Limit Cycles for the Competitive Three Dimensional Lotka–Volterra System

Xiao, Dongmei; Li, Wenxia

doi:10.1006/jdeq.1999.3729

Cited by 79 publications

(54 citation statements)

References 7 publications

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“…when such differential systems have first integrals (see for instance [1,2,4,5,6,7,8,17,18,22]),or • in their periodic orbits (see for example [9,10,11,13,16,20,24,25,26]). …”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the dynamics of a class of Kolmogorov systems

Llibre

Salhi

2013

Applied Mathematics and Computation

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“…when such differential systems have first integrals (see for instance [1,2,4,5,6,7,8,17,18,22]),or • in their periodic orbits (see for example [9,10,11,13,16,20,24,25,26]). …”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the dynamics of a class of Kolmogorov systems

Llibre

Salhi

2013

Applied Mathematics and Computation

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“…When there are no periodic orbits, the global dynamics are as shown, except that the interior equilibrium in classes 26 and 27 may be either attracting or repelling [24]. Classes 26-29 admit at least two isolated periodic orbits [13,17], but an upper bound on the number of isolated periodic orbits remains open [23]. We do not address the question of how many isolated periodic orbits can occur in system (3) under assumptions (4).…”

Section: Geometric Preliminariesmentioning

confidence: 96%

Disease Induced Oscillations between Two Competing Species

Driessche¹,

Zeeman²

2004

SIAM J. Appl. Dyn. Syst.

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Abstract. The interaction of disease and competition dynamics is investigated in a system of two competing species in which only one species is susceptible to disease. The model is kept as simple as possible, combining Lotka-Volterra competition between the species with disease dynamics of susceptible and infective individuals within one of the species. It is assumed that pure vertical disease transmission (from parent to offspring) dominates horizontal transmission (by contact between infective and susceptible individuals) and that infective individuals have the same competition strength as susceptibles but a lower intrinsic growth rate. These assumptions yield three-dimensional competitive Lotka-Volterra dynamics modeling the disease-competition interaction. It is proved that if in the absence of disease there is competitive exclusion between the two species, then the presence of disease can lead to stable or oscillatory coexistence of both species. The case of oscillatory coexistence can be viewed either as disease induced oscillations between competing species or as competition induced oscillations in an endemic disease. By contrast, conditions are found under which, if the two species coexist in the absence of disease, then the introduction of disease does not induce oscillations, and the long-term dynamics are determined by the basic reproduction number.

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“…There have been a series of achievements and unprecedented challenges on the theme even if system (1) is competitive system (cf. [7,11,12,14,21,29,30,31]). In [3], Bobieński andŻoladek gave four components of center variety in the three-dimensional LotkaVolterra class and studied the existence and number of isolated periodic solutions by certain Poincaré-Melnikov integrals of a new type.…”

Section: Introductionmentioning

confidence: 99%

Limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of three-dimensional differential systems

Llibre¹,

Xiao²

2014

Proc. Amer. Math. Soc.

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Abstract. In this paper we study the limit cycles bifurcating from a nonisolated zero-Hopf equilibrium of a differential system in R 3 . The unfolding of the vector fields with a non-isolated zero-Hopf equilibrium is a family with at least three parameters. By using the averaging theory of the second order, explicit conditions are given for the existence of one or two limit cycles bifurcating from such a zero-Hopf equilibrium. To our knowledge, this is the first result on bifurcations from a non-isolated zero-Hopf equilibrium. This result is applied to study three-dimensional generalized Lotka-Volterra systems in [3]. The necessary and sufficient conditions for the existence of a non-isolated zero-Hopf equilibrium of this system are given, and it is shown that two limit cycles can be bifurcated from the non-isolated zero-Hopf equilibrium under a general small perturbation of three-dimensional generalized Lotka-Volterra systems.

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Limit Cycles for the Competitive Three Dimensional Lotka–Volterra System

Cited by 79 publications

References 7 publications

On the dynamics of a class of Kolmogorov systems

On the dynamics of a class of Kolmogorov systems

Disease Induced Oscillations between Two Competing Species

Limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of three-dimensional differential systems

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