Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance

2020 ◽  
Vol 13 (07) ◽  
pp. 2050066
Author(s):  
Xiaofeng Zhang ◽  
Rong Yuan
Keyword(s):  
Linear Chain ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
Degenerate System ◽  
Random Disturbance ◽  
Chemostat Model ◽  
Boundary Equilibrium ◽  
Global Positive Solution ◽  
Stochastic Properties ◽  
Theoretical Results

In this paper, stochastic properties of solution for a chemostat model with a distributed delay and random disturbance are studied, and we use distribution delay to simulate the delay in nutrient conversion. By the linear chain technique, we transform the stochastic chemostat model with weak kernel into an equivalent degenerate system which contains three equations. First, we state that this model has a unique global positive solution for any initial value, which is helpful to explore its stochastic properties. Furthermore, we prove the stochastic ultimate boundness of the solution of system. Then sufficient conditions for solution of the system tending toward the boundary equilibrium point at exponential rate are established, which means the microorganism will be extinct. Moreover, we also obtain some sufficient conditions for ergodicity of solution of this system by constructing some suitable stochastic Lyapunov functions. Finally, we provide some numerical examples to illustrate theoretical results, and some conclusions and analysis are given.

STABILITY ANALYSIS OF A RATIO-DEPENDENT CHEMOSTAT MODEL WITH TIME DELAY AND VARIABLE YIELD

2010 ◽  
Vol 03 (02) ◽  
pp. 243-253 ◽  
Author(s):  
ZHE LI ◽  
RUI XU
Keyword(s):  
Time Delay ◽  
Lyapunov Functional ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Chemostat Model ◽  
Variable Yield ◽  
Boundary Equilibrium ◽  
Comparison Arguments ◽  
Ratio Dependent ◽  
Theoretical Results

A chemostat model with time delay, variable yield and ratio-dependent functional response is investigated. By analyzing the corresponding characteristic equations, the local stability of a boundary equilibrium and a positive equilibrium is discussed and the existence of Hopf bifurcation is established. By using the comparison arguments, sufficient conditions are obtained for the global stability of the boundary equilibrium. By constructing a suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the positive equilibrium. Finally, numerical simulations are carried out to illustrate the theoretical results.

ASYMPTOTIC BEHAVIOR OF A STOCHASTIC DELAYED CHEMOSTAT MODEL WITH NUTRIENT STORAGE

2018 ◽  
Vol 26 (02) ◽  
pp. 225-246 ◽  
Author(s):  
SHULIN SUN ◽  
XIAOFENG ZHANG
Keyword(s):  
Asymptotic Behavior ◽  
Time Delay ◽  
Function Analysis ◽  
Sufficient Conditions ◽  
Classical Approach ◽  
Nutrient Storage ◽  
Chemostat Model ◽  
Global Positive Solution ◽  
The Mean ◽  
Simulation Results

In this paper, a stochastic delayed chemostat model with nutrient storage is proposed and investigated. First, we state that there is a unique global positive solution for this stochastic system. Second, using the classical approach of Lyapunov function analysis, this stochastic delayed chemostat model is discussed in detail. We establish some sufficient conditions for the extinction of the microorganism, furthermore, we prove that the microorganism will become persistent in the mean in the chemostat under some conditions. Finally, the obtained results are illustrated by computer simulations, and simulation results reveal the effects of time delay on the persistence and extinction of the microorganism.

DYNAMICS OF A MODEL FOR VIRULENT PHAGE T4

2008 ◽  
Vol 16 (04) ◽  
pp. 597-611 ◽  
Author(s):  
ZHIPENG QIU
Keyword(s):  
Population Density ◽  
Reproduction Number ◽  
Bacteriophage T4 ◽  
Sufficient Conditions ◽  
Chemostat Model ◽  
Virulent Phage ◽  
E Coli ◽  
Phage T4 ◽  
Theoretical Results ◽  
The Basic Reproduction Number

In this paper, the asymptotical behavior of a chemostat model for E. coli and the virulent phage T4 is analyzed. The basic reproduction number R0 is proved to be a threshold which determines the outcome of the virulent phage T4. If R0 < 1, the virus dies out; if R0 > 1, the virus persists. Sufficient conditions for the Hopf bifurcation are also established. The theoretical results show that increasing the input of nutrient will result in an increase in the equilibrium population density of the virulent bacteriophage T4, but will have no effect on the equilibrium population density of E. coli. The results also show that increasing the input of nutrient or increasing the average lytic time for the infected E. coli can destabilize the interaction between E. coli and T4.

scholarly journals Dynamics of a stochastic HIV/AIDS model with treatment under regime switching

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Miaomiao Gao ◽  
Daqing Jiang ◽  
Tasawar Hayat ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Stationary Distribution ◽  
Regime Switching ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Aids Model ◽  
Telegraph Noise ◽  
Spread Dynamics ◽  
Global Positive Solution ◽  
Theoretical Results ◽  
Hiv Aids

<p style='text-indent:20px;'>This paper focuses on the spread dynamics of an HIV/AIDS model with multiple stages of infection and treatment, which is disturbed by both white noise and telegraph noise. Switching between different environmental states is governed by Markov chain. Firstly, we prove the existence and uniqueness of the global positive solution. Then we investigate the existence of a unique ergodic stationary distribution by constructing suitable Lyapunov functions with regime switching. Furthermore, sufficient conditions for extinction of the disease are derived. The conditions presented for the existence of stationary distribution improve and generalize the previous results. Finally, numerical examples are given to illustrate our theoretical results.</p>

Dynamics of a stochastic rabies epidemic model with markovian switching

2021 ◽  
pp. 2150032
Author(s):  
Hao Peng ◽  
Xinhong Zhang ◽  
Daqing Jiang
Keyword(s):  
Lyapunov Function ◽  
White Noise ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Telegraph Noise ◽  
Global Positive Solution ◽  
Theoretical Results

In this paper, we analyze a stochastic rabies epidemic model which is perturbed by both white noise and telegraph noise. First, we prove the existence of the unique global positive solution. Second, by constructing an appropriate Lyapunov function, we establish a sufficient condition for the existence of a unique ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for the extinction of diseases. Finally, numerical simulations are introduced to illustrate our theoretical results.

scholarly journals Dynamical Behavior of a Stochastic SIRC Model for Influenza A

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 745 ◽  
Author(s):  
Tongqian Zhang ◽  
Tingting Ding ◽  
Ning Gao ◽  
Yi Song
Keyword(s):  
Lyapunov Function ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Epidemic Model ◽  
Influenza A ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Global Positive Solution ◽  
Theoretical Results ◽  
Stochastic Lyapunov Function

In this paper, a stochastic SIRC epidemic model for Influenza A is proposed and investigated. First, we prove that the system exists a unique global positive solution. Second, the extinction of the disease is explored and the sufficient conditions for extinction of the disease are derived. And then the existence of a unique ergodic stationary distribution of the positive solutions for the system is discussed by constructing stochastic Lyapunov function. Furthermore, numerical simulations are employed to illustrate the theoretical results. Finally, we give some further discussions about the system.

scholarly journals Dynamics of an Impulsive Stochastic Nonautonomous Chemostat Model with Two Different Growth Rates in a Polluted Environment

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Yajie Li ◽  
Xinzhu Meng
Keyword(s):  
Growth Rates ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Positive Periodic Solution ◽  
Stochastic Noise ◽  
Polluted Environment ◽  
Chemostat Model ◽  
Lyapunov Functions Method ◽  
Theoretical Results

This paper proposes a novel impulsive stochastic nonautonomous chemostat model with the saturated and bilinear growth rates in a polluted environment. Using the theory of impulsive differential equations and Lyapunov functions method, we first investigate the dynamics of the stochastic system and establish the sufficient conditions for the extinction and the permanence of the microorganisms. Then we demonstrate that the stochastic periodic system has at least one nontrivial positive periodic solution. The results show that both impulsive toxicant input and stochastic noise have great effects on the survival and extinction of the microorganisms. Furthermore, a series of numerical simulations are presented to illustrate the performance of the theoretical results.

scholarly journals Synchronization control of stochastic delayed Lotka–Volterra systems with hardware simulation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Lan Wang ◽  
Yiping Dong ◽  
Da Xie ◽  
Hao Zhang
Keyword(s):  
Sufficient Conditions ◽  
Synchronization Control ◽  
Simulation Tool ◽  
Volterra System ◽  
Stochastic Effects ◽  
Volterra Systems ◽  
Field Programmable ◽  
Global Positive Solution ◽  
System With Time Delay ◽  
Theoretical Results

AbstractIn this paper, the synchronization control of a non-autonomous Lotka–Volterra system with time delay and stochastic effects is studied. The purpose is to firstly establish sufficient conditions for the existence of global positive solution by constructing a suitable Lyapunov function. Some synchronization criteria are then derived by designing an appropriate full controller and a pinning controller, respectively. Finally, an example is presented to illustrate the feasibility and validity of the main theoretical results based on the Field-Programmable Gate Array hardware simulation tool.

Stability in distribution of stochastic Lotka–Volterra delay system under regime switching

2018 ◽  
Vol 18 (05) ◽  
pp. 1850041
Author(s):  
Yanchao Zang ◽  
Pingjun Hou ◽  
Yuzhu Tian
Keyword(s):  
Regime Switching ◽  
Random Perturbation ◽  
Sufficient Conditions ◽  
Classical Case ◽  
Delay System ◽  
Global Positive Solution ◽  
Stability In Distribution ◽  
The Stability ◽  
System With Time Delay ◽  
Theoretical Results

In this paper, we consider a class of stochastic competitive Lotka–Volterra system with time delay and Markovian switching. We prove that there exists a global positive solution under the random perturbation. Some sufficient conditions for the stability in distribution of the system are established which improved the classical case. An example is given to illustrate theoretical results.

scholarly journals Dynamics of a single predator multiple prey model with stochastic perturbation and seasonal variation

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 535-549
Author(s):  
Hong-Wen Hui ◽  
Lin-Fei Nie
Keyword(s):  
Seasonal Variation ◽  
Sufficient Conditions ◽  
Stochastic Perturbation ◽  
Analysis Method ◽  
Ultimate Boundedness ◽  
Persistence In The Mean ◽  
Global Positive Solution ◽  
The Mean ◽  
Stochastically Ultimate Boundedness ◽  
Theoretical Results

Considering various factors are stochastic rather than deterministic in the evolution of populations growth, in this paper, we propose a single predator multiple prey stochastic model with seasonal variation. By using the method of solving an explicit solution, the existence of global positive solution of this model are obtained. The method is more convenient than Lyapunov analysis method for some population models. Moreover, the stochastically ultimate boundedness are considered by using the comparison theorem of stochastic differential equation. Further, some sufficient conditions for the extinction and strong persistence in the mean of populations are discussed, respectively. In addition, by constructing some suitable Lyapunov functions, we show that this model admits at least one periodic solution. Finally, numerical simulations clearly illustrate the main theoretical results and the effects of white noise and seasonal variation for the persistence and extinction of populations.

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