The geometry of transient crashes and their dependence on demographic rates

In theory and practice, many ecological communities possessing a stable coexistence steady state may, in actuality, be difficult to assemble species-by-species, due to temporarily low densities of one or more preexisting species after the introduction of a new community member. This article elucidates some of the conditions under which these transient near-extinctions are likely to occur. These conditions are given in terms of such geometric quantities as the position of the new coexistence state after an invasion, relative to the original coexistence state, and the severity of the post-invasion community's oscillations, as well as more directly biological quantities such as the demographic rates of the invader and the potential victim of a crash. Through graphical reasoning, numerical examples, and preliminary experimental results, the importance of these conditions is demonstrated.