Modeling 445 



start growing and feeding on algae, bacteria, and detritus after about 4 to 

 5 weeks, but towards the end of the season (after 6 weeks) about 90% of the 

 population dies. Thus, after about 4 to 5 weeks of the season we have two 

 generations of these organisms and these two types have different feeding 

 rates. For this reason we have followed them as two separate variables. 



In modeling the dynamics of the consumer species we have assumed 

 that the feeding rate for a single food source is different from the feeding 

 rate when multiple food sources exist in the same environment. Thus, for 

 each food type, the consumer possesses a potential rate of ingestion. The 

 actual rate of ingestion of a particular food, in the multiple food situation, 

 is a function of this potential rate and of a Michaelis-Menten form of the 

 sum of all the potential rates of all the food types each divided by its own 

 ^'max. For example, consider the rate of ingestion of algae by Daphnia. 

 Since bacteria and detritus are also available in the water, the actual rate 

 of ingestion of algae (see Table 10-6) is 



RB203 = RP201(RP201/C20i)/ [(RP201/c2oi) 



+(RP206/c207)+(RP211/c2io)] (16) 



where 



RP203 =rateof ingestion of algae, 



RP201 =potential rate of ingestion of algae in the absence of 



another food type, 

 RP206 = potential rate of ingestion of bacteria in the absence of 



another food type, 

 RP21 1 = potential rate of ingestion of detritus in the absence of 



another food type, 

 C201 = V„,ax for the ingestion of algae, 

 C207 = V„ax for the ingestion of bacteria, and 

 C210 = V ^ax for the ingestion of detritus. 



The potential rate of ingestion is a function of the biomass of algae 

 and of Daphnia and the temperature. Experimental evidence from these 

 ponds (Chisholm et al. 1975) suggests that there is an endogenous feeding 

 rhythm in these Daphnia with a cycle length of 12 hours that varies by a 

 factor of 2. Therefore, to include this rhythmic effect on the feeding rate 

 we have added a sine function, which generates these oscillations, to the 

 V„ax for each food type. Thus the V„ax for each food type is not a constant 

 but varies as a periodic function with a cycle length of 12 hours varying by 

 a factor of 2. As an example, the potential rate of ingestion of algae is 



RP201=^2oi>'i/(>'i+C202)]c2o [1.5+0.5 sin ( yj +1.047)]r, (17) 



