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Harvesting and viability. Harvesting ecology. The birth-immigration-death-emigration approach (BIDE). 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? - PowerPoint PPT Presentation
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Harvesting and viability
Harvesting ecology
The birth-immigration-death-emigration approach (BIDE)
ππ‘+ 1=ππ‘+π΅+πΌ βπ·βπΈ
Basic questions are:
β’ How does population size change after harvesting?β’ How stable are harvested poplations? β’ How much harvesting is save?β’ How to predict future stock population size? β’ What is the minimal size of a population to survive? β’ What is the maximum sustainable yield
ππππ‘ =ππ (1β ππΎ )
The continuous Pearl β Verhulst logistic growth model
Maximum sustainable yield
The maximum sustainable yield is the largest catch that can be taken from a fishery stock over an indefinite period.
K
K/2
Min
At K/2 a population growths fastest. This is the point of maximum sustainable yield.
Minimum viable population size
Carrying capacity
Optimum sustainable yield
ππππ‘ =ππ (1β ππΎ )
Pearl β Verhulst logistic growth model
ππππ‘ =ππ (1β ππΎ )βπ
The stationary point is given by:
ππππ‘ =0=ππ (1β ππΎ )βπ
ππ (1β ππΎ )=π π=π ββπ2β 4πππΎ
2ππΎπ2β 4 πππΎ >0
π>4ππΎ
The classical constant harvesting model of fisheries
Population increase is reduced by a constant amount m.
Under constant harvesting a population is only stable if the reproduction rate r is larger than 4m/kThis is a necessary but not a sufficient condition.
π<ππΎ4
r = 2.6; K = 1000; m = 0 r = 2.6; K = 1000; m = 200
The effect of constant harvesting
Constant harvesting below the extinction threshold tends to stabilize populations.
r = 2.6; K = 1000; m = 240
4m/K = 0.96Even below the 4m/K threshold populations might go extinct.
r K m 4m/K2.6 1000 200 0.8
t N DeltaN0 100 341 134 101.71442 235.7144 268.39813 504.112516 449.956
4 954.068543 -86.06345 868.005114 97.88781
6 965.892927 -114.3467 851.546756 128.6787
8 980.22544 -149.6039 830.62261 165.7906
10 996.413203 -190.70811 805.705425 207.0149
12 1012.72033 -233.49413 779.22678 247.2843
14 1026.51103 -270.756=B19+1 =C19+D19 =$B$2*C20*(1-C20/$C$2)-$D$2
0 20 40 60 80 1000
20
40
60
80
100
120
Time
N
Going Excel
ππ‘+1=ππ‘+π ππ‘ (1β ππ‘
πΎ )βπThe discrete form of logistic growth
The effect of proportional harvesting
ππππ‘ =ππ (1β ππΎ )β ππ
Proportionality term
ππππ‘ =0=ππ (1β ππΎ )β ππ
π=πΎ (1β ππ )
π <π
Under proportional harvesting a population is only stable if the harvesting rate is smaller than the reproduction rate
This is a necessary but not a sufficient condition.
r = 2.6; K = 1000; f = 0
r = 2.6; K = 1000; f = 0.7
Proportional harvesting might also stabilize populations.
r K m noise1.8 1000 2.5 50
t N DeltaN0 100 -88 -1.809551 10.19044703 -7.320234302 19.079622 21.9498354 -16.23211627 -5.936263 -0.218539649 0.152891787 13.898214 13.83256185 -10.02720487 -19.97655 -16.17109766 10.84906044 24.488736 19.1666934 -14.07793722 11.572537 16.66129058 -12.16258089 23.003388 27.50209161 -20.61292121 -16.55649 -9.667181497 6.598809131 -8.63398
10 -11.70234759 7.945142425 13.56866=B15+1 =C15+D15+E15 =$B$2*C16*(1-C16/$C$2)-$D$2*C16 =+$E$2*LOS()-$E$2/2
The influence of noise (stochastic environmental fluctuations)
We model stochastic fluctuations by an additional noise term
ππ‘+1=ππ‘+π ππ‘ (1β ππ‘
πΎ )βπ+πππ(βπ ,π)
r = 1.8; K = 1000; f = 2.5; noise = 50r = 1.8; K = 1000; f = 2.5; noise = 0
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
The problem of recruitmentππππ‘ =ππ (1β ππΎ )β ππ
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 r0 14071 760 -647 2.6698542 672 -88 -0.31673 1152 480 1.6225254 805 -347 -7.467435 865 60 0.2276666 681 -183 -0.759577 1271 589 2.0012458 510 -761 10.157629 1387 878 2.994006
10 1360 -27 0.12443111 914 -446 2.45425112 1274 360 1.65318513 792 -482 6.12922214 967 175 0.64982515 547 -419 -2.23196
Mean 1.327212
K1200
We use long term abundance data to estimate r.
π=πΎ βπ
π (πΎβπ )
π =ππ=3.77
ππππ‘ =π π πΎβπ
πΎ =ππ (1β ππΎ )=ππ β ππΎ π2
The continuous form of the growth function
π=π0πΎππ (π‘β π‘0 )
(πΎ βπ 0)+π 0ππ (π‘ βπ‘ 0)
First order quadratic differential equation
π=πΎ
1+πΎ βπ 0
π0πβπ (π‘β π‘0 ) π β πΎ
1+πΎπβπ (π‘ βπ‘ 0)
N0=1, K>>1
ππππ‘ =π π (1β(ππΎ )
π) The generalized logistic growth model
π=πΎ
(1+ππβπ π(π‘ βπ‘ 0))1π
-1
Applications:Cancer cell growth (bβ0)
π=πΎ πβππβπ (π‘β π‘ 0)
Gompertz growth functionbβ0
Human demographyIndustrial growth processes
The parameter b describes the degree of asymmetry around the inflection point.
=2.8b=1
=2.8b=0.8
=2.8b=1.07
A high degree of asymmetry destabilizes populations
The Gompertz model is not symmetric around the inflection point.
πΎ(1+π)1 /π
π=πΎ πβππβπ (π‘β π‘ 0)
The application of simple population models in fisheries
Safe fishing
Buffer
Stock overfishing
Spaw
ning
bio
mas
s
Mortality due to fishery
Fleet over-
fishing
Harvesting control rule
Stock overfishing is a biological overfishing and occurs at a negative fish growth rate
Fleet overfishing includes the economic aspect and occurs when proft of fishering decreases
ππ‘+1=ππ‘ ππ (1β π π‘
πΎ )
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
ln (πΒΏΒΏ π‘+1)= ln (πΒΏΒΏπ‘ )+πβ ππΎ π π‘ ΒΏΒΏ
This linear function can easily be solved for r and K.
= 2.9K = 500
The Ricker model of fisheries is the discrete counterpart of the continuous logistic model
ππ‘+1=ππ‘ π(πβ π ππ‘
πΎ)+πππ(βπ ,π)
r = 2.0; K = 1000; noise = 0
r = 2.0; K = 1000; noise = 200
ππ‘+ 1=ππ‘ π(πβ π ππ‘
πΎ)+πππ(βπ ,π)β π ππ‘
The Ricker model with proportional harvesting
r = 2.0; K = 1000; f = 0.7; noise = 200
Proportional harvesting predicts equilibrium population sizes below
the carrying capacities.
Overfishing of atlantic cod at the East coast of New Foundland
Population collapse in 1992
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.
Garus morhua
Population bottlenecks
Popu
latio
n siz
e
Time
Population bottleneck
Extinction
Survival
Population bottlenecks might have population consequences
β’ Populations might become too small for successful mating and reproduction
β’ Populations might becomes depressed by inbreeding causing reduced fitness
β’ Populations might become vulnerable to infectuous diseases
β’ Populations might be less able to cope with local habitat changes
Bottlenecks are sharp declines in population size
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.
http://www.vortex9.org/vortex.html
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
Eigenvalue Eigenvector1.046 0.995
0.0950.0180.004
Eggs Larva Imago 1Imago 2Eggs 0 0 50 30Larva 0.1 0 0 0
Imago 1 0 0.2 0 0Imago 2 0 0 0.25 0
N0 N1 N2 N3 N4Eggs 600 1400 675 690 1490Larva 60 60 140 67.5 69Imago 1 10 12 12 28 13.5Imago 2 30 2.5 3 3 7
N0 N1 N2 N3 N4Eggs 600 1050 622.5 690 1140Larva 60 60 105 62.25 69Imago 1 3 12 12 21 12.45Imago 2 30 0.75 3 3 5.25
In this example the population survives
The second year population dies out in the second year. The whole population survives.
Leslie matrix approach
We reduce initial population sizes until the population dies out.
Persistence times and viability analysis in r strategists
What is the probability that a population dies out?
π=2 ln (π )π2 (π ) ( ln (πΎ )β ln (π )
2 )If the population is not density regulated the average persistence time is
Population size
Pers
isten
ce ti
me
r > s2(r)
r < s2(r)
Persistence times increase with population size and decrease with reproductive stochasticity
Orb web spiders on Bahama islands (Schoener 1992) Persistence times of Scandinavian Metrioptera grashoppers (Berggren 2001)
finally extinct finally
surviving
r 2.26s2(r) 2.07K 800
π (π‘+1 ) /π (π‘ )=π
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.
r is calculated at times, where population sizes increase.
π=2 ln (π )π2 (π ) ( ln (πΎ )β ln (π )
2 )
generationsAbsolute population sizes
Typical arthropod species survive locally a few to a few tens of generations.Most species do not have stable populations.
π=2 ln (912)2.07 ( ln (800 )β ln (912)
2 )=22Annual extinction probability π=1βπ
β 1π π=1βπβ 122=0.04
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