38 research outputs found
On mathematical theory of selection: Continuous time population dynamics
Mathematical theory of selection is developed within the frameworks of
general models of inhomogeneous populations with continuous time. Methods that
allow us to study the distribution dynamics under natural selection and to
construct explicit solutions of the models are developed. All statistical
characteristics of interest, such as the mean values of the fitness or any
trait can be computed effectively, and the results depend in a crucial way on
the initial distribution. The developed theory provides an effective method for
solving selection systems; it reduces the initial complex model to a special
system of ordinary differential equations (the escort system). Applications of
the method to the Price equations are given; the solutions of some particular
inhomogeneous Malthusian, Ricker and logistic-like models used but not solved
in the literature are derived in explicit form.Comment: 29 pages; published in J. of Mathematical Biolog
On the asymptotic behavior of the solutions to the replicator equation
Selection systems and the corresponding replicator equations model the
evolution of replicators with a high level of abstraction. In this paper we
apply novel methods of analysis of selection systems to the replicator
equations. To be suitable for the suggested algorithm the interaction matrix of
the replicator equation should be transformed; in particular the standard
singular value decomposition allows us to rewrite the replicator equation in a
convenient form. The original -dimensional problem is reduced to the
analysis of asymptotic behavior of the solutions to the so-called escort
system, which in some important cases can be of significantly smaller dimension
than the original system. The Newton diagram methods are applied to study the
asymptotic behavior of the solutions to the escort system, when interaction
matrix has rank 1 or 2. A general replicator equation with the interaction
matrix of rank 1 is fully analyzed; the conditions are provided when the
asymptotic state is a polymorphic equilibrium. As an example of the system with
the interaction matrix of rank 2 we consider the problem from [Adams, M.R. and
Sornborger, A.T., J Math Biol, 54:357-384, 2007], for which we show, for
arbitrary dimension of the system and under some suitable conditions, that
generically one globally stable equilibrium exits on the 1-skeleton of the
simplex.Comment: 23 pages, 1 figure, several small changes are added, together with
the new titl