  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.

Probability;Dynamical Systems