1948) and subsequently used in numerous 

 studies of human and animal populations. In the 

 present setting, the Leslie matrix takes the form: 



L(r„) 











6»-i 0. 



Here, a, is the number of female eggs produced 

 annually by a female Mya arenaria in class i(age 

 i — 1 to i), and 6, is the probability of a clam in 

 class i — 1 surviving to class i for fe2. The sur- 

 vivorship from age-class 1 to age-class 2 (P„ in 

 the notation of Vaughan and Saila (1976)) has 

 been divided into two factors, r s and 6i. The fac- 

 tor r, is the settlement rate, or the probability 

 that an egg will develop into a clam with a 2 mm 

 shell length (0-2 mo of age); 6i is the probability 

 that a clam with a 2 mm shell length will survive 

 the remainder of the year (about 10 mo). 



The intrinsic growth rate is an increasing 

 function of the settlement rate, r.,. Therefore, the 

 equilibrium value, r., , 0<r s <1, can be cal- 



eq 



culated for the population assuming steady state 

 conditions. Since the values of a, and 6, have been 

 determined the Mya are n aria (Brousseau 1978a, 

 b), r.s eq can be determined for this species. Sim- 

 ilar calculations are described by Vaughan and 

 Saila (1976) and Van Winkle et al. (1978). 



Calculation of Equilibrium Settlement Rate 

 and Stable Age Structure 



Under conditions of the Perron-Frobenius 

 Theorem, the population will reach an equilib- 

 rium state (starting with any initial reproducing 

 population) if A = 1 is the dominant eigenvalue 

 of L(r s ). Thus, r Vq is the unique solution to 

 the equation 



\L(r s )-l\ =0. 



By induction on the size of the matrix (w), 



\L(r s ) —II = ±(1 — ai — r s bi<i2 - r,bi6 2 a 3 - ... 

 — r s 6i& 2 ... &„.ia„). 



The required settlement rate is given by: 



1 - o, 



Using the data in Table 1, the required settle- 

 ment rate for Mya arenaria is r» eq = 0.001462% 

 or 1 egg out of about 68,400 survive to 2 mm. Thus 



1 egg out of 384,000 must survive the first year to 

 maintain a steady population. Errors of up to 5% 

 in the fecundity and survivorship values will 

 yield equilibrium settlement rates of between 

 0.000983% (1 egg in 101,700 surviving to 2 mm 

 size) and 0.00218% (1 egg in 45,800 surviving to 



2 mm size). In addition, the eigenvector corre- 

 sponding to the eigenvalue A = 1 gives the stable 

 age structure for the population. Postsettlement 

 population structures were determined for only 

 the r Seq calculated above. The results are given in 

 Table 2. 



Table 1.— Life history statistics used in the derivation of 

 the Leslie Matrix for Mya arenaria. (Data from Brousseau 

 1978b.) 



'Fecundity = number of female eggs produced per individual, 

 assuming a 1:1 sex ratio. 



Table 2.— Calculated stable age structures for the pop- 

 ulation of Mya arenaria, based on the entire population 

 and the adult population (<30 mm) only. 



r« 



>■■< 



biaz + 616203 + ... + 6162 ... 6/,-iO„ 



'This age class represents clams 2-29 9 mm in shell length 



Discussion 



In his classic work on marine invertebrate 

 communities, Thorson (1950) stated the defini- 

 tive "number of eggs and larvae produced per 

 pair of adult animals per lifetime to maintain the 

 population is. ..one pair of larvae." More simply, 

 to remain at equilibrium, a replacement rate of 

 one must be maintained. For a population of Mya 



643 



