24-Bio Econ 3 PDF

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Bio Econ 3...

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These bioeconomic models that we have developed are all for long-run equilibrium situations. To examine the short-run dynamics we need a model for changes in fishing effort. Clark (1985) discusses the following simple model for a fishery system.

 

B

Change in Biomass:

dB dt

rB 1

Change in Effort:

df dt

a (p q B

K

qf B

c) f

The differential equation for biomass is just the normal Graham-Schaefer model using the assumption that the instantaneous rate of fishing mortality is proportional to fishing effort (which here is the number of active fishing operations). In the differential equation for fishing effort the rate of entry and exit of effort is proportional to the current flow of profits, and parameter a is the constant of proportionality. It is sometimes described as a parameter because its role in the equations is similar to the stiffness of a spring. These two differential equations are said to be , meaning that both equations involve both dependent variables. The system of equations will be at equilibrium when dB/dt=0 and df/dt=0. dB dt df dt

0

0

 

==>

rB 1

==>

pq B

The two lines defined by differential equations

dB

dt dB df 

0 and

B K

==>

qf B

==>

c

df dt

0 are called

ef

Be

 r B e q  q K r

c pq

. Any solution to the system of

 will also be a solution to the ratio of the two equations. Define a  dt dt 

df . A solution to this new differential equation will be a curve on the (f,B) dB plane. Any solution f(B) will be moving in a particular direction when it crosses an isocline.

new differential equation

For example, when a solution f(B) crosses the isocline for because in

df dB

0 , it will be moving horizontally,

the numerator is zero. When a solution crosses the isocline for

moving vertically, because in

df dB

On the next page is the so called dB dB 0 isoclines. 0 and dt dt

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df dt

dB dt

0 , it will be

the denominator is zero. for our fishery system and the lines for the

BioEcon3 - Page 156

Phase plane diagram df dt

We could also draw in isoclines for df/dB=±1. A solution will cross the -1 isocline moving in a 45º direction and it will cross the +1 isocline moving in a 135º direction.

Fishing Effort

r/q

0

(Boae ,foae )

dB dt

0

Biomass 0

K

c/(p·q)

The isoclines divide the phase plane into four regions. Within each region we can identify the sign (+ or -) for dB/dt and df/dt and determine whether biomass and effort are increasing or decreasing. df dt

At combinations of (B,f) above the diagonal isocline for B, natural growth is less than removals by the fishery and the biomass declines.

0

r/q

Fishing Effort

When B is less than c/(p·q),and to the left of the vertical isocline for f, the fishing boats do not cover their fishing costs and boats exit the fishery.

df dt

0

dB dt dB dt

0

0 Biomass

0

c/(p·q)

K

If we specify a starting position for B and f at some initial starting time, we can trace out the path of f(B). We are using a graphical technique to solve the differential equation df dB

a (p q B r B 1



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B K

c) f qf B

BioEcon3 - Page 157

Fishing Effort

Fishing Effort

Here are two examples, each with different values for [ B(0),f(0) ].

Biomass

Biomass

Biomass

Here are graphs of the solutions that correspond to the left hand phase plane diagram.

Fishing Effort

time

time

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BioEcon3 - Page 158

Here are two more examples, each with different values for the entry/exit parameter a. One quarter the original "a" value

Fishing Effort

Fishing Effort

Half the original "a" value.

Biomass

Biomass

Biomass

Here are graphs of the solutions that correspond to the left hand phase plane diagram. With small values for a there is slower entry and exit of boats into the fishery and less tendency for the system to overshoot the equilibrium levels.

Fishing Effort

time

time

Explore the dynamic behavior of the fishery model using the Excel demonstration.

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BioEcon3 - Page 159...