population dynamics
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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Population Dynamics
Mortality, Growth, and More
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Fish Growth• Growth of fish is indeterminate• Affected by:
– Food abundance– Weather– Competition– Other factors too numerous to
mention!
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Fish Growth• Growth measured in length or
weight• Length changes are easier to
model• Weight changes are more
important for biomass reasons
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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
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Growth in length
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Growth in length & weight
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von Bertalanffy growth model
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Von Bertalanffy growth model
€
ΔlΔt=K(L∞ − l)
lt = L∞[1− e−K ( t−t0 )]
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Ford-Walford Plot
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Bluegill in Lake Winona
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8Age (years)
Total length (inches)
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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:
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Weight works, too!
€
W = aLb
wt =W∞[1− e−K (t−t0 )]3
b often is near 3.0
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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
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Fish Mortality Rates• Sources of mortality
– Natural mortality• Predation• Diseases• Weather
• Fishing mortality (harvest)
Natural mortality +Fishing mortality= Total mortality
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Fish Mortality Rates• Lifespan of exploited fish
(recruitment phase)
• Pre-recruitment phase - natural mortality only
• Post-recruitment phase - fishing + natural mortality
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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
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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
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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
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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
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Brown Trout Survivorship
0
200
400
600
800
1000
1200
1 2 3 4 5Age (years)
Number of fish
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Recalling survivorship
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Brown Trout Survivorship
1
10
100
1000
1 2 3 4 5Age (years)
Number of fish
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Recalling survivorship
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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
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Mortality rates: catch dataAge(t)
1 2 3 4 5 6
Number(Nt)
100
150
95 53 35 17
2nd edition p. 144
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0
20
40
60
80
100
120
140
160
0 1 2 3 4 5 6 7Age
Number
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1
10
100
1000
0 1 2 3 4 5 6 7Age
Number
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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
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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
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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
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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
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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)
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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
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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)
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Brown troutGilmore Creek - Wildwood1989-2010
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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
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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
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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
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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)
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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
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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
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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)
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Abundance estimates• Often correlated with actual population
estimates to allow prediction of population size from CPUE
CPUE
Populationestimate
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Population structure• Length-frequency distributions• Proportional stock density
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Proportional stock density• Index of population balance
derived from length-frequency distributions
€
PSD(%) =number ≥ qualitylengthnumber ≥ stocklength
• 100
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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
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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
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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
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Weight-length relationships
• and b is often near 3
€
W = aLb
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Condition factor
€
K =W • XL3
K = condition factorX = scaling factor to make K an integer
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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?
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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
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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
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Yield• Portion of fish population
harvested by humans
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Yield• Major variables
– 1) mortality– 2) growth– 3) fishing pressure (type, intensity,
length of season)• Limited by:
– Size of body of water– Nutrients available
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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
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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
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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
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Yield
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Yield models
Yield
Total Stock BiomassB∞
½ B∞