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Algebraic analysis

The equilibrium resource level is obtained by equating the left-hand side of equation 34 to zero:

 

A positive resource equilibrium is only achieved when the maximum production b is larger than the capital depreciation rate during the same period, , otherwise the capital keeps on decaying. The equilibrium resource gets bigger and bigger as the maximum production in money gets closer to the capital depreciation per unit time.

Now that both and are known the stability analysis can be done around the equilibrium.

The following expression is obtained for the stability parameter :

 

where . Note that the dynamical behavior only depends upon and , but not on c.

Damping of oscillations, related to the real part of , is increased by prices by a factor

 

Prices do attenuate the oscillations, and thus resource depletion, but since the fraction should be smaller than one for equilibrium to exist, the damping factor is at most 2.

The region of oscillation is decreased by prices. Oscillations occur when

 

In fact expression 33 refers to a situation when demand is large and when prices are adjusted by classical supply demand adjustment mechanisms. This is generally the case for artisanal fisheries of highly valued species such as sole, cod, haddock, shrimps and lobsters. But for some fisheries, such as french industrial fisheries of hake , herring, anchovy, market demand can be low with respect to production. In this context, in order to preserve the economic fishing sector, landing prices are maintained by institutions such as government or producer organizations. We can further complicate the monetary coefficient function by introducing a minimum price a for fish (see figure 8), to model the case of a minimum price maintained by some institution:

 

A full algebraic analysis of this system, reported in the Appendix, has been done which gives expressions which are difficult to interpret; but we can still predict the two extreme dynamical regimes. When the minimum price is low, the major difference is that the restriction on b no longer holds since a production equilibrium always exists. At low equilibrium production the minimum price can be neglected and the above analysis (beginning of this section) makes it possible to predict the dynamics. In the large production region the standard analysis of section 2 applies with as if the price were simply . Large resource depletion is observed, and the equilibrium resource is maintained at a level inversely proportional to a.

  
Figure 9: Time plot of the resource population and capital for the model of section 4, which takes into account the role of prices. Prices damp the oscillations, but the effect is lessened by the existence of a minimum price. The parameters of equation 39 are b=2 and c=0.5, corresponding to a maximum price of 4, and a minimum price of 0.1 (for the most damped oscillation), 1 and 2 (for the biggest oscillations).



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weisbuch
Fri Feb 7 13:18:37 GMT+0100 1997