A biological reference point based on the Leslie matrix

Manuscript accepted 17 August 1999. Fish. Bull. 98:75–85 (2000). Abstract.–For fish populations with an annual breeding cycle, a biological reference point based on the Leslie matrix is presented and compared with percent maximum spawning potential (%MSP) and Fmed reference points. For deterministic population projections, the reference point is defined as the level of fishing mortality (Fst) that results in a Leslie matrix with a dominant eigenvalue (i.e. finite rate of increase or λ) of 1.0. It is shown that for the same input data, Fst is similar to a reference point based on a %MSP approach. For populations that are growing or declining, however, populations with the same λ but with different age-specific selectivities have different levels of %MSP. Previous applications of this reference point are extended to include situations where recruitment is a stochastic process. In stochastic projections, Fst is defined as the level of fishing mortality that results in an average finite rate of increase of 1.0. In an example with Georges Bank haddock, a deterministic analysis with mean birth and death rates resulted in an estimate of Fst of 0.52. The same estimate of Fst was obtained in a stochastic projection in which the growth rate of the mean population size was used. Stochastic projections using the mean of the finite rates of increase resulted in a lower estimate of Fst (0.45). When the value of recruits per unit of spawning stock biomass used in the %MSP analysis was calculated as Σrecruits/Σspawning stock biomass, the estimated reference point was the same as the stochastic projection. On the basis of these results, I recommend calculating the reference point based on a stochastic projection for which the mean of the simulated growth rates is used. A reference point based on a %MSP approach using the Σrecruits/Σspawning stock biomass results in an equivalent estimate of the reference point but does not convey important information on the expected population growth rate at higher or lower rates of fishing mortality. Advice to fishery managers on desirable harvest or exploitation rates is ideally based on full knowledge of fishery dynamics, including information on the fish population’s stock-recruitment relationship, growth and maturation schedule, and bioeconomic considerations. With a lack of such complete information, guidance to fishery managers often takes the form of providing an estimate of fishing mortality and a comparison of that rate to one or more biological reference points (e.g. Clark, 1991; Anonymous1). Numerous biological reference points exist, each concerned with a somewhat different aspect of population response to harvesting. One class of reference points focuses on yield per recruit as a function of fishing mortality. The general goal of this class of reference points is to optimize harvest rates in relation to natural mortality and growth (i.e. prevent growth overfishing; Beverton and Holt, 1957). For example, Fmax is the fishing mortality rate at which yield per recruit is maximized (Beverton and Holt, 1957). A related reference point is F0.1, which is the fishing mortality rate where the slope of the yield per recruit curve is 10% of the slope at the origin (Gulland and Boerema, 1973). Fishing at Fmax or F0.1 results in maximal or nearly maximal yield from a fishery when recruitment is independent of stock size. A limitation of this class of reference points, however, is that reductions in recruitment are often evident when stocks are depleted to low levels (e.g. Overholtz et al., 1986). Thus, management advice based on Fmax or F0.1 can result in declines in abundance through recruitment overfishing (Sissenwine and Shepherd, 1987), ultimately resulting in reduced total yield from a stock. As a counterpart to reference points based on yield per recruit, several reference points based on stockrecruitment considerations have been developed. The goal of these reference points is to provide a measure of fishing mortality that will likely avoid recruitment overfishing. An example of this type of reference point is Fmed which is based on the median of the observed levels of recruits produced per unit of spawning stock biomass (R/SSB) (Sissenwine and Shepherd, 1987). The rationale behind this reference point is that fish abundance is maintained when the spawning stock biomass produced by a cohort over its lifetime is equal to the spawning stock biomass of the parent population when the cohort was spawned. Related to Fmed is a set of reference points based on the spawning stock biomass per recruit (SSB/R) in relation to the SSB/R that would be produced if the