Projecting viability of Totoaba macdonaldi, a population with unknown age-dependent variability

For the endangered fish totoaba (Totoaba macdonaldi), as for other endangered species, the relative amount of environmental variability in demographic rates at various life stages is uncertain. Because of the ontogenetic migratory pattern of this species, four life history stages, (a) prerecruits, (b) juveniles, (c) pre-adults, and (d) adults, experience different environmental conditions. We used a diffusion equation approximation of random Leslie matrices to explore the effects on extinction risk of environmental variability in each of these life stages and correlation between stages. Variability in pre-adults and adults had a greater effect on the probability of extinction than did variability in prerecruits and juveniles. In general, correlation between stages increases the rate of increase of the variance of the logarithm of abundance and, hence, the probability of extinction. For example, when all life stages vary coherently, the rate of increase of the variance is the square of a weighted sum of elasticities, while when all stages vary independently, the rate of increase is the same weighted sum of the squared elasticities. The maximum extinction rate occurs when all life history rates in a Leslie matrix vary coherently. Specifically, when the coefficient of variation is the same in each parameter, the rate of increase of the standard deviation of the logarithm of abundance equals the coefficient of variation of the environmental variability. We also evaluated the accuracy of the diffusion equation approximation by comparing predicted extinction rates with those from Monte Carlo simulations of totoaba with variability in each of the four stages. The diffusion equation approximation accurately predicted probabilities of extinction in all cases except one, random variability in recruitment, where it underestimated simulation results substantially. This is apparently due to the large random jumps in abundance in this case. The diffusion equation approximation accurately predicted quasi-extinction of the adult portion of the population.