FISHERY BULLETIN: VOL. 70. NO. 3 



Table 1. — Summary of Young's results. 



^ Cs/K for digae token as Cs/K for water. 



' Concentration factor involves values for specific invertebrates within the range shown. 



Ks = a, coefficient of conversion of matter (or 

 energy) in food into nonliving but re- 

 trievable form (e.g., organic detritus 

 or dissolved organic matter) 



For each component 



Ki + K2 + K3 = l (1) 



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



These coefficients are assumed to apply to the 

 nutrient processes of all heterotrophs in the 

 system, microorganisms, invertebrates, and ver- 

 tebrates alike. Further they are assumed to 

 have constant mean values for each ingested com- 

 ponent of a food web in which plant production 

 is the only primary source. 



Under these assumptions, some of the princi- 

 pal characteristics of a generalized food web can 

 be represented by a matrix such as Figure 1. 

 Each division along an abscissa represents the 

 conversion of a fraction, Kt, of any arbitrary 

 unit of food into living tissue. Therefore, the 

 numbers on the horizontal axis are powers of Ki. 

 Each division along an ordinate similarly rep- 

 resents the conversion of a fraction, Kz, of any 

 arbitrary unit of food into retrievable but non- 

 living organic matter, and the numbers on the 

 vertical axis are, thus, powers of Kz. These 

 numbers can be used to name points in the mat- 

 rix as in any set of cartesian coordinates. One 

 use of the matrix can be illustrated by consid- 

 ering a unit of food. Mm, (e.g., in a copepod or in 

 an edible nonliving particle) at 2, 3. If it is con- 

 sumed by a heterotroph, the fraction Ki is con- 

 verted into tissue and the fraction K^ into non- 

 living retrievable matter. The sum of these two 

 fractions Mm {Ki + Ks) is now found at 3, 4. 



Its trajectory in the matrix is always to the right 

 and down.^ All points on a line at right angles 

 to this trajectory (e.g., a diagonal line) can be 

 considered to represent organic matter that has 

 undergone the same number of steps of con- 

 version from its origin as tissue of autotrophic 

 organisms. The expression at the end of each 

 diagonal line in Figure 1 is the sum of all frac- 

 tions of an original unit of organic matter along 

 that diagonal. 



The number of possible paths from the origin 

 to each point in the matrix is easily counted and 

 is the sum of the two numbers of units shown 

 at the point, the lower right-hand number being 

 the number of paths resulting in living matter 

 (or energy) at that point, and the upper left- 

 hand number being the number of paths result- 

 ing in nonliving but recoverable matter (or en- 

 ergy) at the point. 



It will be seen that this doubly infinite matrix 

 is composed of two superimposed binomial 

 (Pascal) triangles, and can be summed along 

 diagonals resulting in two infinite series of bi- 

 nomials of the form: 



a + a{a + b) + a{a + b)~ + a{a + 6)^ . . 

 a(a+ b)". 



The sum of such a series is 



a 



1—ia + b) (2) 



and for the sum of living matter (or energy), 



^ Cases may be considered in which either Ki or 

 K^ ^ and motion is either only to the right or only 

 down. Ki = is, of course, not a permanently viable 

 condition. 



1054 



