The limit cycles of a general Kolmogorov system

Yuan, Yueding; Chen, Haibo; Du, Cunbin; Yuan, Yuejin

doi:10.1016/j.jmaa.2012.02.065

Cited by 17 publications

(8 citation statements)

References 26 publications

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“…From the Implicit Function Theorem applied to the right-hand side term of (19), there exists a unique curve T ′ 3 such that λ E11 2 λ E12 2 = 0 on T ′ 3 , for |µ| sufficiently small and δµ 2 > 0, given by…”

Section: The Jacobian Matricesmentioning

confidence: 99%

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Analysis of a class of Kolmogorov systems

Tigan

Lăzureanu

Munteanu

et al. 2021

Nonlinear Analysis: Real World Applications

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Section: The Jacobian Matricesmentioning

confidence: 99%

“…For example, an autonomous Lotka-Volterra type competitive system for N −species to control the gut microbiota by antibiotics has been presented recently in [5]. Many environmental, engineering, economics and mechanical models can be reduced to some kind of Kolmogorov systems [19].…”

Section: Introductionmentioning

confidence: 99%

Analysis of a class of Kolmogorov systems

Tigan

Lăzureanu

Munteanu

et al. 2021

Nonlinear Analysis: Real World Applications

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“…The existence of the limit cycle comes from the Poincaré-Bendixson annular region Theorem while its uniqueness is proven in [6]: the stability is a consequence of the uniqueness as shown in [13] and in the more classical [17]. For a general result on the limit cycles of Kolmogorov systems like (6) see [22].…”

Section: A Appendix: Disease-free Modelmentioning

confidence: 99%

Uniform persistence in a prey–predator model with a diseased predator

Dondè

2019

J. Math. Biol.

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Following the well-extablished mathematical approach to persistence and its developments contained in [19] we give a rigorous theoretical explanation to the numerical results obtained in [4] on a certain prey-predator model with functional response of Holling type II equipped with an infectious disease in the predator population.The proof relies on some repelling conditions that can be applied in an iterative way on a suitable decomposition of the boundary. A full stability analysis is developed, showing how the "invasion condition" for the disease is derived. Some counterexamples and possible further investigations are discussed.

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“…Concerning the Kolmogorov systems, most of the studies were limited to study the existence of limit cycles for classes of these systems see [17,20,21,[25][26][27]. To our knowledge, the exact analytic expressions of the limit cycles for a given Kolmogorov system is still unknown except in algebraic case.…”

Section: Introductionmentioning

confidence: 99%

A Class of Quintic Kolmogorov Systems with Explicit Non-algebraic Limit Cycle

Bendjeddou¹,

Grazem

Бенджедду³

et al. 2019

Journal of Siberian Federal University. Mathematics &Amp; Physics

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Various physical, ecological, economic, etc phenomena are governed by planar differential systems. Sub- sequently, several research studies are interested in the study of limit cycles because of their interest in the understanding of these systems. The aim of this paper is to investigate a class of quintic Kolmogorov systems, namely systems of the form x=xP4 (x;y); y= y Q4 (x; y) ; where P4 and Q4 are quartic polynomials. Within this class, our attention is restricted to study the limit cycle in the realistic quadrant {(x; y) 2 R2; x > 0; y > 0}. According to the hypothesises, the existence of algebraic or non-algebraic limit cycle is proved. Furthermore, this limit cycle is explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our result

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The limit cycles of a general Kolmogorov system

Cited by 17 publications

References 26 publications

Analysis of a class of Kolmogorov systems

Analysis of a class of Kolmogorov systems

Uniform persistence in a prey–predator model with a diseased predator

A Class of Quintic Kolmogorov Systems with Explicit Non-algebraic Limit Cycle

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