Abstract
The objective of this paper is to present some concepts of stability which originate from ecological considerations, discuss their relationship with existing concepts of stability, and furthermore to establish sufficient conditions for stability which are both robust in the sense of Levins and Slobodkin, and constructive. The essential notion of stability for unperturbed ecosystems, applied in most cases, is that although fluctuations in particular variables exist, each remains within certain bounds (which may be time-varying). The use of this notion within a finite-time horizon is the framework inwhich total, essential, and terminal stability for both perturbed and unperturbed ecosystems are defined. These concepts are studied and sufficient conditions for stability are given. In particular, theorems 2 and 3 provide conditions to ensure that each variable stays within bounds and theorem 1 a way to identify the occurrence of large fluctuations outside the bounds, by examining the system at “boundary” points only. The strength of these results lies in the criteria for stability and instability: in addition to being robust and constructive, they can be applied directly to the differential (or difference) equations, without any knowledge of the solutions. Relationships between our concepts and the existing concepts of asymptotic and neighborhood stability, and θ-persistance are also stated.
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