Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems

Zeeman, Mary Lou

doi:10.1080/02681119308806158

Cited by 178 publications

(241 citation statements)

References 28 publications

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“…where A kj , B kj ℝ(k, j N) and all A kj , B kj are polynomial functions of coefficients of the system (8) or (2). System (9) is also called the equations on the center manifold or reduction system of (8).…”

Section: The Focal Values On Center Manifoldmentioning

confidence: 99%

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The equivalence between singular point quantities and Liapunov constants on center manifold

Wang

Huang

2012

Adv Differ Equ

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The algorithm of singular point quantities for an equilibrium of three-dimensional dynamics system is studied. The explicit algebraic equivalent relation between singular point quantities and Liapunov constants on center manifold is rigorously proved. As an example, the calculation of singular point quantities of the Lü system is applied to illustrate the advantage in investigating Hopf bifurcation of threedimensional system.

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Section: The Focal Values On Center Manifoldmentioning

confidence: 99%

“…The systems (1) usually involve many important nonlinear dynamical models such as Lotka-Volterra system [1,2] and Lorenz system [3,4]. As far as Hopf bifurcation of the origin of systems (1) is concerned, the Jacobian matrix A at the origin should have a pair of purely imaginary eigenvalues and a negative one.…”

Section: Introductionmentioning

confidence: 99%

The equivalence between singular point quantities and Liapunov constants on center manifold

Wang

Huang

2012

Adv Differ Equ

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“…Following M. L. Zeeman [15], the invariant compact set Σ is referred to as the carrying simplex for (E). In the ecological interpretation, the carrying simplex can be thought of as expressing the balance between the growth of small populations ({0} is a repeller) and the competition of large populations (dissipativity).…”

Section: At Each Rest Pointmentioning

confidence: 99%

On smoothness of carrying simplices

Mierczyński

1999

Proc. Amer. Math. Soc.

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Abstract. We consider dissipative strongly competitive systemsẋ i = x i f i (x) of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface Σ, called the carrying simplex. In this note we give an amenable condition for Σ to be a C 1 submanifold-with-corners. We also provide conditions, based on a recent work of M. Benaïm (On invariant hypersurfaces of strongly monotone maps,

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“…It is interesting to compare the relative strengths of the hypotheses of Theorem 2.1 through Corollary 2.6, by considering the application of each to three-dimensional autonomous competitive Lotka-Volterra systems. These systems were studied in [14], and classified into 33 open equivalence classes called nullcline classes. In [15] it was shown that systems in nullcline class 1 are precisely those satisfying the hypotheses of Corollary 2.6, and that this result is far from being sharp, since nullcline classes 2,3,7 and 8 also consist of systems in which all but one species are driven to extinction.…”

Section: Then Every Trajectory With Initial Condition In Intr Nmentioning

confidence: 99%

Extinction in nonautonomous competitive Lotka-Volterra systems

Zeeman

1996

Proc. Amer. Math. Soc.

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Abstract. It is well known that for the two species autonomous competitive Lotka-Volterra model with no fixed point in the open positive quadrant, one of the species is driven to extinction, whilst the other population stabilises at its own carrying capacity. In this paper we prove a generalisation of this result to nonautonomous systems of arbitrary finite dimension. That is, for the n species nonautonomous competitive Lotka-Volterra model, we exhibit simple algebraic criteria on the parameters which guarantee that all but one of the species is driven to extinction. The restriction of the system to the remaining axis is a nonautonomous logistic equation, which has a unique solution u(t) that is strictly positive and bounded for all time; see Coleman (Math. Biosci. 45 (1979), 159-173) and Ahmad (Proc. Amer. Math. Soc. 117 (1993), 199-205). We prove in addition that all solutions of the n-dimensional system with strictly positive initial conditions are asymptotic to u(t).

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Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems

Cited by 178 publications

References 28 publications

The equivalence between singular point quantities and Liapunov constants on center manifold

The equivalence between singular point quantities and Liapunov constants on center manifold

On smoothness of carrying simplices

Extinction in nonautonomous competitive Lotka-Volterra systems

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