Surplus Production Models and Equilibrium Harvest
Textbook bioeconomics (here considering fisheries) assumes logistic population growth and short term harvest production to be linear in stock biomass and fishing effort. The first assumption is expressed by the growth equation (red curve), being the stock biomass and the parameters and the intrinsic growth rate and the environmental saturation level, respectively. is the per period of time surplus production. is the natural equilibrium in the absence of fishing. The second assumption is the bilinear harvest equation (green line), being the fishing effort, while the parameter is known as the catchability coefficient. The fishing activity disturbs the natural equilibrium () and each level of fishing effort includes new equilibria determined by . This Demonstration illustrates how the collection of all existing equilibria found by varying values of describes the equilibrium catch , (blue curve). Depensation (decrease in marginal growth by decreasing population biomass) may give rise to two equilibrium points, one stable and the other unstable. The dashed blue curve indicates where unstable equilibrium points are found. Positive critical depensation levels are found when depensation causes negative biomass growth. The term depensation refers to a situation where decline in biomass is not compensated by increased per unit of biomass production, as in the logistic growth equation, where is a down-sloping line for .
The term "depensation" in population growth refers to a situation where stock is unable to compensate reduced stock size by increasing growth per unit of biomass, which is normally assumed. Depensation may lead to critical low growth rates, causing stock collapses.