Boundary Conditions in Evolutionary Equations in Biology

Abstract

The talks are devoted to the most important examples of boundary conditions in evolutionary equations that model biological phenomena. The first notable one is the boundary condition in the so-called McKendrick equation, modeling births in an age-structured population. Currently, the McKendrick equation and its generalizations are often used as a building block of more complicated models, for example those involving quiescence or several linked populations. Analytically, the related boundary condition is still of importance, being at the same time of interesting form and having a clear biological meaning. Other boundary conditions of interest describe behavior of diffusion processes at the boundaries. As developed by W. Feller in the 1950, stochastic processes in population genetics, including the famous Wright’s diffusion being an approximation of the Wright–Fisher model of genetic drift, suggest boundary conditions that were not known before. The seminal works of W. Feller, A.D. Wentzell and P. Levy have led mathematicians and biologists to the general form of such boundary conditions, and to a thorough understanding of their probabilistic and analytical meaning.