8 ULANOWICZ 



throughs since May's comment to characterize the domain of 

 attraction for a given stationary point. The task is as intriguing as it is 

 important and formidable, however, and should continue to demand 

 the attention of ecologists for several years to come. Nor will the 

 problem remain of interest only to theoreticians. Lawton, Bedding- 

 ton, and Bower (1974) and Sutherland (1974), for example, have 

 shown from empirical data that switching behavior occurs even 

 among invertebrates. 



Preliminary is perhaps the best word to describe the investiga- 

 tions on finite perturbations to date. An example of the nonreversi- 

 bility of a nonlinear system is provided by McQueen (1975). His 

 model for competition between two species of cellular slime mold 

 exhibits two ranges of persistence in the sense of Rolling. When he 

 made the birthrate of one of the species highly dependent on climate 

 and then shocked the model with a short burst of favorable climate, 

 the model underwent transition from one domain to another where 

 the favorably perturbed population was higher. Most interesting, 

 however, was the model behavior that 



. . . suggests that a population might fluctuate for long periods of time at a 

 low level as it tracks the lower-stable [stationary] point, but given a short 

 burst of favorable climatic conditions it could escape and rapidly grow to 

 an upper-stable [stationary] level. From that point on, the population will 

 remain at a high level tracking the high-stable [stationary point] as it 

 moves in response to changing climate. Return to a low level is only 

 possible when negative forces increase or when climate is very unfavorable 

 to birthrate. 



We can easily envision the reverse situation occurring in a collapsing 

 ecosystem. 



Although we cannot yet say with confidence how an ecosystem 

 will be structured after it has undergone transition, there are two 

 notable attempts to answer this question. 



In the first. Smith (1975) occupied himself with species 

 extinctions and the realm of possible stable subsystems that a known 

 system may possess. He defined stability in a very fundamental way 

 (see also Ulanowicz, 1972); a system is considered stable with regard 

 to some defined stress if none of the component species become 

 extinct as a result of that stress. In general, when one or more species 

 of an ecosystem is removed, either a subsystem that is itself stable (in 

 the sense just mentioned) results or one or more of the remaining 

 species will drive the subsystem to collapse. Smith began by 

 enumerating all possible subsystems by dropping various combina- 

 tions of state variables in turn. To test whether any of these is a 

 stable subsystem, we must view the behavior. of the system when 



