On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex

Ганиходжаев, Насир Набиевич; Zanin, Dmitriy

doi:10.1070/rm2004v059n03abeh000744

Cited by 80 publications

(41 citation statements)

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“…Suppose the evolution of the population system is given by (6). Then the population system is similar to that of population system given by (5).…”

Section: Theorem 421mentioning

confidence: 98%

“…One of the central problems of this theory is to study the limiting behavior of trajectories of such operators. [6][7][8][9] Recall that an evolutionary operator of a free population is a (quadratic) mapping of the simplex 1 x V x in the next generation, then to the state '' 2 ( ) ( ( )) = =…”

Section: Introductionmentioning

confidence: 99%

“…Focusing now on the behavior of6 V on the invariant set ′ S , we have the following lemma and theorem.…”

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confidence: 98%

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ON DYNAMICS OF ξ^S QUADRATIC STOCHASTIC OPERATORS

Mukhamedov

Saburov

Jamal

2012

Int. J. Mod. Phys. Conf. Ser.

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In this research we introduce a new class of quadratic stochastic operators called ξ s -QSO which are defined through coefficient of the operator from measure-theoretic (namely we are looking the coefficient as the measures which are absolute continuous or singular) point of view. We also study the limiting behaviour of ξ s -QSO defined on 2D-simplex. We first describe ξ s -QSO on 2D-simplex and classify them with respect to the conjugacy and renumeration of the coordinates. We find six non-isomorphic classes of such operators. Moreover, we investigate the behaviour of each operator from three classes and prove convergence of trajectories of these classes and study their certain properties. We showed trajectories of two classes converge to the equilibrium. For the third class, it is established only the negative trajectories converge to the equilibrium.

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“…Suppose the evolution of the population system is given by (6). Then the population system is similar to that of population system given by (5).…”

Section: Theorem 421mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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ON DYNAMICS OF ξ^S QUADRATIC STOCHASTIC OPERATORS

Mukhamedov

Saburov

Jamal

2012

Int. J. Mod. Phys. Conf. Ser.

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“…is an LV-operator if and only if p ijk = 0 whenever k ∈ {i, j}. In the paper [6], a necessary condition was provided for ergodicity of q.s.o. defined on the 2D simplex.…”

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confidence: 99%

A class of nonergodic Lotka-Volterra operators

Saburov

2015

Math Notes

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On the basis of some numerical calculations, Ulam has conjectured that the ergodic theorem holds for any quadratic stochastic operator acting on a finite-dimensional simplex. However, Zakharevich showed that Ulam's conjecture is false in general. Later, Ganikhodzhaev and Zanin generalized Zakharevich's example to the class of quadratic stochastic Volterra operators acting on a 2D simplex. In this paper, we provide a class of nonergodic Lotka-Volterra operators which includes all previous operators used in this context.

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“…A fixed point set and an omega limiting set of quadratic stochastic operators defined on the finite dimensional simplex were deeply studied in [13,14,15,16,20,21,22]. Ergodicity and chaotic dynamics of quadratic stochastic operators on the finite dimensional simplex were studied in the papers [8,11,12,23,24,25,26,32]. In the paper [9], the nonlinear Poisson quadratic stochastic operators over the countable state space was studied.…”

Section: Introductionmentioning

confidence: 99%