scholarly journals Note on the Stability Property of a Cooperative System Incorporating Harvesting

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Xiangdong Xie ◽  
Fengde Chen ◽  
Yalong Xue
Keyword(s):  
Iterative Method ◽  
Equilibrium Point ◽  
Stability Property ◽  
Global Attractivity ◽  
Positive Equilibrium ◽  
Cooperative System ◽  
Cooperative Model ◽  
Interior Equilibrium ◽  
The Stability

The stability of a kind of cooperative model incorporating harvesting is revisited in this paper. By using an iterative method, the global attractivity of the interior equilibrium point of the system is investigated. We show that the condition which ensures the existence of a unique positive equilibrium is enough to ensure the global attractivity of the positive equilibrium. Our results significantly improve the corresponding results of Wei and Li (2013).

scholarly journals Global Stability of a Discrete Mutualism Model

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Kun Yang ◽  
Xiangdong Xie ◽  
Fengde Chen
Keyword(s):  
Iterative Method ◽  
Comparison Principle ◽  
Local Stability ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Global Asymptotical Stability ◽  
Interior Equilibrium ◽  
Linear Approximation Method ◽  
Mutualism Model

A discrete mutualism model is studied in this paper. By using the linear approximation method, the local stability of the interior equilibrium of the system is investigated. By using the iterative method and the comparison principle of difference equations, sufficient conditions which ensure the global asymptotical stability of the interior equilibrium of the system are obtained. The conditions which ensure the local stability of the positive equilibrium is enough to ensure the global attractivity are proved.

Dynamics of an Innovation Diffusion Model with Time Delay

2017 ◽  
Vol 7 (3) ◽  
pp. 455-481 ◽  
Author(s):  
Rakesh Kumar ◽  
Anuj K. Sharma ◽  
Kulbhushan Agnihotri
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Population Density ◽  
Differential Equations ◽  
Equilibrium Point ◽  
Innovation Diffusion ◽  
Critical Value ◽  
Normal Form Theory ◽  
Interior Equilibrium ◽  
The Stability

AbstractA nonlinear mathematical model for innovation diffusion is proposed. The system of ordinary differential equations incorporates variable external influences (the cumulative density of marketing efforts), variable internal influences (the cumulative density of word of mouth) and a logistically growing human population (the variable potential consumers). The change in population density is due to various demographic processes such as intrinsic growth rate, emigration, death rate etc. Thus the problem involves two dynamic variables viz. a non-adopter population density and an adopter population density. The model is analysed qualitatively using the stability theory of differential equations, with the help of the corresponding characteristic equation of the system. The interior equilibrium point can be stable for all time delays to a critical value, beyond which the system becomes unstable and a Hopf bifurcation occurs at a second critical value. Employing normal form theory and a centre manifold theorem applicable to functional differential equations, we derive some explicit formulas determining the stability, the direction and other properties of the bifurcating periodic solutions. Our numerical simulations show that the system behaviour can become extremely complicated as the time delay increases, with a stable interior equilibrium point leading to a limit cycle with one local maximum and minimum per cycle (Hopf bifurcation), then limit cycles with more local maxima and minima per cycle, and finally chaotic solutions.

Stability Analysis of a Prey-Predator Model by Using Homotopy Analysis Method

2012 ◽  
Vol 166-169 ◽  
pp. 2855-2858
Author(s):  
Hong Yan Cao
Keyword(s):  
Nonlinear Dynamics ◽  
Equilibrium Point ◽  
Homotopy Analysis Method ◽  
Zero Point ◽  
Positive Equilibrium ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Predator Model ◽  
Positive Equilibrium Point ◽  
The Stability

A prey-predator model was considered. Using the methods of the modern nonlinear dynamics and homotopy analysis method (HAM), its stability was discussed. Firstly, we found the system’s positive equilibrium point and shifted it to zero point through transformation. Secondly, we analyzed the stability of the system at the equilibrium point. Lastly, we analyzed the transformed system by HAM. We support our analytical findings with numerical simulation.

scholarly journals Stability Property for the Predator-Free Equilibrium Point of Predator-Prey Systems with a Class of Functional Response and Prey Refuges

2012 ◽  
Vol 2012 ◽  
pp. 1-5 ◽  
Author(s):  
Fengde Chen ◽  
Yumin Wu ◽  
Zhaozhi Ma
Keyword(s):  
Equilibrium Point ◽  
Functional Response ◽  
Stability Property ◽  
Analytical Approach ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
The Stability ◽  
Prey Refuges

We investigate the stability property for the predator-free equilibrium point of predator-prey systems with a class of functional response and prey refuges by using the analytical approach. Under some very weakly assumption, we show that conditions that ensure the locally asymptotically stable of the predator-free equilibrium point are consistent with that of the globally asymptotically stable ones. Our result supplements the corresponding result of Ma et al., 2009.

scholarly journals Dynamics Analysis of a Delayed Rumor Propagation Model in an Emergency-Affected Environment

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Chunru Li ◽  
Zujun Ma
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Propagation Model ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Rumor Propagation ◽  
Boundary Equilibrium ◽  
Normal Form Method ◽  
Positive Equilibrium Point ◽  
The Stability

Rumors influence people’s decisions in an emergency-affected environment. How to describe the spreading mechanism is significant. In this paper, we propose a delayed rumor propagation model in emergencies. By taking the delay as the bifurcation parameter, the local stability of the boundary equilibrium point and the positive equilibrium point is investigated and the conditions of Hopf bifurcation are obtained. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, some numerical simulations are also given to illustrate our theoretical results.

scholarly journals An Eco-Epidemiological Model Incorporating Harvesting Factors

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2179
Author(s):  
Kawa Hassan ◽  
Arkan Mustafa ◽  
Mudhafar Hama
Keyword(s):  
Equilibrium Point ◽  
Functional Response ◽  
Biological System ◽  
Numerical Verification ◽  
Interior Equilibrium ◽  
Dynamical Fluctuations ◽  
Predator Model ◽  
The Stability ◽  
Analytical Section ◽  
Predator System

The biological system relies heavily on the interaction between prey and predator. Infections may spread from prey to predators or vice versa. This study proposes a virus-controlled prey-predator system with a Crowley–Martin functional response in the prey and an SI-type in the prey. A prey-predator model in which the predator uses both susceptible and sick prey is used to investigate the influence of harvesting parameters on the formation of dynamical fluctuations and stability at the interior equilibrium point. In the analytical section, we outlined the current circumstances for all possible equilibria. The stability of the system has also been explored, and the required conditions for the model’s stability at the equilibrium point have been found. In addition, we give numerical verification for our analytical findings with the help of graphical illustrations.

scholarly journals Global Attractivity of a Diffusive Nicholson's Blowflies Equation with Multiple Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Xiongwei Liu ◽  
Xiao Wang
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Global Attractivity ◽  
Neumann Boundary ◽  
Positive Equilibrium ◽  
Multiple Delays ◽  
Open Problems ◽  
Nicholson’S Blowflies Equation ◽  
Trivial Equilibrium ◽  
The Stability

The present paper considers a diffusive Nicholson's blowflies model with multiple delays under a Neumann boundary condition. Delay independent conditions are derived for the global attractivity of the trivial equilibrium and the positive equilibrium, respectively. Two open problems concerning the stability of positive equilibrium and the occurrence of Hopf bifurcation are proposed.

Dynamics of a prey predator disease model with Holling type functional response and stochastic perturbation

2020 ◽  
pp. 471-488
Author(s):  
M. N. Srinivas ◽  
G. Basava Kumar ◽  
V. Madhusudanan
Keyword(s):  
Equilibrium Point ◽  
Disease Model ◽  
Stochastic Perturbation ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Asymptotically Stable ◽  
Unique Positive Solution ◽  
Stochastic Nature ◽  
Research Article ◽  
Positive Equilibrium Point

The present research article constitutes Holling type II and IV diseased prey predator ecosystem and classified into two categories namely susceptible and infected predators.We show that the system has a unique positive solution. The deterministic and stochastic nature of the dynamics of the system is investigated. We check the existence of all possible steady states with local stability. By using Routh-Hurwitz criterion we showed that the positive equilibrium point $E_{7}$ is locally asymptotically stable if $x^{*} > \sqrt{m_{1}}$ .Moreover condition of the global stability of positive equilibrium point $E_{7}$ are also entrenched with help of Lyupunov theorem. Some Numerical simulations are carried out to illustrate our analytical findings.

scholarly journals Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Computer Network ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Form Theory ◽  
Worm Model ◽  
The Stability

A delayed SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify the theoretical analysis.

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