MODELING LIFE-STAGE-SPECIFIC INSTANTANEOUS 



MORTALITY RATES, AN APPLICATION TO 



NORTHERN ANCHOVY, ENGRAULIS MORDAX, EGGS AND LARVAE 



Nancy C. H. Lo 1 



ABSTRACT 



Life-stage-specific instantaneous mortality rates (IMRs) are often estimated individually for each life 

 stage of an organism using regression analysis. A single estimation procedure for all life stages may 

 be preferable because it would increase the overall precision of the IMRs and also provide a more realistic 

 mortality model. Two such procedures were developed in this paper. One is single-equation model where 

 regression estimates of all IMRs are obtained by fitting a single survivorship function to the entire data 

 set. The other is the maximum likelihood estimator. These models were compared using northern an- 

 chovy egg and larval data. The survivorship functions of each were, respectively, exponential and Pareto 

 functions. 



The mortality of marine fish can be described by its 



survival probability S(t) = P(T > t) - exp [- f 



•Jo 

 X(u)du], where T is the age of the fish and X(t) is the 



instantaneous mortality rate (IMR) at age t. Dur- 

 ing their early life history, pelagic marine fishes pass 

 through a series of life stages: eggs, yolk-sac lar- 

 val, feeding pelagic larval, juvenile and adult stages. 

 The IMR X(t) could be different for some life stages. 

 Therefore, for / life stages, there may be G distinc- 

 tive IMRs where G<I. The IMR X(t) is then a piece- 

 wise function (Gross and Clark 1975, p. 20-21; 

 Johnson and Kotz 1976, p. 272-273) 



' kjit) < t < u x 

 X 2 (t) u x < t < u 2 



and the survival probability S(t) = (P T>t) will be 



m - 



Kit) 



u g-l < 



t < u n 



MO U G-1 < t < U G 



where u g is the maximum age of mortality stanza 

 g. X g (t) # X g (t) for g ± g. For example, X x (t) may be 

 the IMR for egg and yolk-sac larval stages, even 

 though each is a different life stage, and X 2 (t) the 

 IMR for feeding larvae. As a result, the conditional 

 survival probability, S g (t) = P (T > t\T > u g _ x ) cor- 

 responding to X g (t), will also be different from S g (t) 



'Southwest Fisheries Center La Jolla Laboratory, National 

 Marine Fisheries Service, NOAA, P.O. Box 271, La Jolla, CA 

 92038. 



S l (u l )S 2 (t) 



u x < t < u 2 



S(t) = 



G-l 



n S d (u d )S G (t) Uq.^Kug 

 <f=l 



The common method for estimating X g (t)'s for 

 marine fishes has been to fit Sg(t) to sample age 

 data separately for each life stage or to assume one 

 common A(£) for all life stages and to fit one S(t) to 

 sample data of all life stages (Hewitt and Brewer 

 1983). For northern anchovy, Engraulis mordax, 

 eggs and larvae <20 d old, the IMR k(t) for eggs and 

 yolk-sac larvae is different from that of the feeding 

 larvae (Lo 1985): 



m = 



X x (t) = a < t < Mi 

 hit) = -f ^i < t < 20 



6 



(1) 



Manuscript accepted August 1985. 

 FISHERY BULLETIN: VOL. 84, No. 2, 1986. 



where u x is either the hatching time (t h ~ 3 d) or 

 the age of yolk-sac larvae (t ys ~ 4.5 d) with the first 

 feeding as the critical period after which mortality 

 decreased. Either t h or t ys has been used in various 

 models under different assumptions. If mor- 

 phological differences cause the changes in mortality 

 rates, t h is a reasonable separation point between 



395 



