We analyze the equilibrium behavior of deterministic haploid mutation-selection models. To this end, both the forward and the time-reversed evolution processes are considered. The stationary state of the latter is called the ancestral distribution, which turns out as a key for the study of mutation-selection balance. We find that the ancestral genotype frequencies determine the sensitivity of the equilibrium mean fitness to changes in the corresponding fitness values and discuss implications for the evolution of mutational robustness. We further show that the difference between the ancestral and the population mean fitness, termed mutational loss, provides a measure for the sensitivity of the equilibrium mean fitness to changes in the mutation rate. The interrelation of the loss and the mutation load is discussed. For a class of models in which the number of mutations in an individual is taken as the trait value, and fitness is a function of the trait, we use the ancestor formulation to derive a simple maximum principle, from which the mean and variance of fitness and the trait may be derived; the results are exact for a number of limiting cases, and otherwise yield approximations which are accurate for a wide range of parameters. These results are applied to threshold phenomena caused by the interplay of selection and mutation (known as error thresholds). They lead to a clarification of concepts, as well as criteria for the existence of error thresholds.
We continue the study, initiated in Part I, of graphical representations and cluster algorithms for various models in (or related to) statistical mechanics. For certain models, e.g. the Blume-EmeryGri ths model and various generalizations, we develop Fortuin Kasteleyn-type representations which lead immediately to Swendsen Wang-type algorithms. For other models, e.g. the random cluster model, that are deÿned by a graphical representation, we develop cluster algorithms without reference to an underlying spin system. In all cases, phase transitions are related to percolation (or incipient percolation) in the graphical representation which, via the IC algorithm, allows for the rapid simulation of these systems at the transition point. Pertinent examples include the (continuum) Widom-Rowlinson model, the restricted 1-step solid-on-solid model and the XY model.
A graphical representation based on duplication is developed that is suitable for the study of Ising systems in external fields. Two independent replicas of the Ising system in the same field are treated as a single four-state (Ashkin-Teller) model. Bonds in the graphical representation connect the Ashkin-Teller spins. For ferromagnetic systems it is proved that ordering is characterized by percolation in this representation. The representation leads immediately to cluster algorithms; some applications along these lines are discussed.Comment: 13 pages amste
Abstract. We analyze a class of models for unequal crossover (UC) of sequences containing sections with repeated units that may differ in length. In these, the probability of an 'imperfect' alignment, in which the shorter sequence has d units without a partner in the longer one, scales like q d as compared to 'perfect' alignments where all these copies are paired. The class is parameterized by this penalty factor q. An effectively infinite population size and thus deterministic dynamics is assumed. For the extreme cases q = 0 and q = 1, and any initial distribution whose moments satisfy certain conditions, we prove the convergence to one of the known fixed points, uniquely determined by the mean copy number, in both discrete and continuous time. For the intermediate parameter values, the existence of fixed points is shown.
We consider the eigenvalue equation for the largest eigenvalue of certain kinds of non-compact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite rank, which constitutes a discretization procedure. For this purpose, two standard methods of approximation theory, the Nyström and the Galerkin method, are generalized. The operators considered describe models for mutation and selection of an infinitely large population of individuals that are labelled by real numbers, commonly called continuum-of-alleles models.
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