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.

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