9.5 Marine Conservation

Step 1: Stochastic Logistic Growth Model

Begin with deterministic logistic growth and add environmental stochasticity by making the growth rate a random variable drawn each year:

$$N_{t+1} = N_t \exp\!\left[r_t\!\left(1 - \frac{N_t}{K}\right)\right], \quad r_t \sim \mathcal{N}(\bar{r}, \sigma_r^2)$$

Step 2: Define Quasi-Extinction Threshold

A quasi-extinction threshold $N_e$ is set (e.g., 50 individuals) below which the population is functionally extinct due to Allee effects, inbreeding depression, or demographic stochasticity. PVA estimates the probability of reaching $N_e$:

$$P_{\text{ext}}(T) = \Pr\!\left(\min_{0 \le t \le T} N_t \le N_e\right)$$

Step 3: Diffusion Approximation for Extinction Risk

For small populations near the threshold, the log-transformed population $x = \ln N$ approximately follows a Wiener process with drift. The mean time to extinction from initial population $N_0$ is:

$$\bar{T}_{\text{ext}} \approx \frac{2}{\sigma_r^2}\left[\ln\!\left(\frac{N_0}{N_e}\right) \cdot \frac{\bar{r}}{\sigma_r^2/2} \;\text{(correction terms)}\right]$$

Step 4: Monte Carlo Simulation Approach

In practice, PVA runs thousands of stochastic simulations with random $r_t$ draws. The extinction probability is estimated as the fraction of simulations reaching $N_e$:

$$\hat{P}_{\text{ext}}(T) = \frac{\text{number of simulations reaching } N_e}{n_{\text{sims}}} \pm \sqrt{\frac{\hat{P}(1-\hat{P})}{n_{\text{sims}}}}$$

Step 1: Define the MVP Criterion

The minimum viable population is the smallest population size with a specified probability $1 - \alpha$ (e.g., 99%) of persisting for $T$ years (e.g., 100 years):

$$\text{MVP} = \min\{N_0 : P_{\text{ext}}(T \mid N_0) \le \alpha\}$$

Step 2: Genetic Considerations -- the 50/500 Rule

Franklin (1980) proposed that an effective population $N_e \ge 50$ avoids short-term inbreeding depression, while $N_e \ge 500$ maintains long-term evolutionary potential. The effective population size relates to census size through:

$$N_e = \frac{4 N_f N_m}{N_f + N_m} \cdot \frac{1}{1 + \sigma_k^2 / \bar{k}}$$

Step 3: Combine Demographic and Genetic Thresholds

The MVP must satisfy both demographic viability (from PVA simulations) and genetic viability. Since $N_e/N$ is typically 0.1--0.5 for marine species, the census MVP is often much larger than the genetic minimum:

$$N_{\text{MVP}} \ge \max\!\left(\frac{500}{N_e/N}, \; N_0^* \text{ from PVA}\right)$$

Step 4: Application to Marine Species

For marine megafauna with $N_e/N \approx 0.2$, maintaining $N_e = 500$ requires $N \ge 2500$ adults. Combined with high environmental stochasticity ($\sigma_r \sim 0.3$), PVA typically yields MVP estimates of 1000--7000 adults for marine mammals and sea turtles, informing IUCN Red List threat assessments and recovery targets.