AMATH 383 Solution #5
Winter 2010
Problem 9.6
6.
For an open
‐
access fishery, the profit a fisherman can earn is governed by the following
equation:
()
Pt
pH cE
Where
H
is the harvest rate governed by:
H
tq
E
N
The change in Effort expended by fisherman is directly proportional to the
amount of profit the will earn.
Subbing in the above equations, you get the
following differential equation based on Effort and the amount of fish:
(, ) (
)
(
)
.
dE
f E N
a pqEN
cE
aE pqN
c
dt
(0.1)
The fish population is governed by the following equation:
(1
)
dN
N
rN
H
dt
K
By subbing in H(t) above, we get the following differential equation governing
the fish population:
(
,
)
(
1)
[
(
dN
N
N
gEN
rN
qEN Nr
qE
dt
K
K
]
.
(0.2)
To determine the equilibrium (E*,N*) of the coupled equations above, we set
f(E,N) = g(E,N)=0
and solve for each variable:
**
*
*
*
(, )0 (
)
f
E N
a pqE N
cE
*
*0
,
o
r
c
EN
pq
(0.3)
*
*
*
(, )0
(
1
)
N
gE N
qEN
K
*
*
,
o
r
*
0 and
*
, or
) and
*
.
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 Spring '10
 HongQian
 dt, Stability theory

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