scholarly journals Study of the dynamics of two chemostats connected by Fickian diffusion with bounded random fluctuations

2021 ◽  
Author(s):  
Tomás Caraballo ◽  
Javier López-de-la-Cruz ◽  
Alain Rapaport
Keyword(s):  
Numerical Simulations ◽  
Existence And Uniqueness ◽  
Fickian Diffusion ◽  
Time Behavior ◽  
Long Time Behavior ◽  
State Variables ◽  
Random System ◽  
Long Time ◽  
Random Fluctuations ◽  
Theoretical Results

This paper investigates the dynamics of a model of two chemostats connected by Fickian diffusion with bounded random fluctuations. We prove the existence and uniqueness of non-negative global solution as well as the existence of compact absorbing and attracting sets for the solutions of the corresponding random system. After that, we study the internal structure of the attracting set to obtain more detailed information about the long-time behavior of the state variables. In such a way, we provide conditions under which the extinction of the species cannot be avoided and conditions to ensure the persistence of the species, which is often the main goal pursued by practitioners. In addition, we illustrate the theoretical results with several numerical simulations.

The asymptotic behavior of a stochastic multigroup SIS model

2018 ◽  
Vol 11 (03) ◽  
pp. 1850037 ◽  
Author(s):  
Chunyan Ji ◽  
Daqing Jiang
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Simulations ◽  
Stationary Distribution ◽  
Endemic Equilibrium ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Sis Model ◽  
Stochastic Perturbations ◽  
Long Time ◽  
Theoretical Results

In this paper, we explore the long time behavior of a multigroup Susceptible–Infected–Susceptible (SIS) model with stochastic perturbations. The conditions for the disease to die out are obtained. Besides, we also show that the disease is fluctuating around the endemic equilibrium under some conditions. Moreover, there is a stationary distribution under stronger conditions. At last, some numerical simulations are applied to support our theoretical results.

scholarly journals Anti-periodic problem for semilinear differential inclusions involving Hille-Yosida operators

2021 ◽  
pp. 1-31
Author(s):  
Nguyen Thi Van Anh ◽  
Tran Dinh Ke ◽  
Do Lan
Keyword(s):  
Periodic Solutions ◽  
Differential Inclusions ◽  
Mild Solutions ◽  
Periodic Problem ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Multivalued Maps ◽  
Integrated Semigroups ◽  
Long Time ◽  
Theoretical Results

In this paper we are interested in the anti-periodic problem governed by a class of semilinear differential inclusions with linear parts generating integrated semigroups. By adopting the Lyapunov-Perron method and the fixed point argument for multivalued maps, we prove the existence of anti-periodic solutions. Furthermore, we study the long-time behavior of mild solutions in connection with anti-periodic solutions. Consequently, as the nonlinearity is of single-valued, we obtain the exponential stability of anti-periodic solutions. An application of theoretical results to a class of partial differential equations will be given.

scholarly journals Long-Time Behavior of Solution for a Reactor Model

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Yantao Guo ◽  
Jianwei Shen ◽  
Qianqian Zheng
Keyword(s):  
Weak Solution ◽  
Numerical Experiments ◽  
Measure Of Noncompactness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Reactor Model ◽  
Kuratowski Measure Of Noncompactness ◽  
Dynamical Behaviors ◽  
Long Time ◽  
Theoretical Results

In this paper, we would consider the dynamical behaviors of the chemical model represented by Satnoianu et al. (2001). Using the Kuratowski measure of noncompactness method, we prove the existence of global attractor for the weak solution semiflow of system. Finally, several numerical experiments confirm the theoretical results.

Dynamic Analysis of a Stochastic Predator–Prey Model With Crowley–Martin Functional Response, Disease in Predator, and Saturation Incidence

2020 ◽  
Vol 15 (7) ◽  
Author(s):  
Conghui Xu ◽  
Yongguang Yu ◽  
Guojian Ren
Keyword(s):  
Functional Response ◽  
Existence And Uniqueness ◽  
Global Attractivity ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Disease In Predator ◽  
Long Time

Abstract This work aims to study some dynamical properties of a stochastic predator–prey model, which is considered under the combination of Crowley–Martin functional response, disease in predator, and saturation incidence. First, we discuss the existence and uniqueness of positive solution of the concerned stochastic model. Second, we prove that the solution is stochastically ultimate bounded. Then, we investigate the extinction and the long-time behavior of the solution. Furthermore, we establish some conditions for the global attractivity of the model. Finally, we propose some numerical simulations to illustrate our main results.

scholarly journals Measure Solutions To The Conservative Renewal Equation

2018 ◽  
Vol 62 ◽  
pp. 68-78 ◽  
Author(s):  
Pierre Gabriel
Keyword(s):  
Existence And Uniqueness ◽  
Renewal Equation ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Exponential Relaxation ◽  
Duality Approach

We prove the existence and uniqueness of measure solutions to the conservative renewal equation and analyze their long time behavior. The solutions are built by using a duality approach. This construction is well suited to apply the Doeblin’s argument which ensures the exponential relaxation of the solutions to the equilibrium.

Permanence and extinction for a nonautonomous schistosomiasis model with saturation incidence rate

2015 ◽  
Vol 08 (02) ◽  
pp. 1550021 ◽  
Author(s):  
Dehui Xie ◽  
Xiangyu Zhang ◽  
Shuixian Yan ◽  
Shujing Gao
Keyword(s):  
Infectious Diseases ◽  
Numerical Simulations ◽  
Incidence Rate ◽  
Real World ◽  
Sufficient Conditions ◽  
Time Behavior ◽  
Long Time Behavior ◽  
The Real ◽  
Long Time

In the real world, quite a few infectious diseases like schistosomiasis spread seasonally. In this paper, a nonautonomous schistosomiasis system is established, in which the saturation incidence rate and the coefficients varying with time are taken into account. The long-time behavior of the model is studied. Under quite weak assumptions, sufficient conditions for the permanence and extinction of infectious population of disease are obtained. Finally, numerical simulations illustrate the validity of our results.

scholarly journals $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection

2021 ◽  
Author(s):  
Changqing Ji ◽  
Dandan Zhu ◽  
Jingli Ren
Keyword(s):  
Large Amplitude ◽  
Small Amplitude ◽  
Initial Conditions ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Free Boundaries ◽  
Kpp Equation ◽  
Advection Term ◽  
Long Time ◽  
Theoretical Results

In this paper, we investigate a $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection. Considering the influence of advection term and initial conditions on the long time behavior of solutions, we obtain spreading-vanishing dichotomy, spreading-transition-vanishing trichotomy, and vanishing happens with the coefficient of advection term in small amplitude, medium-sized amplitude and large amplitude, respectively. Then, the appropriate parameters are selected in the simulation to intuitively show the corresponding theoretical results. Moreover, the wave-spreading and wave-vanishing cases of the solutions are observed in our study.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

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