The net growth in the stock per unit of time (for example per year) is the difference between the natural growth and the catch:
which gives an equilibrium biomass
Inserting the equilibrium biomass in the harvest equation yields the long term catch equation.
The Gordon-Schaefer model is the combined catch equilibrium model Schaefer proposed on the basis of the logistic growth model (by Verhulst) and simple economic assumptions. The combined model was proposed by the Canadian economists Scott Gordon and Anthony Scott in the mid fifties.
The logistic growth model is expressed by
and gives the growth in terms of biomass per unit of time when X is the stock biomass at the beginning of the period, r the intrinsic growth rate and K the environmental saturation level expressed as maximum equilibrium biomass the environment can hold.
The short term catch (H) equation is, due to Schaefer, assumed to be a bi-linear function of the two variables X and fishing effort E:
q being the catchability coefficient.