152 research outputs found
The Evolution of Dispersal in Random Environments and The Principle of Partial Control
McNamara and Dall (2011) identified novel relationships between the abundance
of a species in different environments, the temporal properties of
environmental change, and selection for or against dispersal. Here, the
mathematics underlying these relationships in their two-environment model are
investigated for arbitrary numbers of environments. The effect they described
is quantified as the fitness-abundance covariance. The phase in the life cycle
where the population is censused is crucial for the implications of the
fitness-abundance covariance. These relationships are shown to connect to the
population genetics literature on the Reduction Principle for the evolution of
genetic systems and migration. Conditions that produce selection for increased
unconditional dispersal are found to be new instances of departures from
reduction described by the "Principle of Partial Control" proposed for the
evolution of modifier genes. According to this principle, variation that only
partially controls the processes that transform the transmitted information of
organisms may be selected to increase these processes. Mathematical methods of
Karlin, Friedland, and Elsner, Johnson, and Neumann, are central in
generalizing the analysis. Analysis of the adaptive landscape of the model
shows that the evolution of conditional dispersal is very sensitive to the
spectrum of genetic variation the population is capable of producing, and
suggests that empirical study of particular species will require an evaluation
of its variational properties.Comment: Dedicated to the memory of Professor Michael Neumann, one of whose
many elegant theorems provides for a result presented here. 28 pages, 1
table, 1 figur
Fundamental Properties of the Evolution of Mutational Robustness
Evolution on neutral networks of genotypes has been found in models to
concentrate on genotypes with high mutational robustness, to a degree
determined by the topology of the network. Here analysis is generalized beyond
neutral networks to arbitrary selection and parent-offspring transmission. In
this larger realm, geometric features determine mutational robustness: the
alignment of fitness with the orthogonalized eigenvectors of the mutation
matrix weighted by their eigenvalues. "House of cards" mutation is found to
preclude the evolution of mutational robustness. Genetic load is shown to
increase with increasing mutation in arbitrary single and multiple locus
fitness landscapes. The rate of decrease in population fitness can never grow
as mutation rates get higher, showing that "error catastrophes" for genotype
frequencies never cause precipitous losses of population fitness. The
"inclusive inheritance" approach taken here naturally extends these results to
a new concept of dispersal robustness.Comment: 17 pages, 1 figur
Simon-Ando decomposability and fitness landscapes
In this paper, we investigate fitness landscapes (under point mutation and recombination) from the standpoint of whether the induced evolutionary dynamics have a “fast-slow” time scale associated with the differences in relaxation time between local quasi-equilibria and the global equilibrium. This dynamical hevavior has been formally described in the econometrics literature in terms of the spectral properties of the appropriate operator matrices by Simon and Ando (Econometrica 29 (1961) 111), and we use the relations they derive to ask which fitness functions and mutation/recombination operators satisfy these properties. It turns out that quite a wide range of landscapes satisfy the condition (at least trivially) under point mutation given a sufficiently low mutation rate, while the property appears to be difficult to satisfy under genetic recombination. In spite of the fact that Simon-Ando decomposability can be realized over fairly wide range of parameters, it imposes a number of restriction on which landscape partitionings are possible. For these reasons, the Simon-Ando formalism does not appear to be applicable to other forms of decomposition and aggregation of variables that are important in evolutionary systems
Proof of the Feldman-Karlin Conjecture on the Maximum Number of Equilibria in an Evolutionary System
Feldman and Karlin conjectured that the number of isolated fixed points for
deterministic models of viability selection and recombination among n possible
haplotypes has an upper bound of 2^n - 1. Here a proof is provided. The upper
bound of 3^{n-1} obtained by Lyubich et al. (2001) using Bezout's Theorem
(1779) is reduced here to 2^n through a change of representation that reduces
the third-order polynomials to second order. A further reduction to 2^n - 1 is
obtained using the homogeneous representation of the system, which yields
always one solution `at infinity'. While the original conjecture was made for
systems of viability selection and recombination, the results here generalize
to viability selection with any arbitrary system of bi-parental transmission,
which includes recombination and mutation as special cases. An example is
constructed of a mutation-selection system that has 2^n - 1 fixed points given
any n, which shows that 2^n - 1 is the sharpest possible upper bound that can
be found for the general space of selection and transmission coefficients.Comment: 9 pages, 1 figure; v.4: final minor revisions, corrections,
additions; v.3: expands theorem to cover all cases, obviating v.2 distinction
of reducible/irreducible; details added to: discussion of Lyubich (1992),
example that attains upper bound, and homotopy continuation method
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