Population Dynamics

Mortality, Growth, and More

Fish Growth• Growth of fish is indeterminate• Affected by:

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

mention!

Fish Growth• Growth measured in length or

weight• Length changes are easier to

model• Weight changes are more

important for biomass reasons

Growth 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

Growth in length

Growth in length & weight

von Bertalanffy growth model

Von Bertalanffy growth model

€

ΔlΔt=K(L∞ − l)

lt = L∞[1− e−K ( t−t0 )]

Ford-Walford Plot

Bluegill in Lake Winona

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8Age (years)

Total length (inches)

More calculations

€

K = −ln(slope)

L∞ =int ercept1− 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!

€

W = aLb

wt =W∞[1− e−K (t−t0 )]3

b often is near 3.0

Exponential 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 Rates• Sources of mortality

– Natural mortality• Predation• Diseases• Weather

• Fishing mortality (harvest)

Natural mortality +Fishing mortality= Total mortality

Fish Mortality Rates• Lifespan of exploited fish

(recruitment phase)

• Pre-recruitment phase - natural mortality only

• Post-recruitment phase - fishing + natural mortality

Estimating 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

Estimating 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)

€

ΔNΔt= −ZN

Estimating 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 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 5Age (years)

Number of fish

Recalling survivorship

Brown Trout Survivorship

1

10

100

1000

1 2 3 4 5Age (years)

Number of fish

Recalling survivorship

Mortality 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: catch dataAge(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 7Age

Number

1

10

100

1000

0 1 2 3 4 5 6 7Age

Number

CalculationsStart 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 + 6.125R2 = 0.9926

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7Age (years)

ln N (number of fish)

slope

Slope = -0.54 = -z z = 0.54

Annual survival, mortalityS = 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 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

ExampleAge 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)

Example

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 estimates• 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 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

Separating 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

Separating natural and fishing mortality

Also: A = u + v

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

€

zA=Fu=Mv

u =FAz

v =MAz

Separating 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 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 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

Abundance 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)

Abundance estimates• Often correlated with actual population

estimates to allow prediction of population size from CPUE

CPUE

Populationestimate

Population structure• Length-frequency distributions• Proportional stock density

Proportional stock density• Index of population balance

derived from length-frequency distributions

€

PSD(%) =number ≥ qualitylengthnumber ≥ stocklength

• 100

Proportional stock density

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

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

€

PSD(%) =number ≥ qualitylengthnumber ≥ stocklength

• 100

Proportional 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

Relative 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

Weight-length relationships

• and b is often near 3

€

W = aLb

Condition factor

€

K =W • XL3

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

Condition factor• Since b is not always 3, K cannot

be used to compare different species, or different length individuals within population

• Alternatives for comparisons?

Relative 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 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

€

log10Ws = −5.316 + 3.191log10 L

Yield• Portion of fish population

harvested by humans

Yield• 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 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

€

yield ≅TotalDissolvedSolids

MeanDepth

Managing 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

Managing 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

Yield

Yield models

Yield

Total Stock BiomassB∞

½ B∞