On the global attractivity for a logistic equation with piecewise constant arguments*1

UESUGI, K

doi:10.1016/s0022-247x(04)00164-7

Cited by 2 publications

(2 citation statements)

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“…In study [17], Cooke and Györi show that differential equation with piecewise constant arguments can be used to obtain good approximate solution of delay differential equations on the infinite interval [0, ∞). Therefore, there has been great interest in studying differential equation with piecewise constant arguments which combine properties of both differential and difference equations [18][19][20][21][22][23][24][25][26]. I. Ozturk et al [18] have modeled bacteria population by using differential equation…”

Section: Introductionmentioning

confidence: 99%

Mathematical modeling and analysis of tumor-immune system interaction by using Lotka-Volterra predator-prey like model with piecewise constant arguments

Kartal¹

2014

PEN

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In this study, we present a Lotka-Volterra predator-prey like model for the interaction dynamics of tumor-immune system. The model consists of system of differential equations with piecewise constant arguments and based on the model of tumor growth constructed by Sarkar and Banerjee. The solutions of differential equations with piecewise constant arguments leads to system of difference equations. Sufficient conditions are obtained for the local and global asymptotic stability of a positive equilibrium point of the discrete system by using Schur-Cohn criterion and a Lyapunov function. In addition, we investigate periodic solutions of discrete system through Neimark-Sacker bifurcation and obtain a stable limit cycle which implies that tumor and immune system undergo oscillation.

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Section: Introductionmentioning

confidence: 99%

Mathematical modeling and analysis of tumor-immune system interaction by using Lotka-Volterra predator-prey like model with piecewise constant arguments

Kartal¹

2014

PEN

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“…In population dynamics, a first model including piecewise constant argument was constructed by Busenberg and Cooke to investigate vertically transmitted diseases. Following this work, several authors have investigated the stability and oscillatory characteristics of difference solutions of logistic differential equations with piecewise constant arguments . May and May and Oster have considered a simple logistic equation for a single species such as

\frac{normald normalN (normalt)}{normald normalt} MathClass-rel= normalr normalN (normalt) ({}, 1 MathClass-bin- \frac{normalN ([normalt])}{normalK}) MathClass-punc,

where r is intrinsic growth rate and K is maximum carrying capacity.…”

Section: Introductionmentioning

confidence: 99%

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

Kartal

Gürcan

2014

Math Methods in App Sciences

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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.

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On the global attractivity for a logistic equation with piecewise constant arguments*1

Cited by 2 publications

References 0 publications

Mathematical modeling and analysis of tumor-immune system interaction by using Lotka-Volterra predator-prey like model with piecewise constant arguments

Mathematical modeling and analysis of tumor-immune system interaction by using Lotka-Volterra predator-prey like model with piecewise constant arguments

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

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