9.1 Fisheries Science

Step 1: Logistic Growth Equation

Begin with a population of size $B$ growing at intrinsic rate $r$ toward carrying capacity $K$. The density-dependent growth is:

$$\frac{dB}{dt} = rB\left(1 - \frac{B}{K}\right)$$

Step 2: Introduce Fishing as a Harvest Term

In the Schaefer model, the catch rate is proportional to both fishing effort $E$ and biomass $B$ through catchability $q$: $C = qEB$. The biomass dynamics with harvest become:

$$\frac{dB}{dt} = rB\left(1 - \frac{B}{K}\right) - qEB$$

Step 3: Find Equilibrium Biomass

At equilibrium, $dB/dt = 0$. Factor out $B$ (discarding $B = 0$):

$$r\left(1 - \frac{B_{eq}}{K}\right) = qE \quad \Longrightarrow \quad B_{eq} = K\left(1 - \frac{qE}{r}\right)$$

Step 4: Derive Equilibrium Yield

Substitute $B_{eq}$ into $Y_{eq} = qEB_{eq}$:

$$Y_{eq} = qEK\left(1 - \frac{qE}{r}\right)$$

Step 5: Maximize Yield to Find MSY

Differentiate $Y_{eq}$ with respect to $E$ and set the derivative to zero:

$$\frac{dY_{eq}}{dE} = qK - \frac{2q^2KE}{r} = 0 \quad \Longrightarrow \quad E_{\text{MSY}} = \frac{r}{2q}$$

Step 6: Compute MSY and $B_{\text{MSY}}$

Substitute $E_{\text{MSY}}$ back into the equilibrium biomass and yield expressions:

$$B_{\text{MSY}} = K\left(1 - \frac{q \cdot r/(2q)}{r}\right) = \frac{K}{2}$$

$$\text{MSY} = qE_{\text{MSY}} \cdot B_{\text{MSY}} = q \cdot \frac{r}{2q} \cdot \frac{K}{2} = \frac{rK}{4}$$

Step 7: CPUE as Abundance Index

The catch per unit effort (CPUE) is directly proportional to biomass:

$$\text{CPUE} = \frac{C}{E} = \frac{qEB}{E} = qB$$

Thus CPUE serves as a relative index of stock abundance; linear regression of CPUE on effort estimates $q$ and $K$.

Step 8: Fishing Mortality at MSY

Define fishing mortality as $F = qE$. At MSY:

$$F_{\text{MSY}} = qE_{\text{MSY}} = q \cdot \frac{r}{2q} = \frac{r}{2}$$

Overfishing occurs when $F > F_{\text{MSY}} = r/2$. The stock is overfished when $B < B_{\text{MSY}} = K/2$.

Step 1: Density-Dependent Juvenile Mortality Assumption

Assume that juvenile survival decreases linearly with population density. The instantaneous mortality rate for juveniles is $\mu(N) = \mu_0 + \mu_1 N$, where $N$ is the number of juveniles competing for resources.

$$\frac{dN}{dt} = -(\mu_0 + \mu_1 N) N$$

Step 2: Solve the Differential Equation

This is a Bernoulli equation. Let $u = 1/N$, so $du/dt = -N^{-2}\,dN/dt$. Substituting:

$$\frac{du}{dt} = \mu_0 u + \mu_1$$

This linear ODE has the solution:

$$u(t) = \left(u_0 + \frac{\mu_1}{\mu_0}\right)e^{\mu_0 t} - \frac{\mu_1}{\mu_0}$$

Step 3: Derive the Recruitment Function

After the juvenile period of duration $\tau$, recruitment $R = N(\tau) = 1/u(\tau)$. With the initial number of eggs proportional to spawning stock $S$: $N(0) = \gamma S$, we obtain:

$$R = \frac{\gamma S \, e^{-\mu_0 \tau}}{1 + \frac{\mu_1 \gamma S}{\mu_0}(1 - e^{-\mu_0 \tau})}$$

Step 4: Standard Beverton-Holt Form

Define $\alpha = \gamma e^{-\mu_0 \tau}$ and $\beta = \frac{\mu_1 \gamma}{\mu_0}(1 - e^{-\mu_0 \tau})$ to obtain the classic form:

$$R = \frac{\alpha S}{1 + \beta S}$$

Step 5: Properties of the Beverton-Holt Curve

At low stock sizes ($S \to 0$): $R \approx \alpha S$ (linear increase). The slope at the origin is the maximum recruits-per-spawner $\alpha$.

$$\lim_{S \to \infty} R = \frac{\alpha}{\beta} \quad \text{(asymptotic maximum recruitment)}$$

Step 6: Steepness Parameterization

Define steepness $h$ as the fraction of unfished recruitment $R_0$ produced when SSB is 20% of unfished SSB $S_0$:

$$h = \frac{R(0.2 S_0)}{R_0} = \frac{\alpha \cdot 0.2 S_0}{(1 + \beta \cdot 0.2 S_0) R_0}$$

Steepness ranges from 0.2 (strong density dependence) to 1.0 (recruitment independent of stock). High steepness means the stock is resilient to fishing.

Step 1: Discrete Biomass Dynamics

Discretize the Schaefer model over annual time steps. The biomass next year equals this year's biomass plus surplus production minus catch:

$$B_{t+1} = B_t + rB_t\left(1 - \frac{B_t}{K}\right) - C_t$$

Step 2: Link CPUE to Biomass

The observed CPUE index $I_t$ is assumed proportional to biomass: $I_t = qB_t$. This allows expressing the equilibrium yield in terms of observables:

$$Y_{eq} = \frac{r}{q} I\left(1 - \frac{I}{qK}\right)$$

Step 3: Observation Model and Likelihood

Assume log-normal observation errors in the CPUE index. The likelihood of the data given parameters $\theta = (r, K, q, \sigma)$ is:

$$\mathcal{L}(\theta | \text{data}) = \prod_{t=1}^{T} \frac{1}{I_t \sigma \sqrt{2\pi}} \exp\left(-\frac{(\ln I_t - \ln qB_t)^2}{2\sigma^2}\right)$$

Step 4: Negative Log-Likelihood

Taking the negative log-likelihood and dropping constants:

$$-\ln \mathcal{L} = \frac{T}{2}\ln\sigma^2 + \frac{1}{2\sigma^2}\sum_{t=1}^{T}(\ln I_t - \ln q - \ln B_t)^2$$

Step 5: Concentrate Out $q$ and $\sigma^2$

The MLE for $q$ is the geometric mean ratio of CPUE to predicted biomass. Define residuals $\epsilon_t = \ln I_t - \ln B_t$:

$$\hat{q} = \exp\left(\frac{1}{T}\sum_{t=1}^{T} \epsilon_t\right), \quad \hat{\sigma}^2 = \frac{1}{T}\sum_{t=1}^{T}(\epsilon_t - \bar{\epsilon})^2$$

Step 6: Management Quantities from Fitted Model

Once $r$ and $K$ are estimated, all management reference points follow directly:

$$\text{MSY} = \frac{rK}{4}, \quad B_{\text{MSY}} = \frac{K}{2}, \quad F_{\text{MSY}} = \frac{r}{2}, \quad E_{\text{MSY}} = \frac{r}{2q}$$

The current stock status is assessed by comparing $B_{\text{current}}/B_{\text{MSY}}$ and $F_{\text{current}}/F_{\text{MSY}}$ on a Kobe plot.