On the comparative static properties of the expected population density in the presence of stochastic fluctuations

We consider the impact of increased stochastic fluctuations on the expected density of an unstructured population evolving according to a regular diffusion process subject to a concave expected growth rate. By relying on the flow nature of the solutions of stochastic differential equations and Girsanov's theorem, we demonstrate that typically increased volatility decreases the expected future population density. As a consequence, we are able to characterize the sensitivity of the expected population density with respect to changes in the diffusion coefficient measuring the size of the stochastic fluctuations. We provide both qualitative and quantitative information about the consequences of a mis-specified volatility structure and, especially, of a deterministic approximation to stochastic population growth. We also consider the effect of uncertainty in the initial density and demonstrate that the sign of the relationship between the expected population density and initial uncertainty is unambiguosly negative.