FISHERY BULLETIN: VOL. 78. NO. 3 



Events A and B are related to densities of adults 

 and juveniles, respectively, during a 2-wk interval. 

 By examination of only those sampling inter- 

 vals for which the (initial) juvenile ( J4.5) density 

 equals 0, one may separate the relation between 

 adult density and fry survival from the compli- 

 cating juvenile-fry interaction. For such intervals, 

 neglecting natural mortality, survival within the 

 refuge area should be approximately 1. That is, 

 when J45 density = 0: Pi A and B) = PiA) = 

 /"(adult density only). Biweekly fry survival rates 

 were estimated as described above and plotted 

 against numerical densities of fish in size cate- 

 gories J5.0 through A 8 in 5.0 mm populations 

 from Phase I and from long-term control popula- 

 tions during Phase II revealing a decreasing trend 

 in survival rates with increasing predator density 

 (Figure 12). Beyond 100 adults, estimated survival 

 was close to zero. The trend appeared roughly 

 exponential (one explanation is based on random 

 encounters between predators and prey, see Ricker 

 1954), so a negative exponential model was used 

 for further analysis: 



PiA) = exp[Si • (adult density)]; Bi<0. 



Since adults were always present when juve- 

 niles were present in refuge areas, it was not 

 possible to separate the effects of the juvenile-fry 

 interaction from adult predation. By analogy, 

 I also used the negative exponential model to 

 describe the relation between juvenile numbers 

 and refuge area fry survival rates, i.e.: 



P{B\A) = exp[B2 • (J4 5 density)]; 5 2 ^0. 



E 

 S 

 T 

 I 



M 

 A 

 T 

 E 

 D 



F 

 R 

 Y 



S 



U 

 R 

 V 

 I 



V 

 A 

 L 



R 

 A 



T 

 E 



20 



40 

 ADULT 



60 80 



PREDATORS 



100 



120 



The full model appropriate for all sampling in- 

 tervals is then: 



P(survive to ^ + 2 given born in ^, / + 2) 



= S = API/EB = exp(BiX, + ^2^2) 



where X^ = number of adults in size categories 

 t/s and above at time t 

 X2 = number of J45 juveniles at t. 



Statistical Analysis 



Two techniques were used to fit the collected 

 guppy population data to the proposed model 

 developed above. Both techniques were based 

 on the same assumed model although the model 

 was expressed in different forms according to 

 analysis technique: 



St,t+2 ^ ■^h,t + ilEBt,t+-2 



= exp(fiiXi^ + B2^2;K and (2) 



APIt^t+2 = EBt,t+2 • exp(SiXi^ + B^X^f). (3) 



Multiple regi'essions, forced through the origin, 

 were fit to a transformation of Equation (2) 

 (adding 1 unit to API to avoid undefined natural 

 logarithms)'*: 



ln[(AP/^, ^+2 + l)IEBt,t+2^ = B^Xif + B^X^r 



Alternative estimates of Sj and B2, the "coeffi- 

 cients of predation" for adults and juveniles, were 

 obtained by nonlinear least-squares regressions 

 based on a Taylor series linearization of Equation 

 (3) (Draper and Smith 1966). In this case one 

 minimizes: 



liAPIt,t+2- AFIt,t+2)^ - l[APIt,t+2 



- EBt,t+2  exp(5iXi^ + ^2^2^)]^ 



to obtain the iterative solutions for B^ and B2- 

 Iteration was continued for these estimates until 

 the last estimate agreed with the previous esti- 

 mate to six decimal places. All population data 

 series were subjected to analysis by the same 

 model. Note that the dependent variable for the 



Figure 12. — Relation between estimated guppy fry survival 

 rates (&t.t + 2' ^^^ number of adult guppy predators (number 

 3= J5 g at t) when no J4 5 juveniles were present at t. Line is 

 drawn by eye. Squares represent multiple observations. 



^ While an interaction term of the form X^X^ might seem a 

 logical addition to the above model, analyses failed to indicate 

 that such an interaction was significantly involved in deter- 

 mining numerical dynamics of the populations. 



570 



