Again On Holling's Puzzle

Evtushenko, Y. & Sokolov, V. (1975). Again On Holling's Puzzle. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-75-047

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Abstract

In a recent IIASA Research Report, C. Holling introduces a new concept of Resilience as an important characteristic of the behavior of complex ecological systems. He writes

(1) In mathematical analyses, stability has tended to assume definitions that relate to conditions very near equilibrium points.

(2) Resilience determines the persistence of relationships within a system and is a measure of the ability of these systems to absorb changes of state variables, driving variables, and parameters, and still persist. In this definition resilience is the property of the system and persistence of probability of extinction is the result.

(3) Stability on the other hand, is the ability of a system to return to an equilibrium state after a temporary disturbance.

(4) The more rapidly it returns, and with the least fluctuation, the more stable it is. In this definition stability is the stability is the property of the system and the degree of fluctuation around specific states the result.

With these definitions in mind a system can be very resilient and still fluctuate greatly, i.e. have low stability.

These forms of definitions are rather vague and underestimate the achievements of modern stability theory. The subsequent examples do little to clarify the definitions. Meanwhile, defining stability as behavior not only near equilibrium but also in the large and allowing for existing oscillations even in stable systems, the concept of stability may be extended to a broader class of problems and in particular to Holling's concept of resilience. These broad definitions are in current use in stability theory.

The vague nature of Holling's approach resulted in the appearance of several mathematical definitions of resilience when this topic was discussed among the IIASA methodology staff in February, 1975.

This note is another attempt to solve a loosely specified problem and it is certainly open for any criticism and comments. As the concepts of stability and resilience appear very often together in Holling's presentation we shall try to relate them directly through rigorous concepts of stability theory.

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