The Cost of Restoration as a Way of Defining Resilience: A Viability Approach Applied to a Model of Lake Eutrophication

Abstract

"Multiple stable states or alternative equilibria in ecological systems have been recognized since the 1960s in the ecological literature. Very often, the shift between alternative states occurs suddenly and the resource flows from these systems are modified. Resilience is the capacity of a system to undergo disturbance and maintain its functions and controls. It has multiple levels of meaning, from the metaphorical to the specific. However, most studies that explore resilience-related ideas have used resilience as a metaphor or theoretical construct. In a few cases, it has been defined operationally in the context of a model of a particular system. In this paper, resilience is defined consistently with the theoretical uses of the term, in the context of ecosystem models within an application to a simple model of lake eutrophication. The theoretical definitions of resilience and the characteristics of the operational definition that are necessary for ensuring consistency are reviewed. A mathematical formulation of resilience is built in the framework of the viability theory. This formulation emphasizes the link between resilience and the cost of the recovery after a disturbance. This cost is first chosen in relation to the time of crisis in the application to a model of lake eutrophication. The resilience values are then obtained by numerical integration. For another choice of the cost function, the viability algorithm is needed to compute the resilience values. These applications demonstrate the usefulness of our operational definition."