ISAACS: FOOD WEBS AND "POLLUTANT ANALOGUES" 



LIVING STEPS K| n- 



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recoverable 

 materlal- 



RULE FOR 



CONSTRUCTION 



OF MATRIX 



* Taken as the number of paths leading to the point. 



■t-^ 



0+ b 



living 

 material 



c + d 



Figure 1. — Matrix representing 

 principal characteristics of a gen- 

 eralized food web. 



a = K\,'b — Ks, and for the sum of recoverable 

 matter (or energy), a = K3, b = K\. 



These series can be further modified and 

 summed in several ways. We will define a term, 

 Mo, where Mo is an increment of initial input 

 periodically introduced into the system at inter- 

 vals equal to the time taken by one average step 

 in the food web. Under the assumption that all 

 steps are equally probable, the diagonals along 

 which the matrix has been summed mark equal 

 intervals of time. Thus, allowing the quantity 

 Mo to be successively introduced at the origin of 

 the matrix, the total quantity of material in the 

 entire web, above the phytoplankton level, (Mt) 

 for steady state becomes: 



Mt = Mt' + Mt" (3) 



where Mt = total in living material, and 



Mt" = total in nonliving recoverable 



Mt' 



Mt'' 

 Mt 



Mo 

 Mo 



material; 



l—(Ki + Ks) 



MoK, 



M0K3 

 K2 



'^2 



(4) 

 (5) 

 (6) 



FURTHER VARIATIONS OF THE 

 FOOD WEB MATRIX ANALYSIS 



Later I will delete earlier steps in the matrix 

 series. In this case, a more general solution of 

 equation (4) involves a summation of the matrix 

 above any step. Here the partial sum, 



1055 



