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