Stochastic Methods in Biology

The use of probabilistic methods in the biological sciences has been so well established by now that mathematical biology is regarded by many as a distinct dis cipline with its own repertoire of techniques. The purpose of the Workshop on sto chastic methods in biology held at Nagoya University during the week of July 8-12, 1985, was to enable biologists and probabilists from Japan and the U. S. to discuss the latest developments in their respective fields and to exchange ideas on the ap plicability of the more recent developments in stochastic process theory to problems in biology. Eighteen papers were presented at the Workshop and have been grouped under the following headings: I. Population genetics (five papers) II. Measure valued diffusion processes related to population genetics (three papers) III. Neurophysiology (two papers) IV. Fluctuation in living cells (two papers) V. Mathematical methods related to other problems in biology, epidemiology, population dynamics, etc. (six papers) An important feature of the Workshop and one of the reasons for organizing it has been the fact that the theory of stochastic differential equations (SDE's) has found a rich source of new problems in the fields of population genetics and neuro biology. This is especially so for the relatively new and growing area of infinite dimensional, i. e. , measure-valued or distribution-valued SDE's. The papers in II and III and some of the papers in the remaining categories represent these areas.

I. Population genetics.- [1] A stochastic model of compensatory neutral evolution.- [2] Some models for treating evolution of multigene families and other repetitive DNA sequences.- [3] A genealogical description of the infinitely-many neutral alleles model.- [4] Equilibrium measures of the stepping stone model with selection in population genetics.- [5] Asymptotic properties for Kimura's diffusion model with altruistic allele.- II. Measure-valued diffusion processes related to population genetics.- [6] The infinitely-many-alleles model with selection as a measure-valued diffusion.- [7] Multi-allelic Gillespie-Sato diffusion models and their extension to infinite allelic ones.- [8] Stationary distribution of a diffusion process taking values in probability distributions on the partitions.- III. Neurophysiology.- [9] Weak convergence of stochastic neuronal models.- [10] Note on the Ornstein-Uhlenbeck process model for stochastic activity of a single neuron.- IV. Fluctuation in living cells.- [11] Fluctuation in living cells: effect of field fluctuation and asymmetry of fluctuation.- [12] Some aspects of Oosawa's equation.- V. Mathematical methods related to other problems in biology, epidemiology, population dynamics, etc..- [13] Problems of epidemic modelling.- [14] Markov semigroups associated with one-dimensional Lévy operators --regularity and convergence--.- [15] On some conditions for diffusion processes to stay on the boundary of a domain.- [16] The point interaction approximation for diffusion in regions with many small holes.- [17] Unimodality and bounds of modes for distributions of generalized sojourn times.- [18] Fluctuation in population dynamics.