9 Environmental Noise Variability in Population Dynamics Matrix Models Michel

The impact of environmental variability on population size growth rate in dynamic models is a recurrent issue in the theoretical ecology literature. In the scalar case, R. Lande pointed out that results are ambiguous depending on whether the noise is added at arithmetic or logarithmic scale, while the matrix case has been investigated by S. Tuljapurkar. Our contribution consists first in introducing another notion of variability than the widely used variance or coefficient of variation, namely the so-called convex orders. Second, in population dynamics matrix models, we focus on how matrix components depend functionaly on uncertain environmental factors. In the log-convex case, we show that, in a sense, environmental variability increases both mean population size and mean log-population size and makes them more variable. Our main result is that specific analytical dependence coupled with appropriate notion of variability lead to wide generic results, valid for all times and not only asymptotically, and requiring no assumptions of stationarity, of normality, of independency, etc. Though the approach is different, our conclusions are consistent with previous results in the literature. However, they make it clear that the analytical dependence on environmental factors cannot be overlooked when trying to tackle the influence of variability.