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Systems with inheritance: dynamics of distributions with conservation of support, natural selection and finite-dimensional asymptotics | | A.N. Gorban
; | | Date: |
19 May 2004 | | Subject: | Statistical Mechanics; Mathematical Physics; Populations and Evolution; Adaptation and Self-Organizing Systems | cond-mat.stat-mech math-ph math.MP nlin.AO q-bio.PE | | Abstract: | A non-linear kinetic system with conservation of supports for distributions generically has (under some additional technical conditions) finite-dimensional asymptotics. This conservation of support has a quasi-biological interpretation: inheritance (if a gene is not present initially in an isolated population without mutations, then it cannot appear at a later time). 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 $t o infty$. 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). The general condition for stable realizability is a stability condition with respect to the corresponding drift equations (a finite system of ordinary differential equations). Models of self-synchronization of cell division are studied, as an example. Appropriate construction of the notion of typicalness in infinite-dimensional space is discussed, and the notion of "completely thin" sets is introduced. Kinetic systems with conservation of supports are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. | | Source: | arXiv, cond-mat/0405451 | | Services: | Forum | Review | PDF | Favorites | |
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