FAS 6932
ASSIGNMENT 5
DUE February 18, 2011
VPA AND STOCK SYNTHESIS: HOW INFORMATIVE ARE AGE
STRUCTURE DATA?
The spreadsheet HERAGE.XLS has data on age composition and spawn survey
abundance for a Pacific herring stock in the Georgia Strait, BC. Develop a VPA
reconstruction of stock size from these data. Use results from the VPA to help develop
an alternative assessment based on a forward (synthesis) model of the historical
dynamics. Fit this model initially using a likelihood function only for the age
composition data, then add a likelihood component for the predicted spawn survey index.
Your forward simulation should predict Na+1,t+1=s(1-va,tut)Na,t where the natural survival
rate s is given on the spreadsheet and exploitation rate ut is calculated each year from the
total observed number of fish caught Ct (sum over ages of Ca,t data) as
ut=min(1,Ct/ava,tNa,t) (Schnute-Richards approach to avoid treating ut as unknown
parameter). Treat the following as unknown parameters to be estimated by Solver:
(1) Na,1: numbers at age in year 1 (1951);
(2) N0,t: numbers at age 0 in all years after 1951, i.e. recruitments;
(3) va,1951-69: vulnerability at age to reduction fishery (assume 1.0 for older fish);
(4) va,1969-97=ma: vulnerability/maturity at age to roe fishery (for spawn biomass
calculations, assume maturity ma stable over all years, again 1.0 for older fish).
Use the VPA results to provide initial estimates for these parameters.
In fitting the model, assume a multinomial log-likelihood function for sampled age
proportions in the catch:
LogLage=at ntpa,tln(Pa,t)
Where
nt=effective sample size for age composition in year t (assume nt=100 or less)
pa,t=observed proportion of age a fish in year t catch
Pa,t=model predicted proportion of age a fish in year t catch (=utva,tNa,t/Ct)
When you include the spawn survey data in the likelihood function, do so by constructing
the toal log likelihood L=logLage+logLsurvey where you calculate the survey log likelihood
component as
LogLsurvey= -(T-1)/2 ln(SS), SS=t(Zt-Zbar)2, Zt=ln(bt/Bt)
Where
T=number of years survey data
bt=observed spawn survey index in year t
Bt=predicted spawn biomass in year t, tmawaNa,t(1-utva,t) (wa=weight at age)
Note you can change the “weight” placed on the age composition versus survey data in
the overall likelihood function by changing the effective sample size nt for age data.
How well do the VPA and synthesis models agree about the stock history, especially
when only the age composition data are used for fitting the synthesis model? Do both
methods agree about the “current” or most recent (1997) spawning biomass? What
happens to your results if you admit age-dependence in the natural survival rate (some
age dependent estimates are included in the spreadsheet)?