Traditionally in population modelling, the mixing of individuals has been assumed to be homogeneous; that is, every individual can come into contact with every other individual. Within the last 40 years, however, a number of population models have been proposed that do not assume homogeneous mixing but rather assume populations are divided into disjoint habitable patches that are separated by uninhabitable space. Populations with this structure are known as metapopulations.
When metapopulation modelling was first proposed, the habitable patches would be classified as either colonised or extinct and the dynamics of coloni- sation and extinction would be the only dynamics accounted for in the model. For example, the logistic model adapted to a metapopulation would be dx =λx(1−x)−μx, dt where x(t) is the proportion of occupied patches at time t, λ is the rate an unoccupied patch becomes colonised when all patches are unoccupied and μ is the rate an occupied patch becomes extinct. Adding more detail to metapopulation models, the model examined in this thesis records the number of individuals in each location, thereby accounting for an individ- ual’s dynamics, such as the rates of births, deaths and migrations, within the model.
The model is a continuous time Markov process that will be used to account for the demographic stochasticity within populations such as births, deaths and migration events. Results by Kurtz, and extended by Pollett, which can be applied to a family of Markov processes, termed asymptotically density dependent, will be used to determine an approximating system of differential equations. These differential equations are then analysed to determine conditions for persistence and extinction. Furthermore, an Allee effect, where the initial conditions of the population determine whether it persists or goes extinct, is confirmed to exist in a two patch system that has a large difference between the migration rate for the two patches.
The model is extended in two ways. The first extension accounts for a deterministically changing environment. This is done by allowing the pa- rameters of the system to depend on time. A new functional limit law is derived which can be applied to time inhomogeneous, asymptotically density dependent Markov processes. This functional limit law is used to derive a nonautonomous system of differential equations. This system is then analysed to provide conditions for persistence and extinction of the metapopulation.
The second extension to the model accounts for a stochastically changing environment. Again, the parameters of the system are allowed to vary in time. However, the parameters vary stochastically according to an under- lying Markov process, designed to model a stochastically changing environ- ment. A functional limit law provides a way to approximate the process with a random dynamical system. The random dynamical system is then analysed to determine a sufficient condition for extinction. While not a com- plete description of the long term behaviour, such an approach facilitates more research into models with a stochastically changing environment.