Definitions of Resilience

During the past year, several research efforts at IIASA have tried to develop a precise mathematical definition of Holling's very general and rich resilience concept. This paper develops a mathematical language for resilience, using the terms and concepts of differential topology. Central to this treatment is the division of the state space of a system- into basins, each containing an attractor. The translation of Holling's concept into this language reads roughly as follows: a system is resilient if, after perturbation, it will still tend to the same attractor as before (or to an "only slightly changed" attractor). The reason for treating changes of state variables and changes of parameters separately is explained. All resilience measures conceived up to now, as defined within this language, are listed as well. The various definitions of resilience are then compared to the well-known concepts of structural stability and of Thorn's catastrophe theory. Finally, the author indicates some -- in his opinion -- important directions for further research into the general resilience concept.