Population Dynamics Mortality, Growth, and More. Fish Growth Growth of fish is indeterminate...

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

€

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

€

K = −ln(slope)

L∞ =intercept

1− slope

For Lake Winona bluegill:

K = 0.327

L∞ = 7.217 inches

€

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

Predicting length of 5-year-old bluegill:

Weight works, too!Weight works, too!

€

W = aLb

wt =W∞[1− e−K (t−t0 )]3

b often is near 3.0

Exponential growth modelExponential growth modelOver short time periods

€

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)

€

ΔNΔt= −ZN

Estimating fish mortality ratesEstimating fish mortality rates

€

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

€

N1N0

= e−z = S

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

€

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:

€

N t = N0e−zt

Take natural log of both sides:

€

ln(N t ) = ln(N0) − zt

Takes form of linear regression equation:

€

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

€

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

€

T = x(Nx )∑

€

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

€

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

€

N t = N0e−zt = N0e

−(F +M )t = N0e−Fte−Mt


Also: A = u + v

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

€

z

A=F

u=M

v

u =FA

z

v =MA

z


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)


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


• 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)


• 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

€

PSD(%) =number ≥ qualitylength

number ≥ stocklength• 100


• 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

€

PSD(%) =number ≥ qualitylength

number ≥ stocklength• 100


• 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

€

W = aLb

Condition factorCondition factor

€

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

€

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

€

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

€

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∞

Population Dynamics Mortality, Growth, and More. Fish Growth Growth of fish is indeterminate...

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