Qualitative Analysis of a Ratio-Dependent Chemostat Model with Holling-(n+1) Type Functional Response

2013 ◽  
Vol 641-642 ◽  
pp. 947-950
Author(s):  
Qing Lai Dong ◽  
Ming Juan Sun
Keyword(s):  
Asymptotic Stability ◽  
Qualitative Analysis ◽  
Functional Response ◽  
Global Asymptotic Stability ◽  
Monod Model ◽  
Chemostat Model ◽  
Dependent Model ◽  
Ratio Dependent ◽  
Lasalle Invariant Principle ◽  
Invariant Principle

In this paper, the ratio-dependent chemostat model with Holling-(n+1) type functional response is considered. The model develops the Monod model and the ratio-dependent model. By use of the Poincar -Bendixson theory we prove the existence of limit cycle. Detailed qualitative analysis about the global asymptotic stability of its equilibria is carried out by using the Lyapunov-LaSalle invariant principle and the method of Dulac criterion.

Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake and a time delay

2014 ◽  
Vol 07 (04) ◽  
pp. 1450045 ◽  
Author(s):  
Qinglai Dong ◽  
Wanbiao Ma
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Qualitative Analysis ◽  
Numerical Simulations ◽  
Global Asymptotic Stability ◽  
Positive Equilibrium ◽  
Substrate Uptake ◽  
Sufficient Condition ◽  
Chemostat Model ◽  
Local Asymptotic Stability

In this paper, we consider a simple chemostat model with inhibitory exponential substrate uptake and a time delay. A detailed qualitative analysis about existence and boundedness of its solutions and the local asymptotic stability of its equilibria are carried out. Using Lyapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Using the fluctuation lemma, the sufficient condition of the global asymptotic stability of the positive equilibrium [Formula: see text] is obtained. Numerical simulations are also performed to illustrate the results.

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 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 Stability and Bifurcation in a Delayed Holling-Tanner Predator-Prey System with Ratio-Dependent Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Juan Liu ◽  
Zizhen Zhang ◽  
Ming Fu
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Center Manifold ◽  
Predator Prey ◽  
Local Asymptotic Stability ◽  
Prey System ◽  
Ratio Dependent ◽  
The Stability

We analyze a delayed Holling-Tanner predator-prey system with ratio-dependent functional response. The local asymptotic stability and the existence of the Hopf bifurcation are investigated. Direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are studied by deriving the equation describing the flow on the center manifold. Finally, numerical simulations are presented for the support of our analytical findings.

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