22 research outputs found
Adaptive Fitness Landscape for Replicator Systems: To Maximize or not to Maximize
Sewall Wright's adaptive landscape metaphor penetrates a significant part of
evolutionary thinking. Supplemented with Fisher's fundamental theorem of
natural selection and Kimura's maximum principle, it provides a unifying and
intuitive representation of the evolutionary process under the influence of
natural selection as the hill climbing on the surface of mean population
fitness. On the other hand, it is also well known that for many more or less
realistic mathematical models this picture is a sever misrepresentation of what
actually occurs. Therefore, we are faced with two questions. First, it is
important to identify the cases in which adaptive landscape metaphor actually
holds exactly in the models, that is, to identify the conditions under which
system's dynamics coincides with the process of searching for a (local) fitness
maximum. Second, even if the mean fitness is not maximized in the process of
evolution, it is still important to understand the structure of the mean
fitness manifold and see the implications of this structure on the system's
dynamics. Using as a basic model the classical replicator equation, in this
note we attempt to answer these two questions and illustrate our results with
simple well studied systems.Comment: 13 pages, 4 figure
Replicator equations and space
A reaction--diffusion replicator equation is studied. A novel method to apply
the principle of global regulation is used to write down the model with
explicit spatial structure. Properties of stationary solutions together with
their stability are analyzed analytically, and relationships between stability
of the rest points of the non-distributed replicator equation and distributed
system are shown. A numerical example is given to show that the spatial
variable in this particular model promotes the system's permanence.Comment: 24 page
On diffusive stability of Eigen's quasispecies model
Eigen's quasispecies system with explicit space and global regulation is
considered. Limit behavior and stability of the system in a functional space
under perturbations of a diffusion matrix with nonnegative spectrum are
investigated. It is proven that if the diffusion matrix has only positive
eigenvalues then the solutions of the distributed system converge to the
equilibrium solution of the corresponding local dynamical system. These results
imply that the error threshold does not change if the spatial interactions
under the principle of global regulation are taken into account.Comment: 16 pages, 1 figure, several typos are fixe