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
l
 K(L  l)
t
lt  L [1  e
K (t t 0 )
]
Ford-Walford Plot
Bluegill in Lake Winona
7
Total length (inches)
6
5
4
3
2
1
0
1
2
3
4
5
Age (years)
6
7
8
More calculations
K  ln( slope)
int ercept
L 
1  slope
For Lake Winona bluegill:
K = 0.327
L∞ = 7.217 inches
Predicting length of 5-year-old bluegill:
l5yrs  7.217[1  e
0.327(5)
]  5.81inches
Weight works, too!
W  aL
b
b often is near 3.0
w t  W  [1  e
K (t t 0 ) 3
]
Exponential growth model
Over short time periods
gt
W t  W 0e
W 0  Initial weight
W t  Weight at time t
g
Gives best results
with weight data,
does not work
well with lengths
Instantaneous growth rate
Wt
g  ln
W0
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
• Assumptions
1) year-to-year production constant
2) equal survival among all age
groups
3) 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)
N
 ZN
t
Estimating fish mortality
rates
N t  N 0e
zt
N t  Number alive at time t
N 0  Number alive initially - at time 0
z
t
Instantaneous total mortality rate
Time since time0
Estimating fish mortality
rates
If t = 1 year
N1
z
e S
N0
S = probability that a fish survives one year
1-S=A
A = annual mortality rate
or
z
1e

A
Brown Trout Survivorship
1200
Number of fish
1000
800
600
400
200
0
1
2
3
Age (years)
4
5
Recalling survivorship
Brown Trout Survivorship
Number of fish
1000
100
10
1
1
2
3
Age (years)
4
5
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 data
Age
(t)
1
2
3
Number 100 150 95
(Nt)
2nd edition p. 144
4
5
6
53
35
17
160
140
120
Number
100
80
60
40
20
0
0
1
2
4
3
Age
5
6
7
1000
Number
100
10
1
0
1
2
4
3
Age
5
6
7
Calculations
Start with:
N t  N 0e
zt
Take natural log of both sides:
ln( Nt )  ln( N0 )  zt
Takes form of linear regression equation:


Y intercept
Y  a  bX
Slope = -z
ln N versus age (t)
6
ln N (number of fish)
5
slope
4
3
y = -0.5355x + 6.125
R2 = 0.9926
2
1
Slope = -0.54 = -z
z = 0.54
0
0
1
2
3
4
Age (years)
5
6
7
Annual 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 estimate
T
S
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
Same data as previous
example, except for age 1
fish (not fully vulnerable)
Example
Age
2
3
4
5
6
Coded 0
age (x)
1
2
3
4
Number 150
(Nx)
95
53
35
17
350 total fish
Example
T   x(N x )
T = 0(150) + 1(95) + 2(53) + 3(35) + 4(17) = 374
374
S
 0.52
350  374 1
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
T 1
V (S)  S(S 
)
n T 2
Brown trout
Gilmore Creek - Wildwood
1989-2010
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 rate
F = instantaneous rate of fishing mortality
M = instantaneous rate of natural mortality
N t  N 0e
zt
 N 0e
(F M )t
Ft Mt
 N 0e e
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
z F M
 
A u
v
FA
u
z
MA
v
z
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
Total mortality
rate (z)
Mortality due to
fishing
M = total mortality
when f = 0
Amount of fishing effort (f)
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 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
Population
estimate
CPUE
Population structure
• Length-frequency distributions
• Proportional stock density
Proportional stock density
• Index of population balance
derived from length-frequency
distributions
number  qualitylength
PSD(%) 
 100
number  stocklength
Proportional stock density
number  qualitylength
PSD(%) 
 100
number  stocklength
• Minimum stock length = 20-26% of
angling world record length
• Minimum quality length = 36-41% of
angling world record length
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
W  aL
b
• and b is often near 3
Brown Trout Length-Weight Relationship
Gilmore Creek (Wildwood) - September 2009
400
350
y = 0.0142x2.9117
R² = 0.98842
Wet weight (g)
300
250
200
150
100
50
0
0
5
10
15
20
Total length (cm)
25
30
35
Condition factor
WX
K
3
L
K = condition factor
X = 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
W  100
Wr 
Ws
W  Weight of individual fish
weight for specimen of measured
W s  Standard
length
Standard weight based upon standard weight-length
relations for each species
Relative weight
• e.g., largemouth bass
log10 Ws  5.316  3.191log10 L
• 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
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
TotalDissolvedSolids
yield 
MeanDepth
• 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
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
½ B∞
Total Stock Biomass
B∞