Positive periodic solution for a two-species ratio-dependent predator–prey system with time delay and impulse

Liu, Xiangsen; Li, Gang; Luo, Guilie

doi:10.1016/j.jmaa.2005.09.022

Cited by 12 publications

(7 citation statements)

References 6 publications

(6 reference statements)

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“…To accommodate these situations, one needs to use the impulsive differential equations [1,2,21]. For works on the existence of periodic solutions of impulsive predator-prey systems, we refer the reader to [5,17,20,22,23,25].…”

Section: §1 Introductionmentioning

confidence: 99%

Permanence and periodic solutions of delayed predator-prey system with impulse

Shi

2010

Appl. Math. J. Chin. Univ.

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The existence of positive periodic solution of a generalized semi-ratio-dependent predator-prey system with time delay and impulse is studied by using the continuation theorem based on the coincidence degree theory. The permanence of the system is also considered. The results partially improve and extend some known criteria. §1 Introduction As Cushing [4] pointed out that it is necessary and important to consider models with periodic ecological parameters or perturbations which might be quite naturally exposed (for example, those due to seasonal effects of weather, food supply, mating habits, hunting or harvesting seasons, etc.). In recent decade, there have been many works which are devoted to the studies of dynamical behaviors for predator-prey systems with various functional responses, including the permanence of system, the existence, uniqueness and global attractivity of positive periodic solution and so on. Along this direction, one can refer to [3,[6][7][8][9][11][12][13]18,26,28] and the references cited therein.In particular, many authors have explored a class of semi-ratio-dependent predator-prey system. In 2003, Wang et al. in [27] studied the following semi-ratio-dependent predator-prey system1) and established some criteria for the existence of positive periodic solution. In 2005, Huo in [19] further considered system (1.1) and improved some results obtained in [27] by dropping some assumptions. Recently, Fazly and Hesaaraki in [15], Fan and Wang in [14] considered the following discrete analogue of system (1.1) 2)Received: 2009-02-03. MR Subject Classification: 34C25, 92D25.

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Section: §1 Introductionmentioning

confidence: 99%

Permanence and periodic solutions of delayed predator-prey system with impulse

Shi

2010

Appl. Math. J. Chin. Univ.

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“…Systems with such kinds of discontinuous changes can be investigated by the theory of impulsive differential equations [1,14]. In the past few years, there have been a number of studies which applied impulsive differential equations to biological problems (see, e.g., [17][18][19][20]23,30,31]). …”

Section: Introductionmentioning

confidence: 99%

“…Under the above assumptions, system (1.2) covers many models appeared in the literature. For instance, g(t, x) can be taken as the logistic growth a − bx [2,[4][5][6]10,11,13,17,18,29], the Gilpin growth a − bx θ [8], and the Smith growth (a − bx)/(D + x) [25]. h(t, x) can be taken as functional responses of the LotkaVolterra type mx [11,13], the Holling type mx n /(A + x n ) [2,6,9,10,27], the Ivlev type m(1 − e −Ax ) [12,26], the sigmoidal type [3], and the Monod-Haldane type mx/( A + x 2 ) [4,24].…”

Section: Introductionmentioning

confidence: 99%

Periodicity in a generalized semi-ratio-dependent predator–prey system with time delays and impulses

Ding

Jiang

2009

Journal of Mathematical Analysis and Applications

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With the help of a continuation theorem based on coincidence degree theory, we establish necessary and sufficient conditions for the existence of positive periodic solutions in a generalized semi-ratio-dependent predator-prey system with time delays and impulses, which covers many models appeared in the literature. When the results reduce to the semi-ratio-dependent predator-prey system without impulses, they generalize and improve some known ones.

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“…In recent years, attention has been given to the study of periodic solutions of these systems, and different methods have been proposed for the study of the existence and qualitative property of periodic solutions. Most of the discussions are for special model problems, and the main approach to the existence problem is based on a continuation theorem in coincidence degree theory [5][6][7]. Some other methods are also used by some researchers, such as Tang [8] and Dou [9] use the monotone iteration technique for the impulsive Lotka-Volterra model, Panetta [10], Lakmeche and Arino [11] investigate an impulsive competition model by a bifurcation method, refer to the monograph [1] for the more details.…”

Section: Introductionmentioning

confidence: 99%

A Monotone-Iterative Method for Finding Periodic Solution of an Impulsive Differential Equations and Application

Dou

2008

Int. J. Biomath.

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In this paper, using the method of upper and lower solutions and its associated monotone iterations, we establish a new monotone-iterative scheme for finding periodic solutions of an impulsive differential equations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for impulsive differential equations initial value problem. This method is constructive and can be used to develop a computational algorithm for numerical solution of the periodic impulsive system. Our existence result improves a result established in [1]. The result is applied to a model of mutualism of Lotka-Volterra type which involves interactions among a mutualist-competitor, a competitor and a mutualist, the existence of positive periodic solutions of the model is obtained.

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Positive periodic solution for a two-species ratio-dependent predator–prey system with time delay and impulse

Abstract: A two-species ratio-dependent predator-prey model with time delay and impulse is investigated. By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for this system is established.

Cited by 12 publications

References 6 publications

Permanence and periodic solutions of delayed predator-prey system with impulse

Permanence and periodic solutions of delayed predator-prey system with impulse

Periodicity in a generalized semi-ratio-dependent predator–prey system with time delays and impulses

A Monotone-Iterative Method for Finding Periodic Solution of an Impulsive Differential Equations and Application

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