjected to coupling between non-Gaussian and Gaussian colored noise as well as nonzero cross-correlation in between. Stationary probability distribution of population model is obtained depending on the Fokker–Planck equation. The mean first-passage time is also calculated in order to quantify the time of transition between survival state and extinction state with Allee effect in population. The intensity of non-Gaussian colored noise can induce phase transition, and population may be vulnerable to extinction due to the increase in the intensity of non-Gaussian colored noise. Whether Allee effect is strong or weak, the increase in Allee threshold will not contribute to the survival and stability of the population. Further, the phenomenon of resonant activation is firstly discovered in the study of population dynamics with Allee effect. These behaviors can be interpreted on the basis of a biological model of population evolution.

Nonlinear Stochastic Optimal Control of Quasi Hamiltonian Systems

2005 ◽  
Author(s):  
W. Q. Zhu
Keyword(s):  
Optimal Control ◽  
Hamiltonian Systems ◽  
Stochastic Optimal Control ◽  
Control Strategies ◽  
First Passage Time ◽  
Stochastic Averaging ◽  
Passage Time ◽  
Stochastic Averaging Method ◽  
Dynamic Programming Principle ◽  
Stochastic Dynamic

In recent years, a class of nonlinear stochastic optimal control strategies were developed by the present author and his co-workers for minimizing the response, stabilization and maximizing the reliability and mean first-passage time of quasi Hamiltonian systems based on the stochastic averaging method for quasi Hamiltonian systems and the stochastic dynamic programming principle. This review summaries the basic idea, procedures and applications of these strategies and pointes out necessary further work.

First-Passage-Time Prototypes for Precipitation Statistics

2014 ◽  
Vol 71 (9) ◽  
pp. 3269-3291 ◽  
Author(s):  
Samuel N. Stechmann ◽  
J. David Neelin
Keyword(s):  
Water Vapor ◽  
Power Law ◽  
First Passage Time ◽  
Threshold Value ◽  
Passage Time ◽  
Precipitation Event ◽  
Prototype Model ◽  
First Passage ◽  
Stratiform Rain ◽  
Time Problem

Abstract Prototype models are presented for time series statistics of precipitation and column water vapor. In these models, precipitation events begin when the water vapor reaches a threshold value and end when it reaches a slightly lower threshold value, as motivated by recent observational and modeling studies. Using a stochastic forcing to parameterize moisture sources and sinks, this dynamics of reaching a threshold is a first-passage-time problem that can be solved analytically. Exact statistics are presented for precipitation event sizes and durations, for which the model predicts a probability density function (pdf) with a power law with exponent −. The range of power-law scaling extends from a characteristic small-event size to a characteristic large-event size, both of which are given explicitly in terms of the precipitation rate and water vapor variability. Outside this range, exponential scaling of event-size probability is shown. Furthermore, other statistics can be computed analytically, including cloud fraction, the pdf of water vapor, and the conditional mean and variance of precipitation (conditioned on the water vapor value). These statistics are compared with observational data for the transition to strong convection; the stochastic prototype captures a set of properties originally analyzed by analogy to critical phenomena. In a second prototype model, precipitation is further partitioned into deep convective and stratiform episodes. Additional exact statistics are presented, including stratiform rain fraction and cloud fractions, that suggest that even very simple temporal transition rules (for stratiform rain continuing after convective rain) can capture aspects of the role of stratiform precipitation in observed precipitation statistics.

Noise-Induced Chaos in a Piecewise Linear System

2017 ◽  
Vol 27 (09) ◽  
pp. 1750137 ◽  
Author(s):  
Chen Kong ◽  
Xian-Bin Liu
Keyword(s):  
Linear System ◽  
White Noise ◽  
First Passage Time ◽  
Piecewise Linear ◽  
Gaussian White Noise ◽  
Stochastic Averaging ◽  
Passage Time ◽  
Singular Perturbation Method ◽  
Generalized Cell Mapping ◽  
Piecewise Linear System

In the present paper, the phenomenon of noise-induced chaos in a piecewise linear system that is excited by Gaussian white noise is investigated. Firstly, the global dynamical behaviors of the deterministic piecewise linear system are investigated numerically in advance by using the generalized cell-mapping digraph (GCMD) method. Then, based on these global properties, the system that is excited by Gaussian white noise is introduced. Then, it is simplified by the stochastic averaging method, through which, a four-dimensional averaged Itô system is finally obtained. In order to reveal the phenomenon of noise-induced chaos quantitatively, MFPT (the mean first-passage time) is selected as the measure. The expression for MFPT is formulated by using the singular perturbation method and then a rather simple representation is obtained via the Laplace approximation, and within which, the concept of quasi-potential is introduced. Furthermore, with the rays method, the MFPT under a certain set of parameters is estimated. However, within the process of analysis, the authors had to face a difficult problem concerning the ill-conditioned matrix, which is the obstacle for the estimation of MFPT, which was then solved by applying one more approximation. Finally, the result is compared with the numerical one that is obtained by the Monte Carlo simulation.

scholarly journals Towards a full quantitative description of single-molecule reaction kinetics in biological cells

2018 ◽  
Vol 20 (24) ◽  
pp. 16393-16401 ◽  
Author(s):  
Denis S. Grebenkov ◽  
Ralf Metzler ◽  
Gleb Oshanin
Keyword(s):  
Single Molecule ◽  
First Passage Time ◽  
Quantitative Description ◽  
Threshold Value ◽  
Mathematical Concept ◽  
Passage Time ◽  
Molecule Reaction ◽  
Cylindrical Annulus ◽  
The Moment ◽  
First Time

The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. We present a robust explicit approach for obtaining the full distribution of FPT to a partially reactive target in a cylindrical-annulus domain.

Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation

2006 ◽  
Vol 59 (4) ◽  
pp. 230-248 ◽  
Author(s):  
W. Q. Zhu
Keyword(s):  
Hamiltonian Systems ◽  
Stochastic Dynamics ◽  
First Passage Time ◽  
Hamiltonian Formulation ◽  
Stochastic Averaging ◽  
Passage Time ◽  
Stochastic Bifurcation ◽  
Dynamics And Control ◽  
Nonlinear Stochastic Dynamics ◽  
And Control

The significant advances in nonlinear stochastic dynamics and control in Hamiltonian formulation during the past decade are reviewed. The exact stationary solutions and equivalent nonlinear system method of Gaussian-white -noises excited and dissipated Hamiltonian systems, the stochastic averaging method for quasi Hamiltonian systems, the stochastic stability, stochastic bifurcation, first-passage time and nonlinear stochastic optimal control of quasi Hamiltonian systems are summarized. Possible extension and applications of the theory are pointed out. This review article cites 158 references.

Piecewise asymmetric exponential potential under-damped bi-stable stochastic resonance and its application in bearing fault diagnosis

2021 ◽  
pp. 2150280
Author(s):  
Gang Zhang ◽  
Chunlin Tan ◽  
Lifang He
Keyword(s):  
Theoretical Analysis ◽  
Stochastic Resonance ◽  
Signal To Noise Ratio ◽  
First Passage Time ◽  
Passage Time ◽  
Adaptive Genetic Algorithm ◽  
Exponential Potential ◽  
Steady State Probability ◽  
Bearing Fault ◽  
Practical Engineering

It is difficult to extract weak signals in strong noise background, therefore a piecewise asymmetric exponential potential under-damped bi-stable stochastic resonance (PAEUBSR) system is proposed. First, the theoretical analysis of the steady-state probability density (SPD), mean first passage time (MFPT) and output signal-to-noise ratio (SNR) are derived under the adiabatic approximation theory. At the same time, the influence of different system parameters on system performance is explored. Then the PAEUBSR system is applied to the fault signal diagnosis of different types of bearings, and the parameters are optimized through the adaptive genetic algorithm (AGA). The test results are compared with the exponential potential over-damped symmetric bi-stable stochastic resonance (EOSBSR) system and the exponential potential under-damped symmetric bi-stable stochastic resonance (EUSBSR) system. Finally, the detection results on two sets of bearing fault data show that the PAEUBSR system has better effects on the enhancement and detection of bearing fault signals. This provides good theoretical support and application value for this system in subsequent theoretical analysis and practical engineering applications.

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