A THEORETICAL TREATMENT OF UNSTRUCTURED FOOD WEBS 



G. D. Lange and a. C. Hurley' 



ABSTRACT 



In a recent paper, Isaacs has proposed a model for an unstructured food web in which the interconnec- 

 tions are so diverse that all heterotrophs in the system can be treated as if they were at the same 

 average trophic position. This paper recasts the original model in terms of a 3 x 3 matrix using three 

 empirical constants. In this form, the model can be easily generalized to one having nine constants and 

 reflecting a more realistic view of the interactions among levels of a community. 



Recent papers by Isaacs (1972, 1973) proposed an 

 alternative to trophic level schemes for represent- 

 ing interactions among species. He termed this an 

 unstructured food web and proposed a "matrix"^ 

 technique (Isaacs 1972) for evaluating the 

 equilibrium distribution of energy (or matter) 

 which would result from these interactions. In this 

 paper we propose an alternative formulation of 

 Isaacs' model which utilizes classical matrix and 

 operator techniques. 



SERIES APPROACH 



Isaacs' model was originally proposed to account 

 for Young's data (Young 1970) from the Gulf of 

 California which indicated that cesium was not 

 found concentrated in ratios one would expect 

 from a simple food chain. Isaacs assumes that the 

 principal interconnections in the marine food web 

 are so diverse that all heterotrophs in the system 

 (from microorganisms to vertebrates) can be 

 treated as if they derived their food from a com- 

 mon source that is only coarsely differentiated. 

 Therefore, the heterotrophs can all be treated as if 

 they were at the same average trophic position. In 

 this unstructured food web, Isaacs visualizes four 

 levels of matter or energy: 1) source, 2) living tis- 

 sue, 3) nonliving but retrievable matter, and 4) 

 irretrievable matter. The source is assumed to be 

 phytoplankton which is added to the system at a 



'Department of Neurosciences, School of Medicine, and Marine 

 Neurobiology Unit, Scripps Institution of Oceanography, 

 University of California San Diego, La Jolla, CA 92037. 



'We have enclosed Isaacs' use of the word "matrix" in quotes 

 because he has used the word in a common rather than in the 

 standard mathematical sense. When the word appears without 

 quotes in this text we are using it in the standard sense of a 

 rectangular array of elements which operates on column vectors 

 from the left to produce new column vectors. 



Manuscript accepted June 1974. 



FISHERY BULLETIN: VOL. 73, NO. 2, 1975. 



378 



constant rate. The living matter consists of all 

 heterotrophs, while the dead retrievable matter 

 may consist of such sources of carbon as organic 

 detritus or dissolved organic matter. The irre- 

 trievable component is that matter (or energy) 

 which is forever lost to the system through such 

 processes as respiratory combustion or 

 mineralization. The "unstructured" nature of the 

 food web comes from a set of coefficients which 

 represent movement of material between these 

 groups. The transitions are not in a trophic level 

 line. Rather, groups two and three interact 

 bilaterally and groups three and four can receive 

 from other levels bypassing intermediates. 



Isaacs calculates the final steady state values for 

 the total living and dead material by summing two 

 infinite series. To obtain these series, he in- 

 troduces a "matrix" which is designed to aid in the 

 formulation of each of the terms. The series take 

 the form: 



M\ = 





\-{K, + K,) 



K, 



M", = Mf,K^ + Mo K^ {K^ ■¥ K^) 



+ M,K,[K,{K, + K,) + K,{K, + K,)] 



M,K, 



1 - (^1 + ^3) 





where Mq = increment of initial input 

 periodically introduced into the 



