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



383 



the northern prawn fishery of Australia. This pattern 

 has been derived from experimental trawls in which 

 the number of individuals in the lowest length class 

 are counted (Wang and Die, 1996). We assume the 

 recruitment and effort patterns are the same in each 

 year (Fig. 1). The effort pattern (dashed line) consists of 

 two constant-fishing periods: 15 May to 15 June, and 1 

 August to 1 December. The unit of effort, E, depends on 

 the unit of catchability, q, because the fishing mortality 

 F=qE must have unit vr 1 : therefore we let £=1 during 

 the fishing season. Note that the proportion of the year 

 that is fished is JE(t)dt=5/12. 



The growth component of our models has ^ = 40 mm 

 and k=3yr~ 1 ; the instantaneous natural mortality is 

 M=2yr~ l \ and the instantaneous fishing mortality, F, 

 during the fishing season is 4yr _1 (i.e., c/ = 4, because in 

 our units, F=q). The resulting annual mortality, Z=fz{t) 

 dt=M+qJE(t)dt=2+4x5/l2=ll/3. The values for mortal- 

 ity come from Somers and Wang (1996). We assume that 

 all recruits have length 19.5 mm. The L ( distribution 

 is normal (standard deviation 4 mm) but is truncated 

 at 19.5 mm. The truncated normal distribution at l is 

 simply a conditional normal distribution conditional on 

 being greater than l . 



We generate twelve length-frequency data sets, one 

 for the beginning of each month. We choose a monthly 

 time interval because the data from our case study in 

 the next section were sampled at roughly monthly in- 

 tervals. In addtion, because the recruitment pattern is 

 periodic it is sufficient to analyze one year of data. 



We obtain each monthly length-frequency data set 

 by taking a sample of size 1000 from the theoretical 

 length distribution f t (l) given by Equation 6, which 

 depends on the recruitment pattern, the effort pattern, 

 and the distribution of L^. That is, for each of the 12 

 time points /, we evaluate numerically the right-hand 



side of Equation 6 over a set of finely spaced / values 

 (i.e., every 0.25 mm), aggregate the f t (l) to 1-mm inter- 

 vals and finally normalize the function by dividing by 

 the sum of f t il). This results in an array of probabilities 

 for an individual's length in each 1-mm interval. It is 

 then straightforward to sample from the corresponding 

 multinomial distribution. 



We then obtain parameter estimates from the twelve 

 months of simulated data. The process is repeated 100 

 times to provide a reasonable estimate of the sampling 

 variance of the parameters. In practice, (k, IJ) can of- 

 ten be estimated from a different study. We therefore 

 consider two models. In model 1, we assume all five 

 parameters are unknown, and, in model 2, we assume 

 that l m and k are known and we estimate M, F, and a. 

 It is also common practice (e.g., Sullivan, 1992) to as- 

 sume that M is known and to estimate the remaining 

 parameters; this is the case in our model 3. 



The results are summarized in Table 1. All the pa- 

 rameters are quite well estimated, even for model 1. 

 Estimates of both natural mortality and fishing mor- 

 tality are quite reliable when growth parameters are 

 assumed known. There is also a modest reduction in the 

 standard deviation when (k, Z x ) are assumed known. 



We have also tested for robustness by performing 

 the estimation process on data generated from a log- 

 normal distribution. The results are shown in Table 1. 

 For model 1 the estimates of M and F have a larger 

 and opposite bias, whereas the absolute bias for Z is 

 somewhat smaller. Model 2 improves the estimates 

 dramatically, despite the fact that an incorrect dis- 

 tribution (the truncated normal) is being used in the 

 model. Note that the variation in the estimates of total 

 annual mortality, Z, is somewhat less than that for F 

 and M; this is because F and M are highly negatively 

 correlated (typically 94%). In model 3 the estimate of 



