Periodic Solutions and Global Asymptotic Stability of a Delayed Discrete Predator-Prey System with Holling II Type Functional Response

2006 ◽  
Vol 19 (4) ◽  
pp. 449-460 ◽  
Author(s):  
Cuimei Zhang ◽  
Wencheng Chen ◽  
Yu Yang
Keyword(s):  
Asymptotic Stability ◽  
Periodic Solutions ◽  
Functional Response ◽  
Global Asymptotic Stability ◽  
Predator Prey ◽  
Prey System

scholarly journals A new study for the global asymptotic stability of a general predator-prey model

2021 ◽  
Author(s):  
Manh Tuan Hoang
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Mathematical Analysis ◽  
Global Asymptotic Stability ◽  
Lyapunov Stability Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Theoretical Results

In a previous paper [L. M. Ladino, E. I. Sabogal, Jose C. Valverde, General functional response and recruitment in a predator-prey system with capture on both species, Math. Methods Appl. Sci. 38(2015) 2876-2887], a mathematical model for a predator-prey model with general functional response and recruitment including capture on both species was formulated and analyzed. However, the global asymptotic stability (GAS) of this model was only partially resolved. In the present paper, we provide a rigorously mathematical analysis for the complete GAS of the predator-prey model. By using the Lyapunov stability theory in combination with some nonstandard techniques of mathematical analysis for dynamical systems, the GAS of equilibria of the model is determined fully. The obtained results not only provide an important improvement for the population dynamics of the predator-prey model but also can be extended to study its modified versions in the context of fractional-order derivatives. The theoretical results are supported and illustrated by a set of numerical examples.

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.

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.

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.

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 Dynamics of a Stochastic Cooperative Predator-Prey System with Beddington-DeAngelis Functional Response

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Dongwei Huang ◽  
Yu Li ◽  
Yongfeng Guo
Keyword(s):  
Asymptotic Stability ◽  
Positive Solution ◽  
Functional Response ◽  
Dynamic Properties ◽  
Deterministic System ◽  
Unique Positive Solution ◽  
Initial Value ◽  
Predator Prey ◽  
Prey System ◽  
Stochastically Bounded

Stochastic cooperative predator-prey system with Beddington-DeAngelis functional response is studied. It presents an investigation of dynamic properties of the system. Our results show that there exists a unique positive solution to the system for any positive initial value, and the positive solution is stochastically bounded. Moreover, under some conditions, we analyze global asymptotic stability of the positive solutions. With small environmental noises, the stochastic system is getting more similar to the corresponding deterministic system. Neither of the species in the system will die out. Finally, simulations are carried out to conform to our result.

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