Risk analysis for populations subject to demographic stochasticity only

In this section we consider small populations (i.e. for which $N <N_c $

) so that as a first approximation one can neglect environmental but not demographic stochasticity. Just because populations are very small, one can often assume that the rate of demographic growth does not depend on density and dynamics is therefore Malthusian. To facilitate understanding of the main concepts that guide the analysis, it is convenient to assume population dynamics in continuous rather than discrete time. However, the results would not change qualitatively if one considered the mathematically more complex case of discrete reproduction. Recall that the instantaneous birthrate $\nu$ and death-rate $\mu$ can be reinterpreted in terms of individual fitness as

$\displaystyle \nu dt$	$\displaystyle =$	probability that a female produces a daughter in the small time interval $\displaystyle dt$
$\displaystyle \mu dt$	$\displaystyle =$	probability for a female to die in the small time interval $\displaystyle dt.$

The analytical approach to demographic stochasticity is not easy. We simply summarize the most important results in the Malthusian case ( $\nu$ = constant, $\mu$ = constant, both independent of $N$

) and then briefly treat the case with density dependence. The reader can refer to Iannelli and Pugliese (2014) for a more thorough analysis.

Subsections