HANKIN: A MULTISTAGE RECRUITMENT PROCESS 



juvenile-fry interactions and the proposed concep- 

 tual model. However, analysis to this point has 

 implicitly assumed a constant input of newly born 

 fry into the refuge areas, modified downward by 

 the presence of interacting juveniles. Since fry 

 were produced by reproductive females whose 

 numbers and weights increased greatly as pop- 

 ulations grew, a complete analysis must clearly 

 include a description of the female reproduc- 

 tive component. Additionally, the influence of 

 adult predators outside the refuge area must be 

 considered. In this section I develop a mathe- 

 matical model of the dynamics of numerical pop- 

 ulation change and subject this model to statis- 

 tical analyses. 



Development of a Mathematical Model 



Population reproductive potential increases 

 with the weight and number of adult guppy 

 females. The total reproductive potential, i.e., the 

 maximum possible number of fry born in a given 

 interval, in a population at the beginning of an 

 interval can be computed from the fecundity 

 relation and the total female weight as: reproduc- 

 tive potential = 22.347 • S t' total weight fe- 



males per size category. Total reproductive poten- 

 tial is not realized in any given interval since only 

 some (variable) fraction of females will actually 

 deliver broods. In order to obtain estimates of 

 biweekly fry production, the probability that a 

 female will deliver a brood during a 14-d experi- 

 mental interval is needed. 



The calculation of this probability requires: 

 1) the expected length of an interbrood interval 

 (time from the last brood when the next brood is 

 delivered), 2) an estimate of the gestation period 

 or minimum time between broods, and 3) a fre- 

 quency distribution for the interbrood interval. In 

 guppies the interbrood interval is roughly 31 d 

 (Breder and Coates 1932; Winge 1937; Rosenthal 

 1952), gestation period has been estimated at from 

 21 to 25 d (Winge 1937; Rosenthal 1952), and a 

 rough frequency distribution may be constructed 

 from the preceding studies. Using "renewal pro- 

 cess" theory (Drake 1967), the probability that the 

 waiting time Y until the next brood of an individ- 

 ual female is delivered will be ^14 d (length of a 

 sampling interval ), when there is no knowledge of 

 her exact stage in the brood cycle (as was the case 

 for these populations), is denoted by P(y^l4 d). 

 Letting T = interbrood interval, and s be a fixed 



but random point in the brood cycle, then T = 

 s + y; that is, the total length of the interbrood 

 interval (T) is equal to the time since the last 

 brood is) plus the waiting time (y) until the next 

 brood is delivered. Using a cumulative density 

 function for the interbrood interval (T), which 

 may be constructed from previous studies, one 

 has (using standard notation): 



Then fyiy) 



PiT^t) = Frit). 



^ [1 - P(T^y)] 

 EiT) 



J4 



, [1 - P[T^y)] 

 and P(y^l4) = J ^^^^ -" dy 



14 



= IIE(T)- / [1 - FT(y)]dy (1) 



where EiT) denotes expected value. 



The probability of an interbrood interval of ^14 d 

 is (gestation period estimates are at least 21 d) 

 so Equation (1) may be reduced to: P(y«14) = 

 lAIEiT) = 14/31 = 0.452. 



This probability may be applied to the esti- 

 mated reproductive potential to obtain an esti- 

 mate of the expected number of births in a 2-wk 

 interval as: expected number of births t, t+2 = 

 0.452 • reproductive potential f Comparison of 

 adjusted population increments (API) with ex- 

 pected number of births (EB) allows estimation 

 of survival rates (S) for fry born in a given 

 interval as: 



St, t+2 - APIt,t+2lEBt^t+2- 



Survival of newly born fry through a 2-wk 

 interval depends on both predation by adults 

 outside the refuge area and juvenile-fry inter- 

 actions within the refuge area. Survival within 

 the refuge area is conditioned upon the event 

 "successful refuge entry," so one has: 



P(surviveto^+ 2) = P(A andS) = P(A) • P(B| A) 



where P(A) 



P(B\A) 



P( "successful refuge entry") 

 P( survive within refuge area 



given A has a successful 



outcome). 



569 



