Hayes: A biological reference point based on the Leslie matnx 



81 



Si 



termination of how rapidly the stock will 

 be rebuilt under various levels of fishing 

 mortality. 



Developing a Leslie matrix 

 representation of harvesting: 

 stochastic case 



One of the major challenges facing fish- 

 ery managers is to determine appropriate 

 reference points for fish populations that 

 show variable recruitment. When one or 

 more elements of the Leslie matrix vary 

 in a stochastic fashion, no general closed- 

 form expressions for the growth rate of 

 the population are available (Tuljapurkar, 

 1989). Results of theoretical studies of sto- 

 chastic Leslie matrices are useful, how- 

 ever, in guiding the analysis and interpre- 

 tation of matrices with entries that vary 

 over time. Two results summarized by 

 Tuljapurkar (1989) that are particularly 

 useful in this analysis are 



1 The analog to lambda for deterministic matrices 

 is the mean population growth rate, in contrast to 

 the growth rate of the average population. This is 

 equivalent to the mean rate of change in the loga- 

 rithm of population size (N). 



2 The distribution of projected population size (AO over 

 time tends towards a lognormal distribution when 

 the dynamics are governed by a stochastic matrix. 



From these results, maintenance fishing mortality can 

 be defined as the fishing mortality that results in an 

 average population growth rate of (equivalent to 

 Ag=l). Because this measure is analogous to lambda, 

 I will use the symbol A for its representation but em- 

 phasize that computationally the measures for deter- 

 ministic and stochastic Leslie matrices differ. An im- 

 portant corollary of the above two results is 



3 A population growing deterministically at the mean 

 growth rate does not produce the mean of the pop- 

 ulation sizes produced in the stochastic represen- 

 tation. Nor does a deterministic matrix composed 

 of the means of the stochastic matrices produce a 

 population with the same dynamics as applying 

 the stochastic matrices. 



To illustrate these theoretical results, I performed a 

 simulation of the Georges Bank haddock stock dy- 

 namics using a stochastic Leslie matrix. In this case, 

 I focused on the effects of stochastic age-0 survival 

 as represented by R/SSB. I performed this simulation 



20 



40 



60 

 % MSP 



80 



100 



Figure 2 



Relationship between rate of population increase (lambda) and percent 

 maximum spawning potential C/rMSP) for a range of age at entry. 



by projecting a starting population forward in time, 

 with the value for R/SSB for each year selected with 

 equal probability from observed values fi-om 1976 to 

 1994. Five thousand replicates were simulated for a 

 150-year period. 



Results of these simulations are in accord with the 

 theoretical assertion that TV, is distributed lognor- 

 mally; for times greater than 110 years, the ln(A^,) 

 did not differ significantly from a normal distribution 

 at a=0.05. It is interesting to note that A^, is lognor- 

 mally distributed, even though the stochastic element 

 (R/SSB) was not normally or lognormally distributed 

 itself The lognormal distribution of A^, arises from the 

 fact that TV, is the result of the process of sequential 

 multiplications of random elements (Aitchison and 

 Brown, 1976). When the distribution of population 

 size is plotted over time (Fig. 3), it is clear that the 

 variance increases rapidly. When year-to-year popu- 

 lation growth rates (i.e. TV^^/TV,) are computed for in- 

 dividual simulation results, the distribution of growth 

 rates shows an initial transient response for the first 

 5 years but thereafter settles into a stable distribution 

 from year to year (Fig. 4). Because of the transient dy- 

 namics, I began the evaluation of long-term dynamics 

 with year 10. 



One of the critical theoretical results is that the 

 growth rate of a population governed by stochastic 

 rates tends towards a single value in the long run. 

 This is what Tuljapurkar (1989) terms the "almost 

 sure population growth rate." When the growth rate 

 is computed over progressively longer intervals, the 

 distribution shows a convergence on nearly the same 



