MODELING ENVIRONMENTAL STRESS 9 



stressed arbitrarily near the extinction of one or more species (stress 

 in the second sense of the word). That is, if 



Xi = fi(Xi,X2,. .., Xm,Xm+,,. ..,Xn) i=l,2, . ..,n (7) 



then we observe 



Xi = fi(Xi , X2 , . . ., Xm , 0, . . ., 0) i = 1, 2, . . ., n (8) 



where variables Xm+i through Xn were chosen (without loss of 

 generality) as those arbitrarily driven to near extinction. Smith then 

 listed four criteria the reduced system must satisfy to be stable. For 

 example, a stable subsystem cannot be obtained if any one of the fi 

 describing an extinct species has become positive as a consequence 

 of extinction, i.e., if any 



Xi = fi(Xi , X2 , . . ., Xm, 0, . . ., 0) > i = m + 1, . . ., n (9) 



He concluded his remarks by performing an analysis on a hypotheti- 

 cal four-species subsystem and identified three possible stable 

 subsystems to which the original system might collapse. 



Concern with environmental degradation in recent years has 

 caused ecologists to become somewhat jaded and to focus on 

 exogenous stress and its consequent simplification of the impacted 

 ecosystem. In far- from -equilibrium nonlinear systems, however, 

 endogenous strains occur which allow a chance perturbation (a 

 mutant or migration) to flourish suddenly and to become an added 

 dimension of the community. In the second example, Prigogine 

 (1976) and Eigen (1971) examined this phenomenon as the crucial 

 element in the prebiotic evolution of polymers, and Allen (1976) 

 extended the analysis to the evolution of new populations in 

 ecosystems. The methods used are similar to those used by Smith for 

 collapsing systems. 



Despite these interesting insights, the basic nature of structural 

 transitions remains an enigma. This is caused in large measure by an 

 inclination to think in terms of linear systems. It is not foreign to 

 think of a system, an input (exogenous stress), and an output which 

 results from that input (stability or instability), but it is discomfiting 

 to be confronted with an output whose origin lies predominately 

 within the system itself. 



MACROSCOPIC TREATMEIMT OF ECOSYSTEM RESPONSE 



In his presentation of evidence for multiple stable points and 

 domains of attraction, Holling (1973) described the system from the 



