Branching Processes

The study of branching processes began in the 1840s with Irénée-Jules Bienaymé, a probabilist and statistician, and was advanced in the 1870s with the work of Reverend Henry William Watson, a clergyman and mathematician, and Francis Galton, a biometrician. In 1873, Galton sent a problem to the Educational Times regarding the survival of family names. When he did not receive a satisfactory answer, he consulted Watson, who rephrased the problem in terms of generating functions. The simplest and most frequently applied branching process is named after Galton and Watson, a type of discrete-time Markov chain. Branching processes fit under the general heading of stochastic processes. The methods employed in branching processes allow questions about extinction and survival in ecology and evolutionary biology to be addressed. For example, suppose we are interested in family names, as in Galton’s original problem, or in the spread of a potentially lethal mutant gene that arises in a population, or in the success of an invasive species. Given information about the number of offspring produced by an individual per generation, branching process theory can address questions about the survival of a family name or a mutant gene or an invasive species. Other questions that can be addressed with branching processes relate to the rate of population growth in a stable versus a highly variable environment. Some background in probability theory is required to apply branching process theory. We present some of this background in the next section. Then we discuss single-type and multi-type Galton-Watson branching processes, and extensions to random environments that will address questions about population survival and growth.