Demographic stochasticity and Allee effect on a scale with isotropic noise

Diffusion models of stochastic population dynamics, including both demographic and environmental stochasticity, can be transformed to a scale with isotropic noise. On this scale, demographic stochasticity contributes a net downward component to population trajectories that is inversely proportional to population size, creating a type of Allee effect. In populations with appreciable demographic stochasticity and a small long-run growth rate, such as many threatened and endangered species, there may be a stochastically unstable equilibrium below which most population trajectories tend to decline toward extinction. Population sizes corresponding to stochastic equilibria on the transformed scale can be obtained directly, without transformation. An explicit transformation and its inverse are derived for illustration in a model with stochastic density-independent growth (including demographic and environmental stochasticity) and deterministic density-dependence.