Wang and Ellis: Maximum likelihood estimate of mortality and growth from multiple length-frequency data 



381 



(1998). See DeLong et al. (2001) for alternative ap- 

 proaches to length-frequency data where individual 

 variability is taken into account. 



In our study, we develop a new framework for analyz- 

 ing length-frequency data. In particular, we incorporate 

 1) individual variability in growth parameters; and 2) 

 an arbitrary recruitment function. The model is flexible 

 enough to incorporate various sizes at recruitment and 

 a fishing selectivity function. However, we did not use 

 these aspects in the analysis of tiger prawn data. Some 

 analytical expressions are derived for these generaliza- 

 tions. A maximum likelihood approach is developed for 

 estimation of mortality and growth parameters. Sepa- 

 ration of fishing mortality from natural mortality is 

 possible only when there is substantial contrast in the 

 effort pattern. We also require a known recruitment 

 pattern, and sampling times are spread out so that 

 the length-frequency data will contain information on 

 growth and mortality. Simulation studies are carried 

 out to determine the performance of the method. The 

 simulated data are generated from the recruitment pat- 

 tern of the brown tiger prawn iPenaeus esculentus) in 

 the northern prawn fishery of Australia. Finally we ap- 

 ply the maximum likelihood method to length-frequency 

 data from grooved tiger prawn data (P. semisulcatus) in 

 the northern prawn fishery of Australia. 



where /',(/IL , =.v, L =s) is the conditional probability 

 density function of L at time t when L x is known to be 

 x and the size at recruitment is s. Note the lower limit 

 of the inner integral is / because L t cannot be less than 

 an individual's length. 



Let the age (again, relative to t ) at recruitment of an 

 individual be A . From Equation 1, we have age a at 

 length / is a = -k~ 1 \og(l-l/L_ j ) and hence the conditional 

 distribution, f t (l/L x =x, L n = s), which may be written 

 as f t (l\x, s) for brevity, can be expressed by using the 

 conditional distribution of age \\tia\L y =x, A =a n ) (see 

 Wang et al.. 1995), as 



f t (l\x,s)- 



k( x-1) 



h, (-/?"' log(l-//.v)|.r,a ). 



(3) 



We now generalize assumptions 2 and 3 by introduc- 

 ing the intensity function of recruitment, r(t ), and the 

 total instantaneous mortality, z(t), which are arbitrary 

 functions of time t. The total mortality would depend on 

 time through the fishing mortality component F, where 

 zit)=M+Fit) and M is the constant natural mortality. 

 The age distribution satisfies 



h,(a IL M =.v,A -a )~exp|-J z(t - a + y)dy\r(t - a + a ). (4) 



Materials and methods 



The model 



We assume that the growth of individuals follow a von 

 Bertalanffy curve so that the length at age a (relative 

 to some origin t ) is given by 



L(a) = LJl 



- e - ha ). 



(1) 



In this study, age is always defined to be relative to t , 

 i.e. t is absorbed into a for the purpose of identifiability. 

 We will consider estimation of (k, l x ) only because t is 

 not estimable from length-frequency data with aging 

 data. Note that this does not mean t is assumed to be 

 0. To provide a general treatment we relax each of the 

 assumptions mentioned in the introduction. First we 

 relax assumption 1 by letting the maximum length, L ,, 

 vary within the population. We denote the density func- 

 tion of L , as p(x), which has a mean of I r and a variance 

 of a 2 . It is possible that recruits to the fishery have a 

 range of sizes. To allow for this range we let the size 

 at recruitment, L , be a random variable with density 

 function u(s). In practice, one may be able to use infor- 

 mation from other studies (such as subadult abundance) 

 to arrive at an approximate parametric form for u(s). 



If f t (l) is the probability density function of L at time 

 t, then 



f t (.l) = J"j"p(x\L =s)f t (l\L„=x,L Q = s)u(s)dxds, (2) 



This equation states that the density of individuals of 

 age a is proportional to the intensity of recruitment at 

 the time when these individuals were recruited, namely 

 t-a+a , multiplied by a reduction factor due to mortality 

 over the intervening period. We therefore have 



h t (a\x,s) = h t (a\L„=x,Ao=-k~ 1 log(.l-s/x)) 



=exp(-| 



(5) 



-k "Mogll-s/.rl 



zit-a + y )dy \r(t- a- k 1 log(l-s/x) 



and Equation 3 becomes (after substituting for a and 

 shifting the dummy variable y) 



x-l 



exp 



-j: 



f t a\x,s) 



-M^f y)dy 



t-k'Hog 



x-s\) (6l 



Let us consider the case of fixed recruitment length, 

 i.e., L =l , and define a parameter vector, p, consisting 

 of th, / x , s), and other parameters quantifying mortality 

 and catchability. Equation 2 then reduces to a single 

 integral over x, 



f l (l\/5)ocj°°p(x)exp 



-L MS) ^4(" r ' log (^))£' 



(7) 



