scholarly journals Dynamical Analysis of a Nitrogen-Phosphorus-Phytoplankton Model

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Yunli Deng ◽  
Min Zhao ◽  
Hengguo Yu ◽  
Yi Wang
Keyword(s):  
Nitrogen Concentration ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Threshold Values ◽  
Globally Asymptotically Stable ◽  
Global System ◽  
Water Ecosystem ◽  
Existence And Stability ◽  
Nitrogen Phosphorus ◽  
Coexistence Equilibrium

This paper presents a nitrogen-phosphorus-phytoplankton model in a water ecosystem. The main aim of this research is to analyze the global system dynamics and to study the existence and stability of equilibria. It is shown that the phytoplankton-eradication equilibrium is globally asymptotically stable if the input nitrogen concentration is less than a certain threshold. However, the coexistence equilibrium is globally asymptotically stable as long as it exists. The system is uniformly persistent within threshold values of certain key parameters. Finally, to verify the results, numerical simulations are provided.

Dynamical analysis of a reaction–diffusion SEI epidemic model with nonlinear incidence rate

2021 ◽  
pp. 2150041
Author(s):  
Jianpeng Wang ◽  
Binxiang Dai
Keyword(s):  
Steady State ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Reaction Diffusion ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Positive Steady State

In this paper, a reaction–diffusion SEI epidemic model with nonlinear incidence rate is proposed. The well-posedness of solutions is studied, including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions. The basic reproduction numbers are given in both heterogeneous and homogeneous environments. For spatially heterogeneous environment, by the comparison principle of the diffusion system, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] if [Formula: see text], the system will be persistent and admit at least one positive steady state. For spatially homogenous environment, by constructing a Lyapunov function, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if [Formula: see text]. Finally, two examples are given via numerical simulations, and then some control strategies are also presented by the sensitive analysis.

Global Dynamics of a Holling-II Amensalism System with Nonlinear Growth Rate and Allee Effect on the First Species

2021 ◽  
Vol 31 (03) ◽  
pp. 2150050
Author(s):  
Demou Luo ◽  
Qiru Wang
Keyword(s):  
Growth Rate ◽  
Allee Effect ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nonlinear Growth ◽  
Trivial Equilibrium ◽  
Existence And Stability ◽  
Stable Equilibria ◽  
Theoretical Results

Of concern is the global dynamics of a two-species Holling-II amensalism system with nonlinear growth rate. The existence and stability of trivial equilibrium, semi-trivial equilibria, interior equilibria and infinite singularity are studied. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, the global dynamics of the model is performed. Next, we incorporate Allee effect on the first species and offer a new analysis of equilibria and bifurcation discussion of the model. Finally, several numerical examples are performed to verify our theoretical results.

scholarly journals Stability and Bifurcation of a Computer Virus Propagation Model with Delay and Incomplete Antivirus Ability

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jianguo Ren ◽  
Yonghong Xu
Keyword(s):  
Hopf Bifurcation ◽  
Threshold Value ◽  
Computer Virus ◽  
Propagation Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Virus Propagation ◽  
Globally Asymptotically Stable ◽  
Existence And Stability ◽  
Computer Virus Propagation Model

A new computer virus propagation model with delay and incomplete antivirus ability is formulated and its global dynamics is analyzed. The existence and stability of the equilibria are investigated by resorting to the threshold valueR0. By analysis, it is found that the model may undergo a Hopf bifurcation induced by the delay. Correspondingly, the critical value of the Hopf bifurcation is obtained. Using Lyapunov functional approach, it is proved that, under suitable conditions, the unique virus-free equilibrium is globally asymptotically stable ifR0<1, whereas the virus equilibrium is globally asymptotically stable ifR0>1. Numerical examples are presented to illustrate possible behavioral scenarios of the mode.

Dynamical analysis for delayed virus infection models with cell-to-cell transmission and density-dependent diffusion

2020 ◽  
Vol 13 (07) ◽  
pp. 2050060
Author(s):  
Shaoli Wang ◽  
Achun Zhang ◽  
Fei Xu
Keyword(s):  
Neumann Boundary ◽  
Basic Reproductive Number ◽  
Dynamical Analysis ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Cell Infection ◽  
Globally Asymptotically Stable ◽  
Single Strain ◽  
Density Dependent ◽  
Number Formula

In this paper, certain delayed virus dynamical models with cell-to-cell infection and density-dependent diffusion are investigated. For the viral model with a single strain, we have proved the well-posedness and studied the global stabilities of equilibria by defining the basic reproductive number [Formula: see text] and structuring proper Lyapunov functional. Moreover, we found that the infection-free equilibrium is globally asymptotically stable if [Formula: see text], and the infection equilibrium is globally asymptotically stable if [Formula: see text]. For the multi-strain model, we found that all viral strains coexist if the corresponding basic reproductive number [Formula: see text], while virus will extinct if [Formula: see text]. As a result, we found that delay and the density-dependent diffusion does not influence the global stability of the model with cell-to-cell infection and homogeneous Neumann boundary conditions.

scholarly journals Dynamical Analysis of a Plateau Pika with Cross-Diffusion under Contraception Control

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Xiaoyan Wang ◽  
Junyuan Yang ◽  
Fengqin Zhang
Keyword(s):  
Steady State ◽  
Local Stability ◽  
Degree Theory ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Plateau Pika ◽  
Globally Asymptotically Stable ◽  
Cross Diffusion ◽  
Contraception Control ◽  
Diffusion Rates

A plateau pika model with spatial cross-diffusion is investigated. By analyzing the corresponding characteristic equations, the local stability of an coexistence steady state is discussed whend21is small enough. However, whend21is large enough, the model shows Turing bifurcation ifB2 -4AC > 0. Furthermore, it is proved that if,R > R0, βK > dand cross-diffusion rates are zero, the positive coexistence steady state is globally asymptotically stable. A nonconstant positive solution bifurcates from the coexistent steady state by the Leray-Schauder degree theory. Numerical simulations are carried out to illustrate the main results.

GLOBAL STABILITY AND HOPF BIFURCATION IN A DELAYED DIFFUSIVE LESLIE–GOWER PREDATOR–PREY SYSTEM

2012 ◽  
Vol 22 (03) ◽  
pp. 1250061 ◽  
Author(s):  
SHANSHAN CHEN ◽  
JUNPING SHI ◽  
JUNJIE WEI
Keyword(s):  
Hopf Bifurcation ◽  
Upper And Lower Solutions ◽  
Neumann Boundary ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Prey System ◽  
The Stability ◽  
Coexistence Equilibrium

In this paper, we consider a delayed diffusive Leslie–Gower predator–prey system with homogeneous Neumann boundary conditions. The stability/instability of the coexistence equilibrium and associated Hopf bifurcation are investigated by analyzing the characteristic equations. Furthermore, using the upper and lower solutions method, we give a sufficient condition on parameters so that the coexistence equilibrium is globally asymptotically stable.

scholarly journals Dynamical Analysis of an SEIRS Model With Incomplete Recovery and Relapse on Networks and Its Optimal Vaccination Control

2021 ◽  
Author(s):  
Lei Zhang ◽  
Maoxing Liu ◽  
Qiang Hou ◽  
Boli Xie
Keyword(s):  
Control Strategy ◽  
Reproduction Number ◽  
Dynamical Analysis ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Seirs Epidemic Model ◽  
Theoretical Results ◽  
Disease Free ◽  
The Basic Reproduction Number

Abstract For some infectious diseases, such as herpes and tuberculosis, there is incomplete recovery and relapse. These phenomena make them difficult to control. In consequence of this status, an SEIRS epidemic model with incomplete recovery and relapse on networks is established and the global dynamics is studied. The results show that when the basic reproduction number R 0 <=1 the disease-free equilibrium is globally asymptotically stable; when R 0 > 1, the endemic equilibrium is globally asymptotically stable. In addition, in consideration of vaccination control strategy, an SVEIRS model is introduced and the optimal control is solved. At last, the theoretical results are illustrated with numerical simulations.

scholarly journals EXISTENCE AND STABILITY OF PERIODIC ORBITS OF PERIODIC DIFFERENCE EQUATIONS WITH DELAYS

2008 ◽  
Vol 18 (01) ◽  
pp. 203-217 ◽  
Author(s):  
ZIYAD ALSHARAWI ◽  
JAMES ANGELOS ◽  
SABER ELAYDI
Keyword(s):  
Difference Equation ◽  
Periodic Orbits ◽  
Difference Equations ◽  
Asymptotically Stable ◽  
Stable Orbit ◽  
Globally Asymptotically Stable ◽  
Stability Of Periodic Orbits ◽  
Existence And Stability ◽  
Sharkovsky's Theorem ◽  
Positive Integers

In this paper, we investigate the existence and stability of periodic orbits of the p-periodic difference equation with delays xn = f(n - 1, xn-k). We show that the periodic orbits of this equation depend on the periodic orbits of p autonomous equations when p divides k. When p is not a divisor of k, the periodic orbits depend on the periodic orbits of gcd(p, k) nonautonomous p/gcd(p, k)-periodic difference equations. We give formulas for calculating the number of different periodic orbits under certain conditions. In addition, when p and k are relatively prime integers, we introduce what we call the pk-Sharkovsky's ordering of the positive integers, and extend Sharkovsky's theorem to periodic difference equations with delays. Finally, we characterize global stability and show that the period of a globally asymptotically stable orbit must be divisible by p.

scholarly journals Dynamical analysis of a fractional order model incorporating fear in the disease transmission rate of COVID-19

2020 ◽  
Vol 99 (99) ◽  
pp. 1-17
Author(s):  
Debasis Mukherjee Debasis Mukherjee ◽  
Chandan Maji
Keyword(s):  
Fractional Order ◽  
Disease Transmission ◽  
Transmission Rate ◽  
Three Dimensional ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Fear Effect ◽  
And Control ◽  
Disease Free

This paper deals with a fractional-order three-dimensional compartmental model with fear effect. We have investigated whether fear can play an important role or not to spread and control the infectious diseases like COVID-19, SARS etc. in a bounded region. The basic results on uniqueness, non-negativity and boundedness of the solution of the system are investigated. Stability analysis ensures that the disease-free equilibrium point is locally asymptotically stable if carrying capacity greater than a certain threshold value.We have also derived the conditions for which endemic equilibrium is globally asymptotically stable that means the disease persists in the system. Numerical simulation suggests that the fear factor is an important role which is observed through Hopf-bifurcation.

NUMERICAL AND STABILITY ANALYSIS OF THE TRANSMISSION DYNAMICS OF SVIR EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE

2019 ◽  
Vol 4 (2) ◽  
pp. 349 ◽  
Author(s):  
Oluwatayo Michael Ogunmiloro ◽  
Fatima Ohunene Abedo ◽  
Hammed Kareem
Keyword(s):  
Reproduction Number ◽  
Disease Spread ◽  
Asymptotically Stable ◽  
Equilibrium Solutions ◽  
Transform Method ◽  
Globally Asymptotically Stable ◽  
Differential Transform ◽  
Mathematical Techniques ◽  
Fourth Order Method ◽  
Theoretical Results

In this article, a Susceptible – Vaccinated – Infected – Recovered (SVIR) model is formulated and analysed using comprehensive mathematical techniques. The vaccination class is primarily considered as means of controlling the disease spread. The basic reproduction number (Ro) of the model is obtained, where it was shown that if Ro<1, at the model equilibrium solutions when infection is present and absent, the infection- free equilibrium is both locally and globally asymptotically stable. Also, if Ro>1, the endemic equilibrium solution is locally asymptotically stable. Furthermore, the analytical solution of the model was carried out using the Differential Transform Method (DTM) and Runge - Kutta fourth-order method. Numerical simulations were carried out to validate the theoretical results. 

Sign in / Sign up

Export Citation Format

Share Document