Open access dynamics

Part 1

Biology

Assume logistic growth of a fish stock population, expressed by

,

r representing intrinsic growth rate, K natural stock biomass equilibrium and X the stock biomass (variable). The short term harvest equation is

,

q being the catchability coefficient and E the fishing effort. The net change in stock biomass per unit f time is then



Now consider the special case when

,

obviously then



This equation has two solutions for X:
the second  equilibrium is asymptotically stable when



This solution is shown in the figure below.




Economics

The economics of the simple Gordon-Schaefer model is described by the revenue function

,

p being a constant unit price of harvest and H the harvest equation presented above. The total cost is given by

,

a being the constant unit cost of effort, including opportunity costs. The resource rent obtained in the fishery is then expressed by the difference between TR and TC,


Since the fishing effort is expected to stabilise when the profit is normal (no resource rent) and positive resource rent attracts more fishers to the fishery while a negative resource rent does the opposite, the change in effort is assumed to be expressed by



when the parameter g is the rate of change in effort on the base of resource rent from the fishery. In the case of



the equation describing change of effort has two solutions


shown in the figure below