A biological reference point based on the Leslie matrix

For fish populations with an annual breeding cycle, a biological reference point based on the Leslie matrix is presented and compared with percent max imum spawning potential (%MSP) and F-med reference points. For deterministi c population projections, the reference point is defined as the level of fi shing mortality (F-st) that results in a Leslie matrix with a dominant eige nvalue (i.e. finite rate of increase or lambda) of 1.0. It is shown that fo r the same input data, F-st is similar to a reference point based on a %MSP approach. For populations that are growing or declining, however, populati ons with the same lambda but with different age-specific selectivities have different levels of %MSP. Previous applications of this reference point ar e extended to include situations where recruitment is a stochastic process. In stochastic projections, F-st is defined as the level of fishing mortali ty 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 de ath rates resulted in an estimate of F-st of 0.52. The same estimate of F-s t was obtained in a stochastic projection in which the growth rate of the m ean population size was used. Stochastic projections using the mean of the finite rates of increase resulted in a lower estimate of F-st (0.45). When the value of recruits per unit of spawning stock biomass used in the %MSP a nalysis was calculated as Sigma recruits/Sigma spawning 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 Sigma recrui ts/Sigma spawning stock biomass results in an equivalent estimate of the re ference paint but does not convey important information on the expected pop ulation growth rate at higher or lower rates of fishing mortality.