population dynamics mortality, growth, and more. fish growth growth of fish is indeterminate...
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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
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
€
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
€
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
€
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
€
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∞
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