Persistence, contractivity and global stability in logistic equations with piecewise constant delays

Muroya, Yoshiaki

doi:10.1016/s0022-247x(02)00095-1

Cited by 58 publications

(50 citation statements)

References 6 publications

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“…In the last few decades, this class of differential equations has attracted considerable attention due to their wide range of applications in biology, control theory, neural networks etc. [3,4,6,8,13,21,25]. However, the reduction to discrete equations and application of numerical methods have been the main instrument of investigation of the differential equations with piecewise constant argument [3,[9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of a predator–prey model with piecewise constant argument of generalized type using lyapunov functions

Aruğaslan

zer

2014

J Math Sci

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We consider a Lotka-Volterra type predator-prey model with piecewise constant arguments of generalized type and investigate the stability of the positive equilibrium point of the proposed model. Although the model includes piecewise constant delays, we do not use Lyapunov functionals. We establish the stability conditions by using Lyapunov functions of the corresponding model of ordinary differential equations. In order to illustrate the validity of our results, we present an appropriate example and numerical simulations.

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Section: Introductionmentioning

confidence: 99%

Stability analysis of a predator–prey model with piecewise constant argument of generalized type using lyapunov functions

Aruğaslan

zer

2014

J Math Sci

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“…where [·] signifies the greatest integer function, was initiated in [13,30] and developed by many authors [1,8,9,25,28,29], [32], [36]- [39]. These systems have been under intensive investigation for the last twenty years.…”

Section: Introductionmentioning

confidence: 99%

Almost periodic solutions of differential equations with piecewise constant argument of generalized type

Akhmet

2008

Nonlinear Analysis: Hybrid Systems

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We consider existence and stability of an almost periodic solution of the following hybrid systemwhere. . , is an identification function, θ i is a strictly ordered sequence of real numbers, unbounded on the left and on the right, p j , j = 1, 2, . . . , m, are fixed integers, and the linear homogeneous system associated with (1) satisfies exponential dichotomy. The deviations of the argument are not restricted by any sign assumption when existence is considered. The problem of positive (almost periodic) solutions of the logistic equation is discussed as an example. A new technique of investigation of equations with piecewise argument, based on integral representation, is developed.

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“…These equations have been investigated widely using the method of reduction to discrete equations by many authors [11][12][13][14][15][16][17][18][19][20][21]. In [16] some limit relations between the solutions of delay differential equations with continuous arguments and the solutions of some retarded delay EPCA have been proved.…”

Section: Introductionmentioning

confidence: 99%

On periodic solutions of differential equations with piecewise constant argument

Akhmet¹,

Büyükadalı²

2008

Computers & Mathematics with Applications

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a b s t r a c t The periodic quasilinear system of differential equations with small parameter and piecewise constant argument of generalized type [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA, 66 (2007) 367-383, M.U. Akhmet, On the reduction principle for differential equations with piecewise argument of generalized type, J. Math. Anal. Appl. 336 (2007) 646-663] is addressed. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. One of the main auxiliary results of our paper is an analogue of Gronwall-Bellman Lemma for functions with piecewise constant and retarded-advanced type arguments. Dependence of solutions on the parameter is investigated. Appropriate examples are given to show our results.

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Persistence, contractivity and global stability in logistic equations with piecewise constant delays

Cited by 58 publications

References 6 publications

Stability analysis of a predator–prey model with piecewise constant argument of generalized type using lyapunov functions

Stability analysis of a predator–prey model with piecewise constant argument of generalized type using lyapunov functions

Almost periodic solutions of differential equations with piecewise constant argument of generalized type

On periodic solutions of differential equations with piecewise constant argument

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