A size-based model for a fish stock assessment
1Saang-Yoon Hyun, 1Jinwoo Gim, and 2Kyuhan Kim
1 Pukyong National University (Busan, Korea)
2 Victoria University of Wellington (Wellington, New Zealand)
4 ~ 8 Nov. 2019
CAPAM workshop
Wellington, New Zealand
Prob(lengths) after ‘1st mortality and then 2nd growth’
- Length distribuiton at recruitment: Â \(P_{y,a=1}(x) = \text{discrete } N(\mu_1,\; \sigma^2_1)\)
- After mortality (at the end of age \(a\)): Â \(P_{y,a, Z}(x) = P_{y,a}(x)\cdot \text{exp}(M+F_{y}(x))\)
- After one growth increment from an individual of length \(x\) to length \(L\):
\[ \color{blue}P_{y+1,a+1}(L|x) = \text{discrete } N(\mu_{a+1}(x),\; \sigma^2_{a+1}) \]
- Finally after both mortality and growth,
\[ P_{y+1,a+1}(L) = \sum_{x}^{} P_{y,a,Z}(x) \cdot \color{blue}P_{y+1,a+1}(L|x) \] \[\because \color{blue}P_{y+1,a+1}(L|x) =\frac{P(L_{y+1,a+1}, \; x_{y,a})} {P_{y,a,Z}(x)} \]
Then, \(P_{y+1, a+1}(L) => P_{y+1, a+1}(x)\), and steps of (2) ~ (4) are repeated over imaginary ages, or along an imaginary cohort.
The growth process
- \(\mu_{a+1}(x)\) and \(\sigma^2_{a+1}\) from Cohen and Fishman (1980) and stochastic LVB equations
\(\mu_{a+1}(x)=f(L_\infty, \color{red}\rho, x)\)
\(\sigma^2_{a+1}=g(\color{red}{\sigma^2_x}, \color{red}\rho, a, \sigma^2_1)\)
where
\(x_{a+1} = h(L_\infty, \color{red}\rho, x_a) + \varepsilon\),  and  \(\varepsilon \text{ ~ } N(0,\color{red}{\sigma^2_x})\)
The other parts
Year- and length- fishing mortality: Â \(F_{y}(x)=\text{sel}\cdot F_{y} = \text{logistic}(x, \color{red} \gamma,\color{red} {L_{0.5}})\cdot\color{red} q\cdot \text{effort}_y\)
           Â
The number of age-\(a\) fish at length \(x\) after mortality: Â \(N_{y,a,Z}(x) = N_{y,a} \cdot P_{y,a,Z}(x)\)
Catch at age \(a\) and length \(x\): Â \(C_{y,a}(x)=N_{y,a} \cdot P_{y,a}(x) \cdot\frac{F_y(x)}{Z_y(x)}\cdot (1-\text{exp}(-Z_y(x)))\)
Biomass and yield at age \(a\): Â \[B_{y,a}=\sum_{x}^{} N_{y,a}(x)\cdot W(x), \; \; \text{and} \; \;
Y_{y,a}=\sum_{x}^{} C_{y,a}(x)\cdot W(x)\] Â Â Â Â Â Â Â Â Â Â Â Â Â Â where \(W(x)\) = body weight at length \(x\).
A total of 27 free parameters, and \(M\)
\(q\): catchability in the fishing mortality function
\(\gamma\) and \(L_{0.5}\): two parameters in the selection (logistic) equation
\(\color{red}\rho\) and \(\sigma^2_x\): two parameters in the growth equations (Cohen and Fishman; LVB)
\(\mathbf N_1\): 22 annual abundances at recruitment (i.e., in 22 years).
\(M\): natural mortality was not treated as a free parameter but its estimate was found with a sensitivity analysis.
Objective function: \(Obj\)
\(\mathbf m_y\), length frequency data in year, \(y\): Â Â \(\mathbf m_y\) ~ Multinomial(\(n_y\), \(\mathbf o_j\))
\(Y_y\), yield data in year, \(y\): Â Â \(\text{log} Y_y\) ~ \(N(\mu_{\text{log}Y_y}\), \(\sigma^2_{\text{log}Y})\)
\(\therefore Obj = -1.0\cdot \bigl[D_1 \cdot \text{log}L(\boldsymbol\theta|\mathbf m_y) + D_2 \cdot \text{log}L(\boldsymbol\theta | \text{log}Y_y) \bigr]\)
where \(D_1\) and \(D_2\) are data weights.
It turned out that \(D_1\) = 0.05, and \(D_2\) = 10.0.
Late Dr. Terry Quinn’s visit at May 2016 
