228 L. B. SLOBODKIN 



The rationale of this equation is that it has the appropriate Hmits, so that 

 F = o impUes that P^ = Pq, and jF = i impUes that P^ = o; has an appro- 

 priate curvihnear shape; and is extremely simple. 



The removal of a few small Daphnia is largely compensated for by an 

 increase in the reproductive rate of the survivors. The lability of reproductive 

 rate is so high that only when F is very high does Pp decline rapidly with F. 



Equation (i) is applicable only to the first quadrant. If negative fishing, 

 i.e. immigration, occurs, it is found that P^' increases linearly with —F and 

 not curvilinearly as would be implied by the equation. It seems likely that 

 alterations in reproductive rate are not significant when immigration is 

 occurriQg. Death rate seems to increase with —F. The immigration studies 

 will be presented in detail elsewhere (Armstrong & Slobodkin MS.). 



Clearly either the reproductive rate or death rate of at least some of the 

 animals in a population must change as a consequence of predation if a 

 steady state population size is to be maintained. Let 



00 







where l^ is the probability of survival up to age x of one individual born at 

 age o and m^ is the fecundity of a single live animal during the interval x, 

 X -{- I. If a population is at a steady state, Rq= i. Given a steady state popu- 

 lation, consider that a system of predation is introduced. The mortality that 

 can be attributed to predation will either compete with other sources of 

 mortality in such a way as to leave the l^ distribution unchanged or, more 

 probably, it will alter the l^ distribution. If the l^ distribution has been 

 altered by predation there must be compensatory alteration in either the l^ 

 or m^ distribution to restore the condition Ro= i. In a sense this is equivalent 

 to the assertion that only populations which are density dependent can 

 achieve a new steady state under predation (Nicholson, 1954). 



When adult animals are removed, the relation between P^ and F is more 

 complex. At low food levels the Daphnia completely consume the algae 

 between feedings and equation (i) seems approximately valid. As the 

 concentration of food increases it is found that equation (i) overestimates the 

 size of the standing crop population (Fig. 3). 



Microscopic examination of the medium shows that uneaten algae are 

 always present when population size is significantly below the value predicted 

 by equation (i). Experimental studies by Richman (1958) show that assimila- 

 tion of ingested algae by Daphnia pulex is less efficient when algal concentra- 

 tions are high. 



If yield in (animals plus eggs) per unit food is plotted against F for the 

 populations from which adults were preferentially removed, there is a wide 



