GLOBAL ASYMPTOTIC STABILITY FOR A TWO-SPECIES DISCRETE RATIO-DEPENDENT PREDATOR–PREY SYSTEM

2013 ◽  
Vol 06 (01) ◽  
pp. 1250064 ◽  
Author(s):  
XIANGLAI ZHUO
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent ◽  
Analysis Of Stability ◽  
Appl Math

The dynamical behaviors of a two-species discrete ratio-dependent predator–prey system are considered. Some sufficient conditions for the local stability of the equilibria is obtained by using the linearization method. Further, we also obtain a new sufficient condition to ensure that the positive equilibrium is globally asymptotically stable by using an iteration scheme and the comparison principle of difference equations, which generalizes what paper [G. Chen, Z. Teng and Z. Hu, Analysis of stability for a discrete ratio-dependent predator–prey system, Indian J. Pure Appl. Math.42(1) (2011) 1–26] has done. The method given in this paper is new and very resultful comparing with papers [H. F. Huo and W. T. Li, Existence and global stability of periodic solutions of a discrete predator–prey system with delays, Appl. Math. Comput.153 (2004) 337–351; X. Liao, S. Zhou and Y. Chen, On permanence and global stability in a general Gilpin–Ayala competition predator–prey discrete system, Appl. Math. Comput.190 (2007) 500–509] and it can also be applied to study the global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present an open question.

scholarly journals Persistence and global stability in a delayed predator-prey system with Holling-type functional response

2004 ◽  
Vol 46 (1) ◽  
pp. 121-141 ◽  
Author(s):  
Rui Xu ◽  
Lansun Chen ◽  
M. A. J. Chaplain
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Functional Response ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Lyapunov Functionals ◽  
Predator Prey ◽  
Prey System ◽  
Holling Type Functional Response

AbstractA delayed predator-prey system with Holling type III functional response is investigated. It is proved that the system is uniformly persistent under some appropriate conditions. By means of suitable Lyapunov functionals, sufficient conditions are derived for the local and global asymptotic stability of a positive equilibrium of the system. Numerical simulations are presented to illustrate the feasibility of our main results.

scholarly journals Global Asymptotic Stability of a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Shengbin Yu
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Linearized Stability ◽  
Appl Math

We study the predator-prey model proposed by Aziz-Alaoui and Okiye (Appl. Math. Lett. 16 (2003) 1069–1075) First, the structure of equilibria and their linearized stability is investigated. Then, we provide two sufficient conditions on the global asymptotic stability of a positive equilibrium by employing the Fluctuation Lemma and Lyapunov direct method, respectively. The obtained results not only improve but also supplement existing ones.

scholarly journals Dynamics of a Predator-Prey System with Beddington-DeAngelis Functional Response and Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Nai-Wei Liu ◽  
Ting-Ting Kong
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Global Asymptotic Stability ◽  
Stage Structure ◽  
Necessary Condition ◽  
Positive Equilibrium ◽  
Sufficient Condition ◽  
Predator Prey ◽  
Prey System ◽  
Hyperbolic Curve

We consider a predator-prey system with Beddington-DeAngelis functional response and delays, in which not only the stage structure on prey but also the delay due to digestion is considered. First, we give a sufficient and necessary condition for the existence of a unique positive equilibrium by analyzing the corresponding locations of a hyperbolic curve and a line. Then, by constructing an appropriate Lyapunov function, we prove that the positive equilibrium is locally asymptotically stable under a sufficient condition. Finally, by using comparison theorem and theω-limit set theory, we study the global asymptotic stability of the boundary equilibrium and the positive equilibrium, respectively. Also, we obtain a sufficient condition to assure the global asymptotic stability.

Dynamical behaviors of a diffusive predator–prey system with Beddington–DeAngelis functional response

2014 ◽  
Vol 07 (03) ◽  
pp. 1450033
Author(s):  
Er-Dong Han ◽  
Peng Guo
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Positive Solutions ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Lower Solution ◽  
Comparison Theory ◽  
Predator Prey ◽  
Prey System

In this paper, we present a diffusive predator–prey system with Beddington–DeAngelis functional response, where the prey species can disperse between the two patches, and there is competition between the two predators. Sufficient conditions for the permanence and extinction of system are established based on the upper and lower solution methods and comparison theory of differential equation. Furthermore, the global asymptotic stability of positive solutions is obtained by constructing a suitable Lyapunov function. By using the continuation theorem in coincidence degree theory, we show the periodicity of positive solutions. Finally, we illustrate global asymptotic stability of the model by a simulation figure.

scholarly journals Global Stability for a Delayed Predator-Prey System with Stage Structure for the Predator

2009 ◽  
Vol 2009 ◽  
pp. 1-24
Author(s):  
Xiao Zhang ◽  
Rui Xu ◽  
Qintao Gan
Keyword(s):  
Global Stability ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
Comparison Argument ◽  
Theoretical Results ◽  
Iteration Technique

A delayed predator-prey system with stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of equilibria of the system is discussed. The existence of Hopf bifurcation at the positive equilibrium is established. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and two boundary equilibria of the system. Numerical simulations are carried out to illustrate the theoretical results.

Global stability of a stochastic predator–prey model with Allee effect

2015 ◽  
Vol 08 (04) ◽  
pp. 1550044 ◽  
Author(s):  
Baodan Tian ◽  
Liu Yang ◽  
Shouming Zhong
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Allee Effect ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Global Positive Solution

In this paper, we study a stochastic predator–prey model with Beddington–DeAngelis functional response and Allee effect, and show that there is a unique global positive solution to the system with the positive initial value. Sufficient conditions for global asymptotic stability are established. Some simulation figures are introduced to support the analytical findings.

scholarly journals Global Stability and Bifurcations of a Diffusive Ratio-Dependent Holling-Tanner System

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Wenjie Zuo
Keyword(s):  
Global Stability ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Prey System ◽  
Existence And Stability ◽  
Ratio Dependent ◽  
Stability And Bifurcations

The dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey system subject to Neumann boundary conditions are considered. By choosing the ratio of intrinsic growth rates of predators to preys as a bifurcation parameter, the existence and stability of spatially homogeneous and nonhomogeneous Hopf bifurcations and steady state bifurcation are investigated in detail. Meanwhile, we show that Turing instability takes place at a certain critical value; that is, the stationary solution becomes unstable induced by diffusion. Particularly, the sufficient conditions of the global stability of the positive constant coexistence are given by the upper-lower solutions method.

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