Population Variance and Deterministic Stability Analysis

Abstract

1. Prompted by the simulation studies of Morrison & Barbosa (1987) and of Taylor (1992), an examination was made of the relationship between population variance and the magnitude of the dominant eigenvalue from stability analyses of linearized stable systems. Explicit expressions were derived for continuous and discrete-time systems and hence it follows that, for determination of local variance, simulations are not required. 2. Drawing on earlier studies it is shown that population variance can increase or decrease as the eigenvalue increases; the latter could be associated with the tracking of 'environmental noise'. Accurate modelling of the process noise is important in this respect. 3. The above suggest that the magnitude of the eigenvalue of the linearized deterministic system will not, in general, allow inference as to the character of population variance or the higher order distributions such as extinction time. As yet, no general rules have been developed that allow identification of whether or not the stability eigenvalue conveys more information than that relating to the existence of an attractive equilibrium and the oscillatory or otherwise approach to the equilibrium.