LANut. and HUKL,tL,i: : uwsi Kuiji UKiiu I'uuu wttis 



system at intervals equal to the 

 time taken by one average step in 

 the food web, 



M'f = total quantity of material in living 

 tissue (level two), 



M"f = total in nonliving recoverable 

 material (level three), 



K^ = a coefficient of conversion of matter 

 (or energy) in food into living tis- 

 sue, 



K2 = acoefficientof conversion of matter 

 (or energy) in food into irretriev- 

 able form (e.g., by respiratory com- 

 bustion or mineralization), and 



K^ = a coefficient of conversion of matter 

 (or energy) in food into nonliving 

 but retrievable form (e.g., organic 

 detritus or dissolved organic mat- 

 ter). 



Restrictions on coefficients are: 



K^ + K2 + Ks= 1, 

 0<Ki< 1, where i = 1, 2, or 3. 



MATRIX APPROACH 



In our representation of the unstructured food 

 web, source, living tissue, and nonliving but re- 

 trievable matter are taken to be components in a 

 vector in a three-dimensional space. This vector 

 can be written 



w = 



where w^ is the amount of matter (or energy) 

 present in phytoplankton, W2 is the amount 

 present in heterotrophs, and Wg is the amount 

 present in retrievable dead material. The fourth 

 level (loss) is the difference between the total in- 

 put and the material present in the three other 

 levels. 



The matrix operator controlling movement of 

 material from one level to another, using Isaacs' 

 coefficients, takes the form: 



1 

 A^.[ K, K, K, 



Each K represents the proportion of material 

 transferred between the levels appropriate to its 



position in a time equivalent to one application of 

 the matrix. 



As Isaacs points out, three constants may not be 

 sufficient. It is probably not reasonable to assume, 

 for example, that all matter is converted to living 

 tissue with the same coefficient of conversion or 

 that both living and dead matter have the same 

 conversion factor to irretrievable form. One ad- 

 vantage of our method is that it can be generalized 

 to a more complex form. This cannot easily be done 

 with Isaacs' original method because crossterms in 

 the ICs rule out viewing the steady state values as 

 simple geometric series. The generalized form of 

 the matrix for an unstructured food web with 

 these additional coefficients is: 



A = 



where k^ = conversion factor from source to 

 living, 



^2 = conversion factor from source to 

 dead retrievable, 



k^ = conversion factor from source to 

 inrretrievable, 



k^ = conversion factor from living to liv- 

 ing, 



k^ = conversion factor from living to dead 

 retrievable, 



kf, = conversion factor from living to 

 irretrievable, 



k'j = conversion factor from dead to liv- 

 ing, 



A;8 = conversion factor from dead to dead 

 retrievable, and 



/cg = conversion factor from dead to irre- 

 trievable. 

 In this case, 



fC 1 ~t" /Co ' /Co ^- X 



fCn t Ko I Kq ^ X 



< ki < 1, where i = 1 to 9. 



When this matrix acts upon the state vector w 

 the result is somewhat more complex: 



\ o\/.A / ;"'! , 



Aw = \kkk ]l w \ = [kiw, + k^W2 + k^w., 



Steady State Results 



The eigenvectors and eigenvalves of a matrix 



379 



