Transcript Lecture 29 Fisheries Mgmt.ppt

Fisheries Management using a population model
dB
 rB  (b  m) B
dt
r is called the intrinsic rate of increase
b is the per capita birth rate, m is per capita death rate
Bt  B 0e (b  m )t or B 0e rt
This is called exponentia l growth
b<m
b=m
b>m
Nt
ln
 (b  m)t
N0
r bm
Density dependent birth and death
Per capita
birth,death
When B  K
b=b0-b1B
m=m0+m1B
b0
Slope=b1
m0
Slope=m1
dB
0
dt
(b  m)
K is called the
carrying capacity
B
K
Biomass
Nt
ln
 rt
N0
We need to define K in terms of the birth and death rates
Since b and m are equal when B  K
and
b  b0  b1B
m  m0  m1B
then b 0  b1K  m0  m1K
and b0  m0  b1  m1K  r 0
r0
K
m1  b1
Now we need to incorporate K into the population growth model
dB
 rB  (b  m) B,
dt
dB
 (b0  b1B   m0  m1B ) B
dt
1 dB
 r 0  b1  m1B
B dt




1 dB
r0


b1  m1
 r 0 1 
B  , since K 


m1  b1
B dt
r0


1 dB
 B
 r 0 1  
B dt
 K
This is called the logistic equation
•per capita rate of
increase slows down
linearly as the biomass
increases and reaches 0
when the carrying
capacity (K) is reached.
1 dB
 B
 r 0 1  
B dt
 K
Is called the logistic equation
1 dB
B dt
per capita rate of increase
reaches an upper limit of
r0 as B approaches 0
r0
slope 
K
It becomes negative
when B>K
B
K
dB
 B
 r 0 B 1  
dt
 K
r0 2
 r0B  B
K
dB
dt
K/2
K
dB
When B  0,
0
dt
dB
When B  K ,
0
dt
K dB
When B  ,
?
2 dt
B
dB
 B
 r 0 B 1  
dt
 K
r0 2
 r0B  B
K
dB
dt
dB
When B  0,
0
dt
dB
When B  K ,
0
dt
K dB r 0 K
When B  ,

2 dt
4
r0K
4
K/2
K
B
What kind of growth curve does this equation generate?—logistic growth
What would happen to a population at K subjected to a harvest rate of C/t
dB
dt
r0K
4
C/t
K/2
*K
B
The population would be reduced which would increase dB/dt
The decrease would continue until it reaches * where
dB/dt increases enough to offset the harvest rate
How great can this harvest rate be and still be compensated for by increased
population growth?
dB
dt
r0K
4
C/t
K/2
Why is
r0K
4
*K
B
called the maximum sustainable yield?
What would happen to a population at K/2 subjected to a harvest rate of
dB
dt
r0K
4
K/2
r0K
4
K
B
is called the Maximum Sustainable Yield (MSY), why?
r0K
4
Constant Quota fishing at levels approaching the MSY shortens the biomass
range the population will recover, and the likelihood of entering the danger zone
increases. Once the danger zone is entered fishing must stop or be severely
curtailed
dB
dt
Danger zone
Stable biomass range
r0K
4
Catch rate
K/2
r0K
4
K
Maximum Sustainable Yield (MSY)
B