Applications in Population Dynamics

In this chapter, we deal in detail with the issue of population growth and intend to point out ideas and thoughts behind the construction of models in population biology. It will help to understand in a better way the theoretical considerations and the outcome of the discrete Master equation approach compared with classical considerations in population dynamics based on continuous population dynamics. The extinction process and life time of single populations are investigated. In a further example, a Master equation City-hinterland model demonstrates how migratory phase transitions may be responsible for the tremendous growth of settlements during the last decades. Predator-prey interaction is one of the fundamental processes of biology. A very simple but often used and discussed model for the description of predator-prey interaction is the so-called famous Volterra-Lotka model. We will consider a special case of the Volterra-Lotka model by allowing the prey species to migrate to a habitat where the predator species cannot follow. In this case the unstable cycles of the Volterra-Lotka model will be replaced by a stable limit cycle, but also a much more complicated pattern may be obtained. The dynamics of a system of interacting populations close this chapter. In case of three interacting populations, a sequence of phase transitions may occur and drive the system from a stable state, via a limit cycle towards chaotic behaviour.