## Production Models

Production models are the simplest form of population dynamics models used in fisheries stock assessment. The detailed processes of new recruitment and mortality are condensed into two terms that describe the intrinsic rate of growth (denoted by r) and the populations' carrying capacity (denoted by K). Change in population biomass over time can be described using a very simple differential equation:

where B is a measure of population biomass or density, Y is the yield removed by the fishery, and m is a parameter that describes the population density at which production is maximized (Figure 3).

The simplest form of the basic production model is to assume that production is maximized at V of the population carrying capacity (i.e., m = 2 in eqn [4]). In fisheries science, this is usually referred to as the Schaefer or logistic production model. Equation [4] forms the underlying theoretical basis for sustainable harvest and a quantitative basis for determining the level of harvest that will maximize yields. If we assume equilibrium conditions (i.e., Bt +1 = Bt or dB/dt = 0 in eqn [4] and fix m = 2 for simplification), then the equilibrium yield is obtained by

and the yield is maximized by setting dY/dB = r(1 - B/K) - rB/K = 0 [6]

from which the population size that maximizes the yield is found by solving eqn [6] for B:

Equation [7] can then be substituted back into eqn [5] to determine the maximum yield:

Regardless of the complexity of the population models, these simple algebraic manipulations of equations (that represent the dynamics of an exploited population) are used to determine how much should be harvested to ensure long-term sustainability.