Population Numbers Count: Tools for Near-Term Demographic Analysis.

How do unit or proportional changes in vital rates affect populations in the short term? We present a new extension to standard methods of matrix model analysis that allows us to answer this question for the first time. By using the sensitivities of all the eigenvalues/vectors, rather than just the leading eigenvalue/vector pair, we can predict the consequences of unit or proportional changes in vital rates to population size and structure at any arbitrary time, not just when populations have neared their stable distribution. These extensions are particularly important in studying populations subject to frequent disturbance, where stable growth rate and stable distribution do not provide sufficient information about the effects of changes in the vital rates; managed populations in which short-term goals are defined; and the adequacy of the underlying matrix model for either short- or long-term understanding. We use analysis of empirical data on the cactus Coryphantha robbinsorum to demonstrate this approach and show that short-term predictions can differ substantially from those based on standard, asymptotic, analysis.