1. Harvesting and viability

2. The birth-immigration-death-emigration approach (BIDE)
Harvesting ecology
The birth-immigration-death-emigration approach (BIDE)
𝑁 𝑡+1 = 𝑁 𝑡 +𝐵+𝐼−𝐷−𝐸
Basic questions are:
How does population size change after harvesting?
How stable are harvested poplations?
How much harvesting is save?
How to predict future stock population size?
What is the minimal size of a population to survive?
What is the maximum sustainable yield

3. Maximum sustainable yield
The continuous Pearl – Verhulst logistic growth model
The maximum sustainable yield is the largest catch that can be taken from a fishery stock over an indefinite period.
𝑑𝑁 𝑑𝑡 =𝑟𝑁(1− 𝑁 𝐾 ) K
Carrying capacity
At K/2 a population growths fastest.
This is the point of maximum sustainable yield.
K/2
Optimum sustainable yield
Min
Minimum viable population size

4. The classical constant harvesting model of fisheries
Pearl – Verhulst logistic growth model
𝑑𝑁 𝑑𝑡 =𝑟𝑁(1− 𝑁 𝐾 )
𝑑𝑁 𝑑𝑡 =𝑟𝑁 1− 𝑁 𝐾 −𝑚
Population increase is reduced by a constant amount m.
The stationary point is given by:
𝑑𝑁 𝑑𝑡 =0=𝑟𝑁 1− 𝑁 𝐾 −𝑚
𝑁= 𝑟− 𝑟 2 − 4𝑟𝑚 𝐾 2𝑟 𝐾
𝑟𝑁 1− 𝑁 𝐾 =𝑚
𝑟 2 − 4𝑟𝑚 𝐾 >0
𝑟> 4𝑚 𝐾
𝑚< 𝑟𝐾 4
Under constant harvesting a population is only stable if the reproduction rate r is larger than 4m/k
This is a necessary but not a sufficient condition.

5. The effect of constant harvesting
r = 2.6; K = 1000; m = 0
r = 2.6; K = 1000; m = 200
Constant harvesting below the extinction threshold tends to stabilize populations.
r = 2.6; K = 1000; m = 240
4m/K = 0.96
Even below the 4m/K threshold populations might go extinct.

6. The discrete form of logistic growth
Going Excel r K m
4m/K
2.6
1000
200
0.8 t N
DeltaN
100 34 1
134 2 3 4 5 6 7 8 9 10 11 12 13 14
=B19+1
=C19+D19
=$B$2*C20*(1-C20/$C$2)-$D$2
𝑁 𝑡+1 = 𝑁 𝑡 +𝑟 𝑁 𝑡 1− 𝑁 𝑡 𝐾 −𝑚
The discrete form of logistic growth

7. The effect of proportional harvesting
r = 2.6; K = 1000; f = 0
𝑑𝑁 𝑑𝑡 =𝑟𝑁 1− 𝑁 𝐾 −𝑓𝑁
Proportionality term
𝑑𝑁 𝑑𝑡 =0=𝑟𝑁 1− 𝑁 𝐾 −𝑓𝑁
𝑁=𝐾 1− 𝑓 𝑟
r = 2.6; K = 1000; f = 0.7
𝑓<𝑟
Under proportional harvesting a population is only stable if the harvesting rate is smaller than the reproduction rate
This is a necessary but not a sufficient condition.
Proportional harvesting might also stabilize populations.

8. The influence of noise (stochastic environmental fluctuations)K m
noise
1.8
1000
2.5 50 t N
DeltaN
100
-88 1 2 3 4 5 6 7 8 9 10
=B15+1
=C15+D15+E15
=$B$2*C16*(1-C16/$C$2)-$D$2*C16
=+$E$2*LOS()-$E$2/2
We model stochastic fluctuations by an additional noise term
𝑁 𝑡+1 = 𝑁 𝑡 +𝑟 𝑁 𝑡 1− 𝑁 𝑡 𝐾 −𝑚+𝑟𝑎𝑛(−𝑎,𝑎)

9. r = 1.8; K = 1000; f = 2.5; noise = 0
r = 1.8; K = 1000; f = 2.5; noise = 50
Environmental stochasticity allows populations to survive even under severe harvesting.
r = 2.6; K = 1000; f = 2.5; noise = 0
r = 2.6; K = 1000; f = 2.5; noise = 50
Noise might also destabilize populations

10. The problem of recruitment
𝑑𝑁 𝑑𝑡 =𝑟𝑁 1− 𝑁 𝐾 −𝑓𝑁 K
1200
How to estimate the intrinsic reproduction rate r, that is the proportion of fish reaching reproductive age.
r varies from year to year. t N DN r
1407 1
760
-647 2
672
-88 3
1152
480 4
805
-347 5
865 60 6
681
-183 7
1271
589 8
510
-761 9
1387
878 10
1360
-27 11
914
-446 12
1274
360 13
792
-482 14
967
175 15
547
-419
Mean
We use long term abundance data to estimate r.
𝑟= 𝐾∆𝑁 𝑁(𝐾−𝑁)
𝑅= 𝑒 𝑟 =3.77

11. First order quadratic differential equation
The continuous form of the growth function
𝑑𝑁 𝑑𝑡 =𝑟𝑁 𝐾−𝑁 𝐾 =𝑟𝑁 1− 𝑁 𝐾 =𝑟𝑁− 𝑟 𝐾 𝑁 2
First order quadratic differential equation
𝑁= 𝐾 1+ 𝐾− 𝑁 0 𝑁 0 𝑒 −𝑟(𝑡− 𝑡 0 )
𝑁= 𝑁 0 𝐾 𝑒 𝑟(𝑡− 𝑡 0 ) (𝐾− 𝑁 0 )+ 𝑁 0 𝑒 𝑟(𝑡− 𝑡 0 )
𝑁≈ 𝐾 1+𝐾 𝑒 −𝑟(𝑡− 𝑡 0 )
N0=1, K>>1
𝑑𝑁 𝑑𝑡 =𝑟𝑁 1− 𝑁 𝐾 𝑏
The generalized logistic growth model
𝑁= 𝐾 1+𝑄 𝑒 −𝑟𝑏(𝑡− 𝑡 0 ) 1 𝑏
Applications:
Cancer cell growth (b→0)
𝑄= 𝑁 𝐾 𝑏 -1
b→0
𝑁=𝐾 𝑒 −𝑏 𝑒 −𝑟(𝑡− 𝑡 0 )
Gompertz growth function
Human demography
Industrial growth processes
The parameter b describes the degree of asymmetry around the inflection point.

12. A high degree of asymmetry destabilizes populations
𝑁=𝐾 𝑒 −𝑏 𝑒 −𝑟(𝑡− 𝑡 0 )
The Gompertz model is not symmetric around the inflection point.
𝐾 (1+𝑏) 1/𝑏
A high degree of asymmetry destabilizes populations
𝑟=2.8
b=1
𝑟=2.8
b=0.8
𝑟=2.8
b=1.07

13. Harvesting control rule
The application of simple population models in fisheries
Harvesting control rule
Stock overfishing is a biological overfishing and occurs at a negative fish growth rate
Safe fishing
Fleet over-fishing
Spawning biomass
Buffer
Fleet overfishing includes the economic aspect and occurs when proft of fishering decreases
Stock overfishing
Mortality due to fishery
𝑟 = 2.9
K = 500
The discrete Ricker model of fisheries is a special case of the Gompertz model and is the standard model of stock forecasting in the EU
𝑁 𝑡+1 = 𝑁 𝑡 𝑒 𝑟 1− 𝑁 𝑡 𝐾
ln⁡(𝑁 𝑡+1 )= ln⁡(𝑁 𝑡 )+𝑟− 𝑟 𝐾 𝑁 𝑡
This linear function can easily be solved for r and K.

14. The Ricker model of fisheries is the discrete counterpart of the continuous logistic model
𝑁 𝑡+1 = 𝑁 𝑡 𝑒 (𝑟− 𝑟 𝑁 𝑡 𝐾 ) +𝑟𝑎𝑛(−𝑎,𝑎)
r = 2.0; K = 1000; noise = 0
𝑁 𝑡+1 = 𝑁 𝑡 𝑒 (𝑟− 𝑟 𝑁 𝑡 𝐾 ) +𝑟𝑎𝑛(−𝑎,𝑎)−𝑓 𝑁 𝑡
The Ricker model with proportional harvesting
r = 2.0; K = 1000; f = 0.7; noise = 200
r = 2.0; K = 1000; noise = 200
Proportional harvesting predicts equilibrium population sizes below the carrying capacities.

15. Overfishing of atlantic cod at the East coast of New Foundland
Population collapse in 1992
Garus morhua
Growth overfishing occurs when fish are harvested at an average size that is smaller than the size that would produce the maximum yield.
Recruitment overfishing occurs when the mature adult population is depleted to a level where it no longer has the reproductive capacity to replenish itself. Ecosystem overfishing occurs when the functioning of the ecological system is altered by overfishing.

16. Bottlenecks are sharp declines in population size
Population bottlenecks
Population bottleneck
Population size
Survival
Extinction
Time
Bottlenecks are sharp declines in population size
Population bottlenecks might have population consequences
Populations might become too small for successful mating and reproduction
Populations might becomes depressed by inbreeding causing reduced fitness
Populations might become vulnerable to infectuous diseases
Populations might be less able to cope with local habitat changes

17. Minimum viable populations
MVPs define the lower bound of population size such that it can survive under natural conditions.
MVP are assessed by viability analysis
The first viability analysis was proposed by Shaffer in 1978 for Yellowstone grizzlys.

18. Viability analyses of K-strategists
The minimum viable population refer to the population size necessary to ensure between 90 and 95% probability of survival between 100 to 1,000 years into the future.
Minimum viable population size is necessary to ensure long term positive population growth: R > 1
Leslie matrix approach
Eggs
Larva
Imago 1
Imago 2 50 30
0.1
0.2
0.25
Eigenvalue
Eigenvector
1.046
0.995
0.095
0.018
0.004 N0 N1 N2 N3 N4
Eggs
600
1400
675
690
1490
Larva 60
140
67.5 69
Imago 1 10 12 28
13.5
Imago 2 30
2.5 3 7
We reduce initial population sizes until the population dies out.
In this example the population survives N0 N1 N2 N3 N4
Eggs
600
1050
622.5
690
1140
Larva 60
105
62.25 69
Imago 1 3 12 21
12.45
Imago 2 30
0.75
5.25
The second year population dies out in the second year. The whole population survives.

19. Persistence times and viability analysis in r strategists
Persistence times increase with population size and decrease with reproductive stochasticity
r > s2(r)
What is the probability that a population dies out?
Persistence time
r < s2(r)
Population size
If the population is not density regulated the average persistence time is
𝑇= 2ln⁡(𝑁) 𝜎 2 𝑟 ln 𝐾 − ln⁡(𝑁) 2
finally extinct
finally surviving
Orb web spiders on Bahama islands (Schoener 1992)
Persistence times of Scandinavian Metrioptera grashoppers (Berggren 2001)

20. 𝑇= 2ln⁡(𝑁) 𝜎 2 𝑟 ln 𝐾 − ln⁡(𝑁) 2
𝑁 𝑡+1 /𝑁 𝑡 =𝑟
r is calculated at times, where population sizes increase. r
2.26
s2(r)
2.07 K
800
𝑇= 2ln⁡(912) ln − ln⁡(912) 2 =22
Absolute population sizes
generations
𝑃=1− 𝑒 − 1 𝑇
𝑃=1− 𝑒 − =0.04
Annual extinction probability
The deciding factor in viability analysis is the variance /mean ratio of the reproduction rate.
The higher the variance /mean ratio is, the higher is the probability of extinction.
The higher the reproduction rate is, the higher is also the probability of extinction.
The carrying capacity is of minor importance for persistence times.
Typical arthropod species survive locally a few to a few tens of generations.
Most species do not have stable populations.