Harvesting and viability

Harvesting ecology

The birth-immigration-death-emigration approach (BIDE)

𝑁𝑡+1=𝑁𝑡+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

𝑑𝑁𝑑𝑡

=𝑟𝑁 (1−𝑁𝐾

)

The continuous Pearl – 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

𝑑𝑁𝑑𝑡

=𝑟𝑁 (1−𝑁𝐾

)

Pearl – Verhulst logistic growth model

𝑑𝑁𝑑𝑡

=𝑟𝑁 (1−𝑁𝐾 )−𝑚

The stationary point is given by:

𝑑𝑁𝑑𝑡

=0=𝑟𝑁 (1−𝑁𝐾 )−𝑚

𝑟𝑁 (1− 𝑁𝐾 )=𝑚 𝑁=𝑟 −√𝑟2− 4𝑟𝑚𝐾

2𝑟𝐾

𝑟2−4 𝑟𝑚𝐾

>0

𝑟>4𝑚𝐾

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/k

This is a necessary but not a sufficient condition.

𝑚<𝑟𝐾4

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

𝑁𝑡+1=𝑁𝑡+𝑟 𝑁𝑡 (1− 𝑁𝑡

𝐾 )−𝑚The discrete form of logistic growth

The effect of proportional harvesting

𝑑𝑁𝑑𝑡

=𝑟𝑁 (1−𝑁𝐾 )− 𝑓𝑁Proportionality term

𝑑𝑁𝑑𝑡

=0=𝑟𝑁 (1−𝑁𝐾 )− 𝑓𝑁

𝑁=𝐾 (1− 𝑓𝑟 )

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.

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

𝑁𝑡+1=𝑁𝑡+𝑟 𝑁𝑡 (1− 𝑁𝑡

𝐾 )−𝑚+𝑟𝑎𝑛(−𝑎 ,𝑎)

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 recruitment

𝑑𝑁𝑑𝑡

=𝑟𝑁 (1−𝑁𝐾 )− 𝑓𝑁How 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.

𝑟=𝐾 ∆𝑁

𝑁 (𝐾−𝑁 )

𝑅=𝑒𝑟=3.77

𝑑𝑁𝑑𝑡

=𝑟 𝑁 𝐾−𝑁𝐾

=𝑟𝑁 (1−𝑁𝐾 )=𝑟𝑁 − 𝑟𝐾𝑁2

The continuous form of the growth function

𝑁=𝑁0𝐾𝑒𝑟 (𝑡− 𝑡0 )

(𝐾 −𝑁 0)+𝑁 0𝑒𝑟 (𝑡 −𝑡 0)

First order quadratic differential equation

𝑁=𝐾

1+𝐾 −𝑁 0

𝑁0

𝑒−𝑟 (𝑡− 𝑡0 ) 𝑁 ≈𝐾

1+𝐾𝑒−𝑟 (𝑡 −𝑡 0)

N0=1, K>>1

𝑑𝑁𝑑𝑡

=𝑟 𝑁 (1−(𝑁𝐾 )𝑏) The generalized

logistic growth model

𝑁=𝐾

(1+𝑄𝑒−𝑟 𝑏(𝑡 −𝑡 0))1𝑏

-1

Applications:Cancer cell growth (b→0)

𝑁=𝐾 𝑒−𝑏𝑒−𝑟 (𝑡− 𝑡

0)

Gompertz growth functionb→0

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.

𝐾(1+𝑏)1 /𝑏

𝑁=𝐾 𝑒−𝑏𝑒−𝑟 (𝑡− 𝑡

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

𝑁𝑡+ 1=𝑁𝑡 𝑒𝑟 (1− 𝑁 𝑡

𝐾 )

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 (𝑁¿¿ 𝑡+1)=ln (𝑁¿¿ 𝑡)+𝑟−𝑟𝐾𝑁 𝑡 ¿¿

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

𝑁𝑡+ 1=𝑁𝑡 𝑒(𝑟−

𝑟 𝑁𝑡

𝐾)+𝑟𝑎𝑛(−𝑎 ,𝑎)

r = 2.0; K = 1000; noise = 0

r = 2.0; K = 1000; noise = 200

𝑁𝑡+ 1=𝑁𝑡 𝑒(𝑟−

𝑟 𝑁𝑡

𝐾)+𝑟𝑎𝑛(−𝑎 ,𝑎)− 𝑓 𝑁𝑡

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 si

ze

Time

Population bottleneck

Extinction

Survival

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

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?

𝑇=2 ln (𝑁 )𝜎2 (𝑟 ) ( ln (𝐾 )− ln (𝑁 )

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

𝑁 (𝑡+1 )/𝑁 (𝑡 )=𝑟

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.

𝑇=2 ln (𝑁 )𝜎2 (𝑟 ) ( ln (𝐾 )− ln (𝑁 )

2 )

generationsAbsolute population sizes

Typical arthropod species survive locally a few to a few tens of generations.Most species do not have stable populations.

𝑇=2 ln (912)2.07 ( ln (800 )− ln (912)

2 )=22Annual extinction probability 𝑃=1−𝑒

− 1𝑇 𝑃=1−𝑒

− 122=0.04