Population Growth

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Population Growth

• Exponential
• Continuous addition of births and deaths at
  constant rates (b d)
• Such that r b - d

Problem no explicit prediction is
made Solution isolate N terms on left, and
integrate
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Result of the integration
r0.05
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Exponential growth relationships
Slope of Curve on left
Density
Slope of this curve Increases with density
Slope of line r
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Exponential growth, log scale
Linear increase of log values with time is a
sign of exponential growth
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Geometric Growth
Time is measured in discrete (contant)
chunks Simplest approach Generations are the
time unit R0 Average number of offspring
produced per individual, per lifetime-- Factor
that a population will be multiplied by for each
generation. Often called the Net Rate of
Increase.
Time is measured in generations in this equation.
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Relationship between R0 and r
A population growing for one generation should
show the same result using either of the
following equations
Discrete, where T1 generation
Continuous, where tt (t generation time)
If these give the same result, then
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R0 and r
So! Information about R and t can lead us to r
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Ways of finding R0 and t
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Cohort study
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Survivorship calculations
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Fecundity calculations
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Age-specific reproduction
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Generation Time, t
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Approximate r
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Assumptions of exponential or geometric growth
projections
Constant lx and mx schedules This implies that
reproduction and survival will not change with
density This also implies that any changes in
physical or chemical environment have no
influence on survival or reproduction No
important interactions with other species if
age-specific data are used, assume stable age
distribution.
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Suppose we let lx, mx and t vary with density
Bottom line let r (per capita growth rate) vary
with N
r
dN/Ndt
0
K
0
N
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Density-dependent growth
r
dN/Ndt
-r/K
0
N
0
K
Y A BX
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Logistic equation
Predictive form
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Human rates of change vs N
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Projection based on Logistic model
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Earlier US projection, similar approach
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Logistic Examples
Full-loop (2x the bacteria)
Half-loop (half that on right)
Paramecium, 2 species, growing for 8 days at high
ltrgt and low ltlgt resource levels. Scale has been
stretched on right to be equivalent to that on
the left
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More logistic examples
Growth of a zooplankton crust- acean, Moina, at
different temperatures
Growth of flour beetles in flower, In containers
holding different amts of flour
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Drosophila studies
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Evolution of K in Drosophila
Post-radiation
Hybrid
Inbred
Control
Results suggest that K responds to an increase in
genetic variation, And that it changes gradually
through time in response to selection.
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Assumptions of Logistic Growth
Constant environment (r and K are
constants) Linear response of per capita growth
rate to density Equal impact of all individuals
on resources Instantaneous adjustment of
population growth to change in N No interactions
with species other than those that are
food Constantly renewed supply of food in a
constant quantity
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Discrete Model for Limited Growth
Same assumptions, except population grows in
bursts with each Generation-- built-in time
lag Models of this sort show the potential
influence that a time lag can have on population
change.
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Simple model, complex behavior
R 0.1, K 1000
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Simple model, complex behavior
R 1.9, K 1000 Damped oscillation
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Simple model, complex behavior
r 2.2, K 1000 Limit cycle
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Simple model, complex behavior
r 2.5, K 1000 4-point cycle
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Simple model, complex behavior
r 2.58, K 1000 8-point cycle
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Simple model, complex behavior
r 2.7, K 1000 Erratic
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Chaos
r 3, K 1000
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Overshoot, Crash, Extinction
r 3.000072, K 1000
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Concerns about Chaos
Biological populations dont appear to have the
growth capacity to generate chaos, but this shows
the potential importance of time lags. More
complicated models can be even more
sensitive Some systems might be completely
unpredictable
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Evolution of Life Histories
Life history features Rates of birth, death,
population growth Patterns of reproduction and
mortality Behavior associated with
reproduction Efficiency of resource use, and
carrying capacity Anything that affects
population growth
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Patterns
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More patterns
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Tradeoff