In quang-huynh/RBassess: An Age- And Length-Based Assessment Model for Rainbow Trout in British Columbia, Canada

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Growth

Growth is assumed to follow a re-parameterized von Bertalanffy function where the mean length at age $a$ ($L_a$) at the time of the survey is: $$L_a = \tilde{L}{a} \exp(-K \Delta_a) + L{\infty}(1 - \exp(-K \Delta_a)),$$ where $\tilde{L}{a}$ is the mean length of the cohort currently of age $a$ at the time of stocking, $L{\infty}$ is the mean asymptotic length, $K$ is the von Bertalanffy growth coefficient, and $\Delta_a$ is the accumulated degree days (converted to years) experienced by the cohort of age $a$.

The variability in length-at-age $\sigma_a$ in the population is assumed to be normally distributed in the absence of fishing and proportional to $L_a$ by a coefficient of variation $CV$: $$\sigma_a = CV \times L_a.$$ Thus, probability of a fish to be within a length bin of mid-point $\ell$ at age $a$ is $$\textrm{P}(\ell|a) = \frac{1}{\sigma_a\sqrt{2\pi}}\int_{\ell-d}^{\ell+d}\exp\left(\frac{(L_a - x)^2}{2\sigma_a^2}\right)dx,$$ where $d$ is the half the width of the length bin.

Population model

Selectivity $v$ of the anglers and the survey gillnet is assumed to be logistic and size-specific, parameterized as $$v^{AN}{\ell} = [1 + \exp(-\gamma^{AN}{\ell - h^{AN}})]^{-1} $$ and $$v^{GN}{\ell} = [1 + \exp(-\gamma^{GN}{\ell - h^{GN}})]^{-1}, $$ where $\gamma$ is the slope of the logistic curve at 50% selectivity, $h$ is the length at 50% selectivity, and the superscript indexes type ($AN$ = angler and $GN$ = gillnet).

The population abundance of length $\ell$ and age $a$ is given by $$N_{\ell,a} = R_{\ell,a}\exp(-\Sigma_{a'=1}^{a-1} [F_{\ell'(a')} + M]),$$ where $M$ is the instantaneous natural mortality rate, $$R_{\ell,a} = N^{stock}a \textrm{P}(\ell|a),$$ where $N^{stock}_a$ is the stocking density and $R{\ell,a}$ would be the abundance of animals of length $\ell$ and age $a$ in the absence of mortality, $$ F_{\ell'(a')} = v_{\ell'(a')}^{AN} F,$$ where $F$ is the apical instantaneous fishing mortality rate, and $$\ell'(a') = \frac{\ell}{L_a}L_{a'}.$$

Equation 6 calculates the current abundance of the age-$a$ cohort as a function of cumulative fishing pressure from previous ages $a' = 1, \ldots, a-1$. We assume larger animals in the cohort remain proportionally large (relative to $L_{a}$) for the entirety of the cohort's lifespan (and similarly, smaller animals remain proportionally small). Since selectivity is size-based, animals in a cohort may not experience equal magnitudes of fishing mortality over time. The summation in Equation 6 accounts for this phenomenon when calculating historical fishing mortality. In other words, Equation 9 calculates a fish at current length $\ell$ to have been length $\ell'$ at age $a'$, and angler selectivity is calculated accordingly in Equation 8.

Iterative solution for F

Fishing mortality $F$ includes a component for retention $F^{retain}$ and mortality from catch-and-release $F^{release}$, $$F = F^{retain} + F^{release}.$$ Total fishing effort, capture efficency, voluntary release, and regulations (i.e., bag limits) affect F. From estimates of effort $E$, catchability $q$, and the probability of harvest $p_h$, and release mortality $m_{rel}$ (proportion of fish that die from catch-and-release), fishing mortality rates $F^{retain}$ and $F^{release}$ are calculated as: $$F^{retain} = qEp_h,$$ and $$F^{release} = qE(1-p_h)m_{rel}.$$

We usually don't have $p_h$ but rather the bag limit $b_{lim}$ and some notion of the rate of voluntary release $p_{vrel}$ by anglers. To calculate $p_h$ from the bag limit and voluntary release, we first calculate the expected catch in length and age $C_{\ell,a}$ based on the population abundance. From the Baranov equation and an initial value of $F$, $$ C_{\ell,a}= \frac{v^{AN}{\ell} F}{v^{AN}{\ell} F+M} N_{\ell,a} [1 - \exp(-v^{AN}{\ell} F-M)] .$$ The expected $CPUE$ is $$CPUE = \frac{\Sigma{\ell}\Sigma_{a} C_{\ell, a}}{E}.$$

Next, the expected value of the CPUE (pertaining to retained catch) subject to the bag limit and voluntary release ($CPUE | b_{lim},p_{vrel}$) is calculated. If the CPUE for individual anglers is a Poisson random variable with rate parameter $\lambda = CPUE$, then $CPUE|b_{lim},p_{vrel}$ can be calculated as: $$CPUE|b_{lim},p_{vrel} = \Sigma_x g(x) f(x;\lambda) $$ where $$ g(x) = \textrm{min}(x[1-p_{vrel}], b_{lim}),$$ and $f(x;\lambda)$ is the probability density function for a Poisson random variable, $$ f(x;\lambda) = \exp(-\lambda) \frac{\lambda^x}{x!},$$ with $x = 0, 1, 2, \ldots$. Generally, the summation is stopped at some arbitrarily large number (e.g., 20).

Then, the probabilty of harvest $p_h$ is $$p_h = \frac{CPUE|b_{lim},p_{vrel}}{CPUE}.$$ From a starting value of $p_h$, $F$ is solved iteratively through Equations 10-18 until the value stabilizes (often at 5 iterations or fewer).

From Equation 16, voluntary release acts upon retention behavior before the bag limit. For example, if the bag limit is 5 fish per day and anglers release half their catch, then then an angler that catches 8 fish would keep four of them.

Finally, the harvest rate $u$ is calculated as the ratio of total catch to total abundance: $$u = \dfrac{\sum_{\ell}\sum_a C_{\ell, a}} {\sum_{\ell}\sum_a N_{\ell, a}}.$$

Likelihood

In the model, the numbers at length and age from the survey $N^{GN}{\ell,a}$ is calculated as $$N^{GN}{\ell,a} = v^{GN}{\ell} N{\ell,a},$$

Assuming a multinomial distribution, the log-likelihood ($\Lambda_1$) for a random sample of length-age observationss is $$\Lambda_1 = O_{\ell,a} \log\left(\dfrac{N^{GN}{\ell,a}}{\Sigma{\ell}\Sigma_a N^{GN}_{\ell,a}}\right),$$ where $O$ is the observed numbers in the sample.

The observed numbers at length is calculated as: $$N^{GN}{\ell} = \Sigma_a N^{GN}{\ell,a}.$$ The likelihood $\Lambda_2$ for a random sample of lengths is $$\Lambda_2 = O_{\ell} \log\left(\dfrac{N^{GN}{\ell}}{\Sigma{\ell}N^{GN}_{\ell}}\right).$$

quang-huynh/RBassess documentation built on May 8, 2019, 7:30 a.m.