 04259 A.N. Gorban
 Systems with inheritance:
dynamics of distributions with conservation of support, natural selection
and finitedimensional asymptotics
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Aug 25, 04

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Abstract. If we find a representation of an infinitedimensional dynamical
system as a nonlinear kinetic system with {\it conservation of
supports} of distributions, then (after some additional technical
steps) we can state that the asymptotics is finitedimensional.
This conservation of support has a {\it quasibiological
interpretation, inheritance} (if a gene was not presented
initially in a isolated population without mutations, then it
cannot appear at later time). These quasibiological models can
describe various physical, chemical, and, of course, biological
systems. The finitedimensional asymptotic demonstrates effects of
{\it ``natural" selection}. The estimations of asymptotic
dimension are presented. The support of an individual limit
distribution is almost always small. But the union of such
supports can be the whole space even for one solution. Possible
are such situations: a solution is a finite set of narrow peaks
getting in time more and more narrow, moving slower and slower. It
is possible that these peaks do not tend to fixed positions,
rather they continue moving, and the path covered tends to
infinity at $t \rightarrow \infty$. The {\it drift equations} for
peaks motion are obtained. Various types of stability are studied.
In example, models of cell division selfsynchronization are
studied. The appropriate construction of notion of typicalness in
infinitedimensional spaces is discussed, and the ``completely
thin" sets are introduced.
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