Mutation and Chaos in Nonlinear Models of Heredity

The present paper focuses on the dynamics of doubly stochastic quadratic operators (d.s.q.o) on a finitedimensional simplex. We prove that if a d.s.q.o. has no periodic points then the trajectory of any initial point inside the simplex is convergent. We show that if d.s.q.o. is not a permutation then it has no periodic points on the interior of the two dimensional (2D) simplex. We also show that this property fails in higher dimensions. In addition, the paper also discusses the dynamics classifications of extreme points of d.s.q.o. on two dimensional simplex. As such, we provide some examples of d.s.q.o. which has a property that the trajectory of any initial point tends to the center of the simplex. We also provide and example of d.s.q.o. that has infinitely many fixed points and has infinitely many invariant curves. We therefore came-up with a number of evidences. Finally, we classify the dynamics of extreme points of d.s.q.o. on 2D simplex.

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“…There are other classes of q.s.o. that are well studied [4][5][6]. Another interesting class of q.s.o.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex

et al. 2016

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“…In particular, in these aforementioned works, equivalent conditions for uniform asymptotic stability of such operators in terms of nonhomogeneous chains of linear Markov operators are expressed. The study of the limit behaviour of quadratic stochastic operators is becoming a more and more important topic, for instance, recently in [11,13,14] non-ergodicity of QSOs was studied.…”

Section: Preliminariesmentioning

confidence: 99%

Weak Stability of Centred Quadratic Stochastic Operators

Bartoszek

Domsta

Pułka

2017

Bull. Malays. Math. Sci. Soc.

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We consider the weak convergence of iterates of so-called centred quadratic stochastic operators. These iterations allow us to study the discrete time evolution of probability distributions of vector-valued traits in populations of inbreeding or hermaphroditic species, whenever the offspring's trait is equal to an additively perturbed arithmetic mean of the parents' traits. It is shown that for the existence of a weak limit, it is sufficient that the distributions of the trait and the perturbation have a finite variance or have tails controlled by a suitable power function. In particular, probability distributions from the domain of attraction of stable distributions have found an application, although in general the limit is not stable.

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“…[13,23]) and computer supported mathematical projects (cf. [3,14,15]) aiming to develop a unified theory illustrating (asymptotic) properties of q.s.o.. This subject has been intensively studied in mathematics and biology for almost three decades.…”

Section: Introductionmentioning

confidence: 99%