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5.5
Ln (Abundance)
5
4.5
Deterministic
Exponential Growth
4
Deviations from exponential
model predicted values due solely
to variations in r.
Stochastic Growth 1
3.5
Stochastic Growth 2
3
Stochastic Growth 3
Growth rate can be estimated by
averaging r’s from smaller
intervals (each time step).
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2.5
Year
0.15
Deviation from mean r
0.1
0.05
0
1980
1985
1990
1995
2000
-0.05
-0.1
-0.15
-0.2
Some final thoughts on
exponential growth:
N
•Exponential growth does not always mean
‘rapid’ growth
•Be clear on the implications of the models you
use / assumptions you make about where
stochasticity occurs.
time
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Population growth models
Carrying capacity (k)
k
N
N
time
time
Classic growth curve,
unlimited resources
Classic growth curve,
limited resources (k)
Population growth
Reproduction, births, natality (B)
Immigration (I)
Emigration (E)
Population
Mortality, death (D)
r or lambda
n t+1 = n t + (B + I) - (D + E)
Gains
Losses
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How Birth rate and/or Mortality could change with density:
Logistic Population Growth (Density dependent growth)
• At higher densities these factors limit
population growth b/c they influence
one or some combination of BIDE.
• This shows up as our measure of
growth rate (lambda, r) changing as a
function of population size (density).
Growth rate is negatively affected by
higher populations.
• And results in a sigmoidal shaped
function of population growth over
time….most typically described as
logistic growth.
• Growth slows as pop size approaches
the maximum number/density that a
given area can sustain—the carrying
capacity (K).
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Logistic Population Growth (Density dependent growth)
There are various models that describe different ways that r changes with
increasing density. We will cover 3.
Ricker (logistic) Model:
Assumes a constant, linear decrease of r as population size increases.
ln(n t+1) = ln (nt) + rmax + b(nt) + F
Where:
b is a parameter measuring the strength of intraspecific competition
rmax is the populations maximum growth rate in the absence of density
dependent competition (what we’ve dealt with up to now)
Rickers (logistic) Growth:
Constant linear decrease in r as nt increases
ln(n t+1)/ ln (nt) = + rmax + b(nt) + F
r max is the y-intercept of this line
0.8000
b is the slope of this line
growth rate (rt)
0.7000
0.6000
0.5000
K (carrying capacity) is the
x-intercept of this line
0.4000
0.3000
0.2000
0.1000
0.0000
0
50
100
150
200
Abundance
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Sometimes useful to think about
recruitment, or yield.
What is added to the population
at each time step (product of rate
and n)
Logistic Population Growth (Density dependent growth)
Gompertz Model:
Similar to Rickers except for underlying assumption of how r changes with
density.
ln(n t+1) = ln (nt) + rmax + b( ln(nt)) + F
Where:
b is a parameter measuring the strength of intraspecific competition
rmax is the populations maximum growth rate in the absence of density
dependent competition (what we’ve dealt with up to now)
Also produces sigmoidal curve
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Gompertz model of DD growth:
Constant linear decrease in r as ln(nt ) increases
ln(n t+1)/ ln (nt) = + rmax + b(ln(nt))+ F
r max is the y-intercept of this line
0.7000
b is the slope of this line
Growth rate (rt)
0.6000
0.5000
0.4000
K (carrying capacity) is the
x-intercept of this line
0.3000
0.2000
0.1000
0.0000
0
1
2
3
4
5
6
ln (Abundance)
Gompertz model of DD growth:
Constant linear decrease in r as ln(nt ) increases
ln(n t+1)/ ln (nt) = + rmax + b(ln(nt))+ F
0.7000
growth rate (rt)
0.6000
Concave relationship between
r and abundance
0.5000
0.4000
0.3000
Where is dd strongest?
0.2000
0.1000
0.0000
0
50
100
150
200
Abundance
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Logistic Population Growth (Density dependent growth)
Theta-logistic model:
DD growth rate is a function of population size raised to the power of θ
ln(n t+1)/ ln (nt) = rmax + b( nt θ) + F
Where:
b is a parameter measuring the strength of intraspecific competition
rmax is the populations maximum growth rate in the absence of density
dependent competition (what we’ve dealt with up to now)
θ is a parameter describing the curvature of
the r and n relationship
Theta-logistic model
The shape of the relationship between growth
rate and population size varies such
that when θ is:
θ=1
0<θ<1
rt
θ>1
rt
n
Rickers
rt
n
n
Lots of DD at
small numbers
Lots of DD at
large numbers
7