Stability Analysis for Discrete Biological Models Using Algebraic Methods

Li, Xiaoliang; Mou, Chenqi; Wei, Ning; Wang, Dongming

doi:10.1007/s11786-011-0096-z

Cited by 43 publications

(19 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, by integrating of system (5) we first obtain a solution and later discuss the boundedness and the local asymptotic stability of system (7). We can rewrite system (5) on an interval of the form t ∈ [n, n + 1) as follows:…”

Section: Boundedness Local and Global Stability Analysismentioning

confidence: 99%

Global behaviour of a predator–prey like model with piecewise constant arguments

Kartal

Gürcan

2015

Journal of Biological Dynamics

View full text Add to dashboard Cite

The present study deals with the analysis of a predator-prey like model consisting of system of differential equations with piecewise constant arguments. A solution of the system with piecewise constant arguments leads to a system of difference equations which is examined to study boundedness, local and global asymptotic behaviour of the positive solutions. Using Schur-Cohn criterion and a Lyapunov function, we derive sufficient conditions under which the positive equilibrium point is local and global asymptotically stable. Moreover, we show numerically that periodic solutions arise as a consequence of Neimark-Sacker bifurcation of a limit cycle.

show abstract

Section: Boundedness Local and Global Stability Analysismentioning

confidence: 99%

Global behaviour of a predator–prey like model with piecewise constant arguments

Kartal

Gürcan

2015

Journal of Biological Dynamics

View full text Add to dashboard Cite

show abstract

“…, We consider that P 3 (λ) is the characteristic polynomial of J(E * ), so, eigenvalues of Nash equilibrium correspond to the roots of P 3 (λ) = 0. The Nash equilibrium is stable if the necessary and sufficient condition for the roots of the polynomial P 3 (λ) to satisfy |λ| < 1 can be obtained by applying Jury's test [12]. The characteristic polynomial has the form:…”

Section: A Asymmetric Scenariomentioning

confidence: 99%

“…where com(·) denotes the comatrice operator. Recalling Jury's test [12] for stability of Nash equilibrium, we get the necessary and sufficient conditions for |λ i | < 1, i = 1, 2, 3:…”

Section: A Asymmetric Scenariomentioning

confidence: 99%

“…. According the Jury criteria, [12], the Nash equilibrium point is locally asymptotically stable if the following condition equations are satisfied : (i)…”

Section: B Symmetric Scenariomentioning

confidence: 99%

“…The stability analysis can be reduced to that of determining conditions to ensure that all the roots of P 8 (λ) lie in the open unit disk |λ| < 1. The stability conditions of Nash equilibrium are given by recalling Jury's conditions [12] which are the necessary and sufficient conditions for |λ k | < 1, k ∈ {1, 2, .., 8}. Consider the characteristic polynomial P 8 (λ) and let:…”

Section: Joint Price and Qos Gamementioning

confidence: 99%

See 2 more Smart Citations

New Insights from a Bounded Rationality Analysis for Strategic Price-QoS War

Baslam

El-Azouzi

Sabir

et al. 2012

Proceedings of the 6th International Conference on Performance Evaluation Methodologies and Tools

View full text Add to dashboard Cite

We present a game theoretic framework for the dynamical behaviors of a duopoly game in telecommunications service providers' context. Competition between two Service Providers (SPs) is assumed to take place in terms of their pricing decisions and the Quality of Service (QoS) they offer. According to the SPs' rationality level, we consider two schemes: 1) Both SPs are rational, and 2) One SP is rational and the second SP is boundedly rational. We describe the competitive interaction and analyze the resulting equilibria. Later, we compute explicitly the steady states of the dynamical system induced by bounded rationality, and establish a necessary and sufficient condition for stability of its Nash equilibria (NEs). We prove that there exists exactly one NE which is fair whereas remaining equilibria are unfair. A special feature is that the stability condition of the Nash equilibrium coincides with the instability condition of the boundary equilibria. Thus the system would never be absorbed by any of the unfair equilibria which solves the equilibrium selection issue ! Moreover, we show that considering the delay case (i.e., assuming a market with memory) increases the stability of the system. Here, the size of the memory could be considered for multi-level rationality, which means that bounded rationality tends to rationality as the memory size increases. We finally show that boundedly rational SPs with delay have a higher chance of reaching the fair Nash equilibrium.

show abstract

Stability and bifurcations analysis of a competition model with piecewise constant arguments

Kartal

Gürcan

2014

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we investigate local and global asymptotic stability of a positive equilibrium point of system of differential equations {alignedrightleftdxdt=r1xMathClass-open(tMathClass-close)1−xMathClass-open(tMathClass-close)k1−α1xMathClass-open(tMathClass-close)yMathClass-open(MathClass-open[t−1MathClass-close]MathClass-close)+α2xMathClass-open(tMathClass-close)yMathClass-open(MathClass-open[tMathClass-close]MathClass-close),right rightleftdydt=r2yMathClass-open(tMathClass-close)1−yMathClass-open(tMathClass-close)k2+α1yMathClass-open(tMathClass-close)xMathClass-open(MathClass-open[t−1MathClass-close]MathClass-close)−α2yMathClass-open(tMathClass-close)xMathClass-open(MathClass-open[tMathClass-close]MathClass-close)−d1yMathClass-open(tMathClass-close), where t ≥ 0, the parameters r1, k1, α1, α2, r2, k2, and d1 are positive, and [t] denotes the integer part of t ∈ [0, ∞ ). x(t) and y(t) represent population density for related species. Sufficient conditions are obtained for the local and global stability of the positive equilibrium point of the corresponding difference system. We show through numerical simulations that periodic solutions arise through Neimark–Sacker bifurcation. Copyright © 2014 John Wiley & Sons, Ltd.

show abstract

Stability Analysis for Discrete Biological Models Using Algebraic Methods

Cited by 43 publications

References 46 publications

Global behaviour of a predator–prey like model with piecewise constant arguments

Global behaviour of a predator–prey like model with piecewise constant arguments

New Insights from a Bounded Rationality Analysis for Strategic Price-QoS War

Stability and bifurcations analysis of a competition model with piecewise constant arguments

Contact Info

Product

Resources

About