# Selection Theorem for Systems with Inheritance

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 2, Issue: 4, page 1-45
- ISSN: 0973-5348

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topGorban, A. N.. "Selection Theorem for Systems with Inheritance." Mathematical Modelling of Natural Phenomena 2.4 (2010): 1-45. <http://eudml.org/doc/222439>.

@article{Gorban2010,

abstract = {
The problem of finite-dimensional asymptotics of infinite-dimensional dynamic
systems is studied. A non-linear kinetic system with conservation of supports
for distributions has generically finite-dimensional asymptotics. Such systems are
apparent in many areas of biology, physics (the theory of parametric wave interaction),
chemistry and economics. This conservation of support has a biological interpretation:
inheritance. The finite-dimensional asymptotics demonstrates effects of “natural”
selection. Estimations of the asymptotic dimension are presented. After some initial time,
solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become
increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not
tend to fixed positions, and the path covered tends to infinity as t$\to$∞. The drift equations
for peak motion are obtained. Various types of distribution stability are studied: internal
stability (stability with respect to perturbations that do not extend the support), external
stability or uninvadability (stability with respect to strongly small perturbations that extend
the support), and stable realizability (stability with respect to small shifts and extensions of
the density peaks). Models of self-synchronization of cell division are studied, as an example
of selection in systems with additional symmetry. Appropriate construction of the notion
of typicalness in infinite-dimensional space is discussed, and the notion of “completely thin”
sets is introduced.
},

author = {Gorban, A. N.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {dynamics; attractor; dimension; evolution; entropy; natural selection},

language = {eng},

month = {3},

number = {4},

pages = {1-45},

publisher = {EDP Sciences},

title = {Selection Theorem for Systems with Inheritance},

url = {http://eudml.org/doc/222439},

volume = {2},

year = {2010},

}

TY - JOUR

AU - Gorban, A. N.

TI - Selection Theorem for Systems with Inheritance

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/3//

PB - EDP Sciences

VL - 2

IS - 4

SP - 1

EP - 45

AB -
The problem of finite-dimensional asymptotics of infinite-dimensional dynamic
systems is studied. A non-linear kinetic system with conservation of supports
for distributions has generically finite-dimensional asymptotics. Such systems are
apparent in many areas of biology, physics (the theory of parametric wave interaction),
chemistry and economics. This conservation of support has a biological interpretation:
inheritance. The finite-dimensional asymptotics demonstrates effects of “natural”
selection. Estimations of the asymptotic dimension are presented. After some initial time,
solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become
increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not
tend to fixed positions, and the path covered tends to infinity as t$\to$∞. The drift equations
for peak motion are obtained. Various types of distribution stability are studied: internal
stability (stability with respect to perturbations that do not extend the support), external
stability or uninvadability (stability with respect to strongly small perturbations that extend
the support), and stable realizability (stability with respect to small shifts and extensions of
the density peaks). Models of self-synchronization of cell division are studied, as an example
of selection in systems with additional symmetry. Appropriate construction of the notion
of typicalness in infinite-dimensional space is discussed, and the notion of “completely thin”
sets is introduced.

LA - eng

KW - dynamics; attractor; dimension; evolution; entropy; natural selection

UR - http://eudml.org/doc/222439

ER -

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