Fisheries Management using a population model growth l exponentia called is This or rat death capita per is rate, birth capita per the is increase of rate intrinsic the called is ) ( 0 ) ( 0 rt t m b t e B e B B m b r B m b rB dt dB b<m b=m b>m
Fisheries Management using a population model
Jan 15, 2016
Fisheries Management using a population model. bm. Density dependent birth and death. Per capita birth,death. b=b 0 -b 1 B m=m 0 +m 1 B. b 0. Slope=b 1. m 0. Slope=m 1. K is called the carrying capacity. K. Biomass. B. - PowerPoint PPT Presentation
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Fisheries Management using a population model
growth lexponentia called is This
or
ratedeath capitaper is rate,birth capitaper theis
increase of rate intrinsic thecalled is
)(
0)(
0rttmb
t eBeBB
mb
r
BmbrBdt
dB
b<m b=m b>m
tmbN
Nt)(ln
0
mbr
B
m0
b0
K
Per capita birth,death
b=b0-b1Bm=m0+m1B
Slope=b1
Slope=m1
Biomass
K is called the carrying capacity
)(
0
When
mbdt
dB
KB
Density dependent birth and death
rtN
Nt
0ln
11
0
01100
1010
10
10
and
then
and
when equal are and Since
bm
rK
rKmbmb
KmmKbb
Bmmm
Bbbb
KBmb
We need to define K in terms of the birth and death rates
K
Br
dt
dB
B
bm
rKB
r
mbr
dt
dB
B
Bmbrdt
dB
B
BBmmBbbdt
dB
BmbrBdt
dB
11
since11
1
)(
,)(
0
11
0 ,
0
110
110
1010
Now we need to incorporate K into the population growth model
This is called the logistic equation
K
Br
dt
dB
B1
10
Is called the logistic equation
dt
dB
B
1
B
K
r0slope
K
•per capita rate of increase slows down linearly as the biomass increases and reaches 0 when the carrying capacity (K) is reached.
per capita rate of increase reaches an upper limit of r0 as B approaches 0
It becomes negative when B>K
200
0 1
BK
rBr
K
BBr
dt
dB
dt
dB
BKK/2
?,2
When
0,When
0,0When
dt
dBKB
dt
dBKB
dt
dBB
200
0 1
BK
rBr
K
BBr
dt
dB
dt
dB
BKK/2
4,
2When
0,When
0,0When
0Kr
dt
dBKB
dt
dBKB
dt
dBB
4
0Kr
What kind of growth curve does this equation generate?—logistic growth
dt
dB
BKK/2
4
0Kr
C/t
What would happen to a population at K subjected to a harvest rate of C/t
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
*
dt
dB
BKK/2
4
0Kr
C/t
How great can this harvest rate be and still be compensated for by increased population growth?
*
4
0KrWhy is called the maximum sustainable yield?
dt
dB
BKK/2
4
0Kr
What would happen to a population at K/2 subjected to a harvest rate of4
0Kr
4
0Kris called the Maximum Sustainable Yield (MSY), why?
dt
dB
BKK/2
4
0Kr
Catch rate
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
4
0KrMaximum Sustainable Yield (MSY)
Danger zone
Stable biomass range
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