Disturbed nonlinear multispecies models in ecology

We analyze a disturbed form of the general Lotka-Volterra model of an ecosystem with m interacting species. The disturbances act on the intrinsic growth rates of the species and are assumed to be bounded but otherwise unknown. We employ a Lyapunov technique and the concept of "reachable set" from control theory to estimate the set of all possible population densities that are attainable as a result of the disturbances. To calculate estimates for this reachable set, a number of numerical methods that entail the solution to one or more global optimization problems are developed. Specific examples involving two, three, and four species are solved. We also derive an explicit analytical expression that represents an estimate for the reachable set in the m-dimensional case. The estimate is conservative but can be evaluated without carrying out any optimization procedure. We show that methods developed in this paper can be applied to certain other types of nonlinear ecosystem models.