Mutation -selection balance for polynomial selection costs and matrix-valued orthogonal polynomials

In the first part, we study a (possibly infinite-dimensional) dynamical system model for genetic mutation and natural selection in the presence of recombination. Some features of the model, such as existence and uniqueness of solutions and convergence to the dynamical system of an approximating sequence of discrete time models, were presented in earlier work by Evans, Steinsaltz, and Wachter for quite general selection costs. Here we establish that the phenomenon of mutation-selection balance occurs in the special case of "polynomial" selection costs under mild conditions.;That is, we show that the dynamical system has a unique equilibrium and that it converges to this equilibrium from all initial conditions. In the second part, we consider the generalization of the theory of orthogonal polynomials and birth-and-death processes to the case of matrix-valued polynomials and so-called "quasi-birth-and-death processes". We derive a formula relating the Stieltjes transforms of the spectral measures of a block-tridiagonal matrix and the matrix of its "0th associated process" and give some examples to illustrate the use of this formula. Additionally, we apply orthogonal polynomial techniques to the study of random walks on sin-graphs and higher dimensional birth-and-death processes, for which the relevant polynomials are multivariate.