Evolution of the age structured population has been simulated on a square lattice. Environment has been changed uniformly. Temporarily dispensable genes accumulate deleterious mutations and they could be responsible for elimination of genomes if the environment again calls for their functions, affecting locally the population density. Those fluctuations of population size enable sympatric speciation.

Can a finite population with variation between individuals persist? What is the probability of (rapid) extinction as opposed to long-time stability? What are the conditions for exponential growth? How will a growing population stabilise? Its age-distribution and its composition in general?

I shall describe simple and general branching processes and answer such classical questions about growth and extinction of populations, but also hint at modern developments beyond. Prerequisites: basic probability theory.

Many of the best known models of population dynamics are formulated on a "phenomenological" (meaning more or less ad hoc) basis. I will set forth seven reasons why such models are unsatisfactory and argue that models must be derived from the properties of individuals and the mechanisms of their interactions. Next, I will present a possible mechanistic underpinning to several famous discrete-time population models including the discrete logistic, Beverton-Holt, Ricker, and Hassell models.

This talk is based on joint work with Stefan Geritz.