6 ULANOWICZ 



developed mathematical tools that can be brought to bear on 

 them — especially linear stability analysis. 



In terms of the effects of low-level stress on a system, there is 

 one key question, "Will the response to the stress remain small, or 

 will it grow to the point of disrupting the integrity of the 

 community?" For linearized systems the procedure for answering 

 this question is well defined and has been reviewed by May (1971). 

 All the eigenvalues of the matrix A must have negative real parts. If 

 the linear ecosystem model has constant coefficients, this test will 

 show whether the model is properly behaved. More frequently, 

 however, the coefficients of A vary because of exogenous driving 

 forces. In this case the eigenvalues vary also, and, under changing 

 conditions, it is possible that some eigenvalue will acquire a positive 

 real part (Hcilfon, 1976). Thus we can map out domains of driving 

 forces for which the systems response is possibly unstable. 



Much of the literature involving linear stability theory in 

 ecosystems has been given over to debating the question of whether 

 diversity will better enable an ecosystem to cope with an applied 

 stress. MacArthur (1955) suggested such a causal link, and May 

 (1973) reviewed the use of linear stability analysis to question this 

 hypothesis. Central to the counter argument is the observation that 

 increasing the dimensionality and connectivity of a randomly 

 assembled system decreases the probability that all eigenvalues will 

 be negative. Gardner and Ashby (1970) sampled randomly con- 

 structed matrixes to illustrate this point. Others have argued that 

 ecosystems are not randomly constructed and that constraints on the 

 form of A can lead to different conclusions (Roberts, 1974; 

 McMurtrie,~1975; Saunders and Bazin, 1975; Jeffries, 1974). 



The diversity— stability controversy is actually a macroscopic 

 issue, and further discussion is best deferred to that section of this 

 paper. What is important here is that linear stability results are 

 neither necessary nor sufficient to determine the persistence of an 

 ecosystem under stress. 



Despite their simplicity, linear models remain a popular medium 

 for modeling total ecosystems (Patten, 1975). In fact, there are 

 instances where linear models seem preferable for simulating total 

 system behavior (Patten, 1976; Ulanowicz et al., 1978). Patten's 

 success with linear descriptions led him to propose linearity as an 

 evolutionary design criterion (Patten, 1975) — a much criticized 

 stance (e.g., Wiegert, 1975). Leaving philosophical considerations 

 aside, we see that the preceding discussion of local models of stress 

 may help illuminate why linear models are such popular tools. As 

 stated earlier, thus far most attempts to model ecosystem response to 



