Quadratic differential systems for interactive population models

Jenks, Richard D.

doi:10.1016/0022-0396(69)90090-4

Cited by 58 publications

(38 citation statements)

References 2 publications

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“…Such operator frequently arise in many models of mathematical genetics [1][2][3][4][5][6][7][8][9][10][11], [13], [15], [16]. The quadratic stochastic operator (QSO) is a mapping of the simplex S n−1 = {x = (x 1 , ..., x n ) ∈ R n : x i ≥ 0,…”

Section: Introductionmentioning

confidence: 99%

On Non-Volterra Quadratic Stochastic Operators Generated by a Product Measure

Rozikov

Shamsiddinov

2009

Stochastic Analysis and Applications

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In this paper we describe a wide class of non-Volterra quadratic stochastic operators using N. Ganikhadjaev's construction of quadratic stochastic operators. By the construction these operators depend on a probability measure µ being defined on the set of all configurations which are given on a graph G. We show that if µ is the product of probability measures being defined on each maximal connected subgraphs of G then corresponding non-Volterra operator can be reduced to m number (where m is the number of maximal connected subgraphs of G) of Volterra operators defined on the maximal connected subgraphs. Our result allows to study a wide class of non-Volterra operators in the framework of the well known theory of Volterra quadratic stochastic operators.

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Section: Introductionmentioning

confidence: 99%

On Non-Volterra Quadratic Stochastic Operators Generated by a Product Measure

Rozikov

Shamsiddinov

2009

Stochastic Analysis and Applications

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“…(See [7,9,13].) The notion of Volterra QSO was introduced in [16], it is defined by (2.2), (2.3) and the additional assumption…”

Section: Preliminariesmentioning

confidence: 99%

On Dynamics of ℓ-Volterra Quadratic Stochastic Operators

Rozikov

Zada²

2010

Int. J. Biomath.

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We introduce a notion of -Volterra quadratic stochastic operator defined on (m − 1)dimensional simplex, where ∈ {0, 1, . . . , m}. The -Volterra operator is a Volterra operator if and only if = m. We study structure of the set of all -Volterra operators and describe their several fixed and periodic points. For m = 2 and 3, we describe behavior of trajectories of (m − 1)-Volterra operators. The paper also contains many remarks with comparisons of -Volterra operators and Volterra ones.This means that the association x 0 → x defines a map V called the evolution operator. The population evolves by starting from an arbitrary state x 0 , then passing to the state x = V (x 0 ) (in the next "generation"), then to the state x = V (V (x 0 )), 143 Int. J. Biomath. 2010.03:143-159. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only. 144 U. A. Rozikov & A. Zada Int. J. Biomath. 2010.03:143-159. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only.

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“…{p s,t ij,k } is of type (A) or (B) if it satisfies the fundamental equations (iii A ) or (iii B ), respectively. In this definition the functions p [s,t] ij,k denotes the probability that under the interaction of the elements i and j at time s the element k comes into effect at time t. Since for physical, chemical and biological phenomena a certain time is necessary for the realization of an interaction, we shall take the greatest such time to be equal to 1 (see the Boltzmann model [5] or the biological model [8]). Thus the probability p [s,t] ij,k is defined for t − s ≥ 1.…”

Section: Introductionmentioning

confidence: 99%