No abstract

We consider an asexual biological population of constant size N evolving in discrete time under the influence of selection and mutation. Beneficial mutations appear at rate U and their selective effects s are drawn from a distribution g(s). After introducing the required models and concepts of mathematical population genetics, we review different approaches to computing the speed of logarithmic fitness increase as a function of N, U and g(s). We present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid. We then discuss approximate approaches to the finite population problem, distinguishing between the case of a single selection coefficient, g(s) = δ (s − s b ), and a continuous distribution of selection coefficients. Analytic estimates for the speed are compared to numerical simulations up to population sizes of order 10 300 .

We show that in the asymmetric simple exclusion process (ASEP) on a ring, conditioned on carrying a large flux, the particle experience an effective longrange potential which in the limit of very large flux takes the simple form U = −2 i =j log | sin π(n i /L − n j /L)|, where n 1 , n 2 , . . . n N are the particle positions, similar to the effective potential between the eigenvalues of the circular unitary ensemble in random matrices. Effective hopping rates and various quasistationary probabilities under such a conditioning are found analytically using the Bethe ansatz and determinantal free fermion techniques. Our asymptotic results extend to the limit of large current and large activity for a family of reaction-diffusion processes with onsite exclusion between particles. We point out an intriguing generic relation between classical stationary probability distributions for conditioned dynamics and quantum ground state wave functions, in particular, in the case of exclusion processes, for free fermions.