SECT. 4] THEORY OF FOOD -CHAIN RELATIONS IN THE OCEAN 445 



The equation for phytoplankton, p, at any given depth z is 



dp ,. 8 A 8p 8p 



- = p{ p h --r p - g h) + - z7 -- v J-, ( ii) 



where A is the coefficient of vertical eddy diffusivity (calculated as eddy 

 conductivity from seasonal temperature changes and assumed to be synony- 

 mous), v is the sinking rate of phytoplankton, and p is the density of the water. 

 The concentration of phytoplankton, p, and other variables at particular depths 

 are denoted by lower case symbols to distinguish them from the former usage 

 in which capital letters indicated the total population per unit area of sea 

 surface. The respiratory coefficient r p has an identifying subscript, as will 

 the respiratory coefficients for other groups. 



Solution by Southwell's method requires a steady-state, finite difference 

 form of equation (11), which is 



. ,. 1 IA\ pi—po A-i po — p-i\ v(pi-p-i) 



Po(p h ~r p -gh) + - i £ -• l KF \ F l) = 0, (12) 



z \ p z p z ) 2z x/ 



where po is the phytoplankton concentration at any given depth and p~± and 

 p\ are the concentrations at equidistant intervals z above and below po. A-i 

 and A i are average eddy coefficients for the depth intervals above and below 

 the level of po respectively. 



There is a series of equations for a controlling nutrient, in this case phosphate, 

 taking the form 



1 /A\ ni — no A-\ no — n-i\ 



" : ; )+v[hr h + cr e -po(p h -r p )] = 0. (13) 



z \ P z P z i 



Vertical exchanges are expressed in the same way as the diffusion term in 

 equation (12). The biological rate of change depends upon phytoplankton 

 productivity and excretion by herbivores, h, and carnivores, c. Excretion is 

 assumed to be proportional to the product of the biomass of animals and their 

 respiratory coefficients r^ and r c . Since the biological populations are expressed 

 in terms of carbon, their inclusion in the nutrient equation requires a pro- 

 portionality factor v. 



Vertical migration of animals is postulated to permit an equal amount of 

 time at each depth. This is equivalent to assuming that there exists over a 

 period of time a statistically uniform distribution, so that a single equation 

 giving the mean concentration is sufficient. In the steady state the quantity of 

 herbivores can be derived from equation (12). Thus we have 



-r 



p{ph-r p )dz-vp z ' / gpdz, (14) 



where z' is the deepest level in the model, and p z ' is the concentration of phyto- 

 plankton at that depth. The sense of the equation is that herbivores eat all of 

 the phytoplankton except that which is lost by sinking. In a few cases it might 

 also be desirable to include a term for loss by eddy flux, but in general, vertical 



