Issue |
Math. Model. Nat. Phenom.
Volume 2, Number 4, 2007
Patterns and waves in ecology and evolution
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Page(s) | 1 - 45 | |
DOI | https://doi.org/10.1051/mmnp:2008024 | |
Published online | 15 June 2008 |
Selection Theorem for Systems with Inheritance
University of Leicester, LE1 7RH Leicester, UK
Corresponding author: ag153@le.ac.uk
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 ∞. 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.
Mathematics Subject Classification: 92D15 / 37L30 / 54H20
Key words: dynamics / attractor / dimension / evolution / entropy / natural selection
© EDP Sciences, 2007
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