An understanding of the underlying dynamics of population systems is an important part of many sciences. In systems that are marginally stable (or near-marginally stable) the effects of noise become especially important. In particular, the intrinsic demographic stochasticity associated with birth and death processes can qualitatively change the dynamics of the system.
In this thesis, two models where such fluctuations are important are presented. The first is the seminal predator-prey model first studied by Lotka and Volterra. This system is marginally stable in the radial direction but becomes stable in the presence of a variety of density-dependent effects. The mean extinction time is calculated both for the marginal and for the stable case. The second system is a model of genetic drift. In this system, noise associated with the genetic drift actually increases the persistence time of the system before it enters an absorbing state. The ability of fluctuations to stabilize a system is an unusual phenomena.
Both of these systems are two-dimensional which often makes analytic treatment difficult. However, both models possess a time-scale separation between a fast motion in one direction and a slow motion in the other. This separation allows both systems to be reduced to one dimension, vastly simplifying calculation.