important for the development and maintenance 

 of species populations. Settlement rates are no- 

 toriously variable in nature, however, making it 

 impossible to determine a fixed r s and hence a 

 fixed A . For this reason, we have studied sensitiv- 

 ities over a range of A's (0.25-3.0). Values of a, 

 and bi used for My a arenaria are empirically 

 derived (Brousseau 1978a, b). 



Sensitivity Formulae 



The population growth rate, A, is the eigen- 

 value of M with maximum modulus. In general, 

 A is unique and is a positive real number. This 

 follows from the Perron-Frobenius Theorem, 

 which may be referenced, for example, in Deme- 

 trius (1969). The sensitivity of A to a life history 

 parameter is defined to be the derivative of A with 

 respect to that parameter. 



Following Caswell (1978), let u and v be col- 

 umn vectors satisfying 



Mv = kv 



u'M = ku' 



(u, v) = 1 



(2) 



(3) 



(4) 



where u' denotes the transpose of u, and (•,) 

 denotes inner product. Statement (2) indicates 

 that v is a right eigenvector, while Statement (3) 

 indicates that u is a left eigenvector, each asso- 

 ciated with A. Statement (4) is used as a normal- 

 ization device. While Statements (2) through (4) 

 do not define u and v uniquely, they are sufficient 

 to make the sensitivity formulae below well de- 

 fined. Explicit calculations for the components of 

 vectors u and v start with 



«i 



ut= 1 a j bj- 1 ...b i k- ( J- i+1 \i>l (5) 



and 



y, =1 



vi = k I 6j- 1 ...6 1 r s , i > 1 



(6) 



and then normalize using Statement (4) above. 

 With these definitions, Caswell (1978) shows 



dkldmij = uivj, ij = l...n, (7) 



where mij is the parameter in the ij position of 

 538 



the Leslie matrix M , ut is the itb. component of 

 vector ui , and vj is the jth component of vector v. 

 Of course, the components of M of interest to us 

 are those in the first row (the fecundity param- 

 eters) and those in the main subdiagonal (the 

 survivorship parameters). Further, since position 

 m 2 i equals r s b x in our notation, the sensitivity 

 formulae become 



dkldai = UiUi , i = l,2,...,n 



(8) 



dkldbt = Ut + ivt , i — 2,3,..., n - 1 (9) 

 dk/dbi = r s dk/dm 2 i = r s u 2 Vi (10) 



dkldr s = bidk/dm 2 i = biU 2 V\ 



(11) 



In particular, notice that A is not equally sensitive 

 to r s and b x unless the two values are equal. 



For the present study, we hold a t and b t fixed 

 and allow A to vary. The settlement rate, r s , then 

 becomes a function of A, specifically, 



r s = (A - ax )/(A~ 1 a 2 bi + k~ 2 a 3 b 2 bi 



+ ... +k n+1 dnbn-i ... bi), (12) 



and is used in the Leslie matrix, M. We then 

 compute u and v satisfying Statements (2)-(4) for 

 the given A , and the sensitivity values Statements 

 (8)-(ll). 



Relationships among the sensitivity formulae 

 above have been derived by Demetrius (1969) and 

 Caswell (1978). Of particular interest are 



dkldai > dk/daj , i <j, A > 1 



(13) 



dkldai < dk/daj , i<j,k<l 



b,dk Idbi > bj d A Idbj , i < j (14) 



dk/dbi A - ai 



d A Ida i b i 



(15) 



Statement (13) can actually be made stronger, as 

 proven by Demetrius (1969, Statement (8) ); State- 

 ment (14), in the case i = 1, and Statement (15) 

 follow from Demetrius (1969, Statement (11) ) and 

 Caswell (1978, Statement (22)); and Statements 

 (5), (10), and (12) above. 



Calculation of Sensitivity Values 



Settlement Rate.— Using the data in Table 1, 

 the sensitivity of the population growth rate of 



