The impact of environmental variation on demographic convergence of Leslie matrix population models: An assessment using Lyapunov characteristic exponents

In a constant environment, the rate of convergence of a density- independent Leslie matrix model to stable age distribution is determined by the damping ratio (the ratio of the absolute magnitudes of the first and second eigenvalues of the projection matrix). In a stochastic environment, the difference between the first two Lyapunov exponents is known to be analogous to the logarithm of the damping ratio, but there has been no systematic investigation of the consequences of environmental variation on convergence rates. In this study, the Lyapunov spectrum has been calculated for a wide variety of density-independent projection matrices subject to random variations in vital rates. This allows the impact of these random variations on convergence rates to be assessed. For rapidly convergent life histories, stochastic variation leads to a decrease in convergence rate. For life histories which are slow to converge, stochastic variation speeds up convergence. These effects are, however, relatively minor, and the value of the damping ratio for the mean matrix is a good predictor of the damping ratio in a stochastic environment. Consequently, when only an approximate indication of convergence rates is needed, the damping ratio for the mean projection matrix gives a very good guide. Detailed calculations of the Lyapunov spectrum would only be necessary to make comparisons between similar life histories or if very precise information on convergence rates were needed.