## Schaefer model

In all production models, production (rate of increase) per unit biomass is highest at low population sizes and decreases at larger ones, becoming zero at the carrying capacity K. A simple way to express this is with the equation

which implies

defining r, the 'intrinsic rate of increase'. From eqn [34], the population production without fishing is that of logistic growth:

B2/K

which forms a parabola. Population production is typically called 'surplus production', meaning the surplus of recruitment and growth above mortality. If the surplus is caught in the fishery rather than contributing to n = 2

Biomass (B )/carrying capacity (K)

Figure 5 Maximum sustainable yield (MSY) from a generalized production model with different values of n. The Schaefer model of logistic population growth corresponds to n = 2. Other values of n chosen so that BMSY/K = {0.3, 0.7}.

Biomass (B )/carrying capacity (K)

Figure 5 Maximum sustainable yield (MSY) from a generalized production model with different values of n. The Schaefer model of logistic population growth corresponds to n = 2. Other values of n chosen so that BMSY/K = {0.3, 0.7}.

population growth, it is denoted 'equilibrium' or 'sustainable' yield (Ye), synonyms for the annual yield that can be taken from a population at equilibrium. Each population has many possible sustainable yields, depending on the biomass level at which it attains equilibrium. That level in turn is controlled by the steady-state fishing mortality rate. Maximum sustainable yield (MSY) is found at the peak of the parabola formed by eqn [36] and is related to model parameters by MSY = rK/4 (Figure 5). The biomass from which MSY can be taken is Bmsy = K/2, and the corresponding fishing mortality rate is FMSY= MSY/ Bmsy = r/2.

For nonequilibrium analyses, an instantaneous rate of fishing mortality per unit time is added to eqn [36]:

By integrating with respect to time over the period t to t + 1, a population projection equation is obtained:

Bt 1

Integration can of course give the biomass at any time during an interval of constant Ft. That makes possible the integration of dY/dt = FtBin the time interval tto t + 1 to obtain the yield (catch in weight) during that interval:

log 1

Applying the model to a typical data set (records of removals and relative abundance) results in estimates of MSY, Fmsy, Smsy, EMsY, and of the trajectories of biomass and fishing mortality rate through time.