MODELING ENVIRONMENTAL STRESS 5 



architecture of the models is in their pairameter specifications. Thus, 

 with changing driving forces, the "normal" state, H*, may vary 

 considerably according to how^ the instantaneous stationary state, 



f(X*,P, t) = (5) 



varies as P changes with exogenous stress. The behavior function, H, 

 will tend to track H* closely. The hypothetical deviation may never 

 be large, and the system is not stressed in the far-field sense of the 

 word. 



Nevertheless, the dramatic responses of some systems commonly 

 referred to as stressed can be shown by a judicious choice of the 

 functional dependence of the parameters on exogenous stress. 

 Bierman et al. (1973), for example, chose Chlorella and Microcystis 

 as two compartments of a nutrient-uptake model. Using separate 

 laboratory information to describe the nutrient-uptake kinetics and 

 the response to temperature, they investigated zones where one of 

 the species dominated starkly. Lassiter and Kearns (1973) simulated 

 an annual progression of six species as they dominated the 

 phytoplankton of a hypothetical limnetic system. Falco and Mulkey 

 (1976), using the law of mass action, anticipated the significant 

 differential effect that pesticides can have on populations of bass and 

 bluegills. 



In these models all the information about the behavior of the 

 system is contained in the functional form f and the parameter 

 dependencies. Even when the population structure is predicted to be 

 quite unbalanced (e.g., practically all blue-green algae), the system 

 may be very near its stationary point, and, hence, by our definition it 

 is only slightly stressed. Furthermore, the response is usually almost 

 reversible (perhaps retrievable is a better word), meaning that, when 

 the external influence is removed, the system returns to near its 

 original condition. 



The introductory examples of local low-stress models were 

 purposely chosen for their nonlinear construction, but we more 

 naturally associate low stress with linearity and, consequently, with 

 linear models. Mathematically speaking, linear models have the form 



X = AX (6) 



where the matrix of coefficients is allowed to vary parametrically, 

 i.e., A = A(E, t). Any well-behaved nonlinear f can be approximated 

 by a~linear system in the neighborhood of a given point in phase 

 space. The tremendous advantage of linear systems is the well- 



