Systems with inheritance: dynamics of distributions with conservation of support, natural selection and finite-dimensional asymptotics

If we find a representation of an infinite-dimensional dynamical system as a nonlinear kinetic system with conservation of supports of distributions, then (after some additional technical steps) we can state that the asymptotics is finite-dimensional. This conservation of support has a quasi-biological interpretation, inheritance (if a gene was not presented initially in a isolated population without mutations, then it cannot appear at later time). These quasi-biological models can describe various physical, chemical, and, of course, biological systems. The finite-dimensional asymptotic demonstrates effects of “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 →∞. The drift equations for peaks motion are obtained. Various types of stability are studied. In example, models of cell division self-synchronization are studied. The appropriate construction of notion of typicalness in infinite-dimensional spaces is discussed, and the “completely thin” sets are introduced.