Stability in Randomly Fluctuating Versus Deterministic Environments

The relation between the dynamics of population models in which all the environmental parameters are strictly deterministic and the corresponding more realistic models with random environmental fluctuations are considered. The connection between the deterministic, mechanical usage of the term "stability" and that usage which associates stability or instability with the degree of random population fluctuation in a stochastic environment is discussed. The eigenvalues of the (deterministic or average) "interaction matrix" or "community matrix" play a key role in both circumstances. In the deterministic environment, only the signs of the real parts of these eigenvalues are needed to know whether the population interactions are stabilizing. In the stochastic environment, where the system may be in tension between the stabilizing population interactions and the destabilizing environmental fluctuations, the magnitudes of the real parts of these eigenvalues must be contrasted with the magnitudes of the characteristic environmental variances. These points are developed in a general way and are illustrated by specific analytic and numerical work on single-species and multispecies models. Some applications of the theory are indicated, both to circumstances where it may be used to justify deterministic calculations and to the opposite extreme circumstances where even the smallest environmental stochasticity leads to a result qualitatively different from the deterministic one.