Population Dynamics

Population Dynamics

Mortality, Growth, and More

Mortality, Growth, and More

Fish GrowthFish Growth

• Growth of fish is indeterminate• Affected by:

– Food abundance– Weather– Competition– Other factors too numerous to

mention!

• Growth of fish is indeterminate• Affected by:

– Food abundance– Weather– Competition– Other factors too numerous to

mention!

Fish GrowthFish Growth

• Growth measured in length or weight

• Length changes are easier to model

• Weight changes are more important for biomass reasons

• Growth measured in length or weight

• Length changes are easier to model

• Weight changes are more important for biomass reasons

Growth rates - 3 basic typesGrowth rates - 3 basic types• Absolute - change per unit time -

l2-l1

• Relative - proportional change per unit time - (l2-l1)/l1

• Instantaneous - point estimate of change per unit time - logel2-logel1

• Absolute - change per unit time - l2-l1

• Relative - proportional change per unit time - (l2-l1)/l1

• Instantaneous - point estimate of change per unit time - logel2-logel1

Growth in lengthGrowth in length

Growth in length & weightGrowth in length & weight

von Bertalanffy growth modelvon Bertalanffy growth model

Von Bertalanffy growth modelVon Bertalanffy growth model

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ΔlΔt=K(L∞ − l)

lt = L∞[1− e−K ( t−t0 )]

Ford-Walford PlotFord-Walford Plot

Bluegill in Lake Winona

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8

Age (years)

Total length (inches)

More calculationsMore calculations

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K = −ln(slope)

L∞ =intercept

1− slope

For Lake Winona bluegill:

K = 0.327

L∞ = 7.217 inches

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l5yrs = 7.217[1− e−0.327(5)] = 5.81inches

Predicting length of 5-year-old bluegill:

Weight works, too!Weight works, too!

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W = aLb

wt =W∞[1− e−K (t−t0 )]3

b often is near 3.0

Exponential growth modelExponential growth modelOver short time periods

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W t =W0egt

W0 =

W t =

g =

g = lnW t

W0

Initial weight

Weight at time t

Instantaneous growth rate

Gives best results with weight data, does not work well with lengths

Used to compare different age classes within a population, or the same age fish among different populations

Fish Mortality RatesFish Mortality Rates

• Sources of mortality– Natural mortality

• Predation• Diseases• Weather

• Fishing mortality (harvest)

Natural mortality +Fishing mortality= Total mortality

• Sources of mortality– Natural mortality

• Predation• Diseases• Weather

• Fishing mortality (harvest)

Natural mortality +Fishing mortality= Total mortality

Fish Mortality RatesFish Mortality Rates

• Lifespan of exploited fish (recruitment phase)

• Pre-recruitment phase - natural mortality only

• Post-recruitment phase - fishing + natural mortality

• Lifespan of exploited fish (recruitment phase)

• Pre-recruitment phase - natural mortality only

• Post-recruitment phase - fishing + natural mortality

Estimating fish mortality ratesEstimating fish mortality rates• Assumptions1) year-to-year production constant2) equal survival among all age

groups3) year-to-year survival constant• Stable population with stable age

structure

• Assumptions1) year-to-year production constant2) equal survival among all age

groups3) year-to-year survival constant• Stable population with stable age

structure

Estimating fish mortality ratesEstimating fish mortality rates• Number of fish of a given cohort

declines at a rate proportional to the number of fish alive at any particular point in time

• Constant proportion (Z) of the population (N) dies per unit time (t)

• Number of fish of a given cohort declines at a rate proportional to the number of fish alive at any particular point in time

• Constant proportion (Z) of the population (N) dies per unit time (t)

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ΔNΔt= −ZN

Estimating fish mortality ratesEstimating fish mortality rates

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N t = N0e−zt

N t =

N0 =

z =

t =

Number alive at time t

Number alive initially - at time 0

Instantaneous total mortality rate

Time since time0

Estimating fish mortality ratesEstimating fish mortality ratesIf t = 1 year

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N1N0

= e−z = S

S = probability that a fish survives one year1 - S = A A = annual mortality rateor

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1− e−z = A

Brown Trout Survivorship

0

200

400

600

800

1000

1200

1 2 3 4 5

Age (years)

Number of fish

Recalling survivorshipRecalling survivorship

Brown Trout Survivorship

1

10

100

1000

1 2 3 4 5

Age (years)

Number of fish

Recalling survivorshipRecalling survivorship

Mortality rates: catch dataMortality rates: catch data

• Mortality rates can be estimated from catch data

• Linear least-squares regression method

• Need at least 3 age groups vulnerable to collecting gear

• Need >5 fish in each age group

• Mortality rates can be estimated from catch data

• Linear least-squares regression method

• Need at least 3 age groups vulnerable to collecting gear

• Need >5 fish in each age group

Mortality rates: catch dataMortality rates: catch data

Age(t)

1 2 3 4 5 6

Number(Nt)

100

150

95 53 35 17

2nd edition p. 144

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6 7

Age

Number

1

10

100

1000

0 1 2 3 4 5 6 7

Age

Number

CalculationsCalculationsStart with:

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N t = N0e−zt

Take natural log of both sides:

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ln(N t ) = ln(N0) − zt

Takes form of linear regression equation:

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Y = a+bXY intercept Slope = -z

ln N versus age (t)

y = -0.5355x + 6.125

R2 = 0.9926

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7

Age (years)

ln N (number of fish)

slope

Slope = -0.54 = -z z = 0.54

Annual survival, mortalityAnnual survival, mortality

S = e-z = e-0.54 = 0.58 = annual survival rate

58% chance of a fish surviving one year

Annual mortality rate = A = 1-S = 1-0.58 = 0.42

42% chance of a fish dying during year

Robson and Chapman Method - survival estimateRobson and Chapman Method - survival estimate

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S =T

n +T −1

n =

T =

Total number of fish in sample (beginning with first fully vulnerable age group)

Sum of coded age multiplied by frequency

ExampleExample

Age 2 3 4 5 6

Coded age (x)

0 1 2 3 4

Number(Nx)

150 95 53 35 17

350 total fish

Same data as previous example, except for age 1 fish (not fully vulnerable)

ExampleExample

T = 0(150) + 1(95) + 2(53) + 3(35) + 4(17) = 374

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T = x(Nx )∑

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S =374

350 + 374 −1= 0.52 52% annual survival

Annual mortality rate A = 1-S = 0.48

48% annual mortality

Variability estimatesVariability estimates

• Both methods have ability to estimate variability

• Regression (95% CI of slope)• Robson & Chapman

• Both methods have ability to estimate variability

• Regression (95% CI of slope)• Robson & Chapman

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V (S) = S(S −T −1

n +T −2)

Brown troutGilmore Creek - Wildwood1989-2010

Separating natural and fishing mortalitySeparating natural and fishing mortality• Usual approach - first estimate total

and fishing mortality, then estimate natural mortality as difference

• Total mortality - population estimate before and after some time period

• Fishing mortality - angler harvest

• Usual approach - first estimate total and fishing mortality, then estimate natural mortality as difference

• Total mortality - population estimate before and after some time period

• Fishing mortality - angler harvest

Separating natural and fishing mortalitySeparating natural and fishing mortality

z = F + M

z = total instantaneous mortality rateF = instantaneous rate of fishing mortalityM = instantaneous rate of natural mortality

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N t = N0e−zt = N0e

−(F +M )t = N0e−Fte−Mt

Separating natural and fishing mortalitySeparating natural and fishing mortality

Also: A = u + v

A = annual mortality rate (total)u = rate of exploitation (death via fishing)v = natural mortality rate

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z

A=F

u=M

v

u =FA

z

v =MA

z

Separating natural and fishing mortalitySeparating natural and fishing mortality

May also estimate instantaneous fishing mortality (F) from data on fishing effort (f)

F = qf q = catchability coefficient

Since Z = M + F, then Z = M + qf(form of linear equation Y = a + bX)(q = slope M = Y intercept)

Need several years of data:1) Annual estimates of z (total mortality rate)2) Annual estimates of fishing effort (angler hours, nets)

Separating natural and fishing mortalitySeparating natural and fishing mortality

Once relationship is known, only need fishing effort data to determine z and F

Amount of fishing effort (f)

Total mortality rate (z)

M = total mortality when f = 0

Mortality due to fishing

Abundance estimatesAbundance estimates

• Necessary for most management practices

• Often requires too much effort, expense

• Instead, catch can be related to effort to derive an estimate of relative abundance

• Necessary for most management practices

• Often requires too much effort, expense

• Instead, catch can be related to effort to derive an estimate of relative abundance

Abundance estimatesAbundance estimates

• C/f = CPUE

• C = catch• f = effort• CPUE = catch per unit effort

• Requires standardized effortstandardized effort– Gear type (electrofishing, gill or trap nets, trawls)– Habitat type (e.g., shorelines, certain depth)– Seasonal conditions (spring, summer, fall)

• C/f = CPUE

• C = catch• f = effort• CPUE = catch per unit effort

• Requires standardized effortstandardized effort– Gear type (electrofishing, gill or trap nets, trawls)– Habitat type (e.g., shorelines, certain depth)– Seasonal conditions (spring, summer, fall)

Abundance estimatesAbundance estimates

• Often correlated with actual population estimates to allow prediction of population size from CPUE

• Often correlated with actual population estimates to allow prediction of population size from CPUE

CPUE

Populationestimate

Population structurePopulation structure

• Length-frequency distributions• Proportional stock density

• Length-frequency distributions• Proportional stock density

Proportional stock densityProportional stock density

• Index of population balance derived from length-frequency distributions

• Index of population balance derived from length-frequency distributions

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PSD(%) =number ≥ qualitylength

number ≥ stocklength• 100

Proportional stock densityProportional stock density

• Minimum stock length = 20-26% of angling world record length

• Minimum quality length = 36-41% of angling world record length

• Minimum stock length = 20-26% of angling world record length

• Minimum quality length = 36-41% of angling world record length

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PSD(%) =number ≥ qualitylength

number ≥ stocklength• 100

Proportional stock densityProportional stock density

• Populations of most game species in systems supporting good, sustainable harvests have PSDs between 30 and 60

• Indicative of a balanced age structure

• Populations of most game species in systems supporting good, sustainable harvests have PSDs between 30 and 60

• Indicative of a balanced age structure

Relative stock densityRelative stock density

• Developed to examine subsets of quality-size fish– Preferred – 45-55% of world record length– Memorable – 59-64%– Trophy – 74-80%

• Provide understandable description of the fishing opportunity provided by a population

• Developed to examine subsets of quality-size fish– Preferred – 45-55% of world record length– Memorable – 59-64%– Trophy – 74-80%

• Provide understandable description of the fishing opportunity provided by a population

Weight-length relationshipsWeight-length relationships

• and b is often near 3• and b is often near 3

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W = aLb

Condition factorCondition factor

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K =W • X

L3

K = condition factorX = scaling factor to make K an integer

Condition factorCondition factor

• Since b is not always 3, K cannot be used to compare different species, or different length individuals within population

• Alternatives for comparisons?

• Since b is not always 3, K cannot be used to compare different species, or different length individuals within population

• Alternatives for comparisons?

Relative weightRelative weight

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Wr =W ×100

Ws

W =

Ws =

Weight of individual fish

Standard weight for specimen of measuredlength

Standard weight based upon standard weight-lengthrelations for each species

Relative weightRelative weight

• e.g., largemouth bass

• 450 mm bass should weigh 1414 g

• If it weighed 1300 g, Wr = 91.9• Most favored because it allows for direct

comparison of condition of different sizes and species of fish

• e.g., largemouth bass

• 450 mm bass should weigh 1414 g

• If it weighed 1300 g, Wr = 91.9• Most favored because it allows for direct

comparison of condition of different sizes and species of fish

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log10Ws = −5.316 + 3.191log10 L

YieldYield

• Portion of fish population harvested by humans

• Portion of fish population harvested by humans

YieldYield

• Major variables– 1) mortality– 2) growth– 3) fishing pressure (type, intensity,

length of season)

• Limited by:– Size of body of water– Nutrients available

• Major variables– 1) mortality– 2) growth– 3) fishing pressure (type, intensity,

length of season)

• Limited by:– Size of body of water– Nutrients available

Yield & the Morphoedaphic IndexYield & the Morphoedaphic Index

• 70% of fish yield variation in lakes can be accounted for by this relationship

• Can be used to predict effect of changes in land use

• 70% of fish yield variation in lakes can be accounted for by this relationship

• Can be used to predict effect of changes in land use

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yield ≅TotalDissolvedSolids

MeanDepth

Managing for YieldManaging for Yield

• Predict effects of differing fishing effort on numbers, sizes of fish obtained from a stock on a continuing basis

• Explore influences of different management options on a specific fishery

• Predict effects of differing fishing effort on numbers, sizes of fish obtained from a stock on a continuing basis

• Explore influences of different management options on a specific fishery

Managing for YieldManaging for Yield

• Predictions based on assumptions:• Annual change in biomass of a stock

is proportional to actual stock biomass

• Annual change in biomass of a stock is proportional to difference between present stock size and maximum biomass the habitat can support

• Predictions based on assumptions:• Annual change in biomass of a stock

is proportional to actual stock biomass

• Annual change in biomass of a stock is proportional to difference between present stock size and maximum biomass the habitat can support

YieldYield

Yield modelsYield models

Yield

Total Stock Biomass

B∞½ B∞