FOX: POPULATION SIMULATOR 



copulated females to total mature females, p(t), 

 giving 



n 



-Z;,t, 



dp{l)ldt = kc ^ (Pmi^if '-' [1-P(0],(19) 

 / = 1 



the solution obtained is 



Pf=[l-e 



-k. 



-Zi 



.S 0;n/^/,(l-e 'i)\Z.\ (20) 

 i = 1 •' 



with t = (0, 1) months and pq = 0. Equation 

 (20) may be further simplified by substitution 

 from equation (9) to give 



P/=[l-c 



-k^^ 



mj 



]. 



(21) 



Therefore, the fraction of the females in any 

 year class that are copulated is the same for all 

 year classes provided that k^ is not age spe- 

 cific. Given the fraction copulated in the previous 

 month, Pj _ I [where Pf _ ]^ = 0], the fraction 

 copulated at the end of the month J is 



pj^[l.il-pj_^)e-^'c^^j]. (22) 



The total number of females in the population 

 bearing fertile eggs at the end of the breeding 

 season is then 



n 



%r^^;,.r/^,-^'^;' 



(23) 



further assuming that one copulation results in 

 fertilization. 



Recounting the assumptions implicit in the 

 simple model, equation (23), they are: 



1. The instantaneous copulation rate per female 

 is linear and proportional to the number of 

 males. 



2. The copulation coefficient is independent of 

 the age of the males and females. 



3. The copulation coefficient is independent of 

 population size. 



4. A single copulation results in fertilization. 



5. Multiple copulations, if they occur, do not 

 alter the fraction of each egg clutch that is 

 fertilized from the first copulation. 



In developing a reproduction model for in- 

 sects, Conway (1969) criticized the first assump- 

 tion as being unrealistic. Empirical evidence 

 and biological induction suggest that the copu- 

 lation rate should be concave downwards, ris- 

 ing to a maximum at some intermediate density 

 and declining at very high densities owing to 

 interference. Conway (1969) dismissed the sec- 

 ond type of models (empirical) as lacking gen- 

 erality for a critical examination of reproduc- 

 tion, and developed three structural models 

 based on nearest neighbor distances. Given the 

 population size, density, and sex ratio, four 

 parameters are contained in his models as com- 

 pared with only one for the simple model. Con- 

 way fits each of his models to data obtained 

 from the literature on insects. The fits he ob- 

 tained are hardly remarkable considering that 

 four parameters were estimated, and a number 

 given, for only eight points. However, the fits 

 are an indication of the flexibility and possible 

 validity of Conway's structural concepts — es- 

 pecially since an irregularity in the data with 

 biological significance was predicted with the 

 fitted model. The decline in copulation rate in 

 the data, however, occurs at ver>' high popula- 

 tion densities. The major use of GXPOPS will 

 likely be for examining the dynamics of popula- 

 tions under exploitation, hence at less than vir- 

 gin population densities. Also, since the two 

 data sets given by Conway are adequately de- 

 scribed by a straight line for the observations 

 at population densities before the decline in 

 copulation rate occurred, and lacking other em- 

 pirical evidence to the contrar>\ equation (22) 

 was adopted as the copulation model in 

 GXPOPS. 



Many exploited animals carry their fertilized 

 eggs so that the number of eggs reaching the 

 hatching period is intimately tied to the sur- 

 vival of the females during the ovigerous peri- 

 od. If it is assumed that all female mortality 

 results in the loss of the eggs as well, then the 

 number of copulated females reaching the hatch- 

 ing period is given by 



^h = Pi's i r ^^fi^ith ^^^^ 



where f/j is the time at hatching. Instantaneous 



1023 



