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Algebraic analysis

Equilibria:

There is only a single equilibrium for this new system. The equilibrium, which existed for all previous models, is no longer a possible equilibrium state since some level of consumption still occurs even when there are no fish and no boats.

Normalizing N and K with respect to the and of section 2, the equilibrium with the new consumption function is at = , where

 

The positive equilibrium is at the same capital level as before, but the equilibrium number of fish is larger than the baseline value of . Clearly this occurs because more of the capital is drained off into consumption and less can be used for harvesting fish. The equilibrium level of the resource, , increases with the minimum consumption, , the width of the transition region (as governed by ), and the fraction of profits are that are consumed, .

Substituting the equilibrium values of the resource and the capital back into the expression for shows that the consumption at the equilibrium point is equal to , or . Thus when the corner in the consumption function becomes sharp, (), the equilibrium is at the corner, with .

Stability:

When the equilibrium point lies near the region of minimum consumption, it is strongly modified by the existence of this minimum consumption. We can still have a stable equilibrium with oscillations, but the size of the parameter region where oscillations do occur is different, and for some parameter values the equilibrium is no longer stable.

In addition, a positive minimum consumption implies that the derivative of K is negative at K=0: when the initial capital is small, the system might collapse even though the equilibrium is attractive (see figure 11b).

Linearizing the system around the equilibrium gives the eigenvalues

 

where

 

  
Figure 11: Time plots of the logarithms of resource, capital, and consumption for the model of section 5. Here N and K are scaled by and , respectively, of section 2, and the consumption C is scaled by . Shown are two different initial conditions for the case , = 0.5, , and . In both cases N starts out at its equilibrium value of . Each time the profit plummets to or less, the consumption bottoms out to . (a) . In this case the initial conditions are in the attracting region of the equilibrium, and capital never goes to zero. (b) . Although this initial condition is near to that of (a), it is just outside the zone of attraction of the equilibrium. The amplitude of the oscillations gradually grows, until capital collapses.

The equilibrium is thus stable as long as , i.e. when equilibrium resource level is lower than twice harvesting equilibrium. For very small values of this condition is equivalent to . When cannot be neglected the transition occurs for smaller values of . As increases, the real part of goes to zero, and attraction gets gradually weaker. However, even before reaches 2, an unstable region can occur, and trajectories can hit the boundary if the capital gets too small (figure 11b). In fact the basin of attraction of equilibrium shrinks in the phase space as increases.



next up previous
Next: Simulation results Up: A model with Previous: A model with



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