12 ULANOWICZ 



Finally, Harte and Levy (1975), borrowing from an analysis 

 popular in physics around the turn of the century, constructed for 

 three hypothetical ecosystems a community function whose exis- 

 tence ensured that the system would be stable to finite perturbations 

 within a given domain. Briefly, if the differential equation 



X = f(X, P) (10) 



possesses a steady state or limit cycle, Xg, and is perturbed to 

 Xg + AX, it may or may not return to the neighborhood of Xg. If we 

 can construct a function L(AX) (called a Liapunov function) which 

 vanishes at the origin, is positive and monotonically increasing with 

 AX in some domain about the origin, and has a negative time 

 derivative, then the system is stable with respect to perturbations 

 within that domain. The function L is not necessarily unique but is a 

 conservative estimator of the stability properties of the system (and 

 likewise a conservative estimator of the domain of stability). For 

 certain classes of functions, f, there are st£indard methods for 

 determining whether or not Liapunov functions exist. For these 

 particular systems the question of stability is unequivocally resolved. 

 In general, however, failure to find a Liapunov function does not 

 imply instability of the system. The function may exist but may defy 

 analytical description. 



Despite these analytical difficulties, the Liapunov method has 

 two things to recommend it. First, it bridges the gap between the 

 microscopic (species level) and the macroscopic (L being a commu- 

 nity function). Second, it offers the hope of ordinating the various 

 domains of stability. Harte and Levy (1975) speculated that if 

 succession is in the direction of ever more resiliency to stress, then 



A= -min (i ^ In l)/AX (11) 



provides a measure of the maturity of the system. 



SUMMARY AIMD SPECULATIONS 



There is a historical duality in the scientific meaning of the term 

 stress, and this is reflected in the various models of ecosystem 

 response to stress. Since strain, the conjugate to stress, is not well 

 defined in ecology, discussion of the topic is difficult. If we assume a 

 measure of system response to stress, two distinct groups of stress 

 analysis arise, local and far-field. 



I have classified most existing models of total ecosystems as local 

 because either they are linear or they track the instantaneous 



