llOll&Iil^l LtU M^LIIH 1. M.^ 



fully characterize its properties. For the matrix 

 representing the Isaacs assumptions the following 

 eigenvalues (X's) and eigenvectors (it's) can be ob- 

 tained: 



A,= l 



^2 ~ ^1 ■'■ ^3 



^3=0 



Ho = 



Any initial state of the system (e.g., w) can be 

 written as a weighted sum of the eigenvectors 



W = CiU^ + C2U2 + C^Uq. 



If we now apply A n times to this vector we obtain 



A'W = CjAf Wj + C2X^U2 + CgAg^Wg. 



After a sufficient time {n very large), the second 

 and third term will vanish, leaving an expression 

 for the final state of the system: 



lim^"w = c-^u^. 



«->00 



In Isaacs' terms Cj = Mq and the limiting values 

 for the second and third compartments are M\ and 

 M"t respectively. Therefore 



M\ =M,K,/K2 

 M'; = MoK,/K2 



which is exactly Isaacs' result. 



For the nine constant model, there is also always 

 a steady state distribution of matter in the sys- 

 tem. By finding the eigenvector corresponding to 

 an eigenvalue of one, we can obtain the following 

 steady state values of M', (total quantity of 

 material in living matter) and M',' (total in 

 nonliving recoverable material) in terms of a con- 

 stant input Mq-. 



{k,-l){k,-l)-k,k, 

 ' {k,-\){k^-\)-k^k^ 



Trophic Level Equations 



In addition to values for total amounts of living 

 and retrievable dead matter, Isaacs develops 

 equations for general trophic levels. His equations 

 can be generated by our approach if our original 

 matrix is broken down into component parts and 

 then applied to the steady state vector. For 

 example, let us consider Isaacs' case (Isaacs 1973) 

 of a subset of trophic levels which are complete and 

 mutually exclusive. He considers strict herbivores, 

 detrital feeders, and full predators to be such a 

 subset. 



Our original matrix A can be written in the 

 following way 



A = A 



s + i? + ^// + ^Z) + ^i 



where As + r = 





 



^^3^3^3> 



^0 

 ,0 



'0 

 A^ = I 0/^1 

 ^0 



/o 0' 



Ap = I K^ 

 \0 0, 



-^s+ij - matrix responsible for the biomass in 

 source and the retrievable dead matter, 

 = matrix responsible for biomass in her- 

 bivores, 

 = matrix responsible for biomass in detrital 

 feeders, and 

 Ap = matrix responsible for biomass in preda- 

 tors. 

 To obtain the potential biomass for each of the 

 trophic levels, we take the appropriate matrix 

 times the steady state vector. Thus, the equation 

 for the potential biomass of herbivores is obtained 

 from 



^H 



^D 



AfjUi 





 A',0 

 



Mn 



= \M,K, 



380 



