20000
Collared Dove in Britain
Number of
Eurasian
Collared Dove
15000
10000
5000
0
1955
10000
Number of
Eurasian
Collared Dove
For purely
geometric or
exponential
growth, log
graphs show
a straight
i h liline
1960
Year
Collared Dove in Britain
1000
100
10
1
1955 1957 1959 1961 1963
Year
Doubling Time is a good way to compare
growth rates of different species:
Tdouble=ln(2)/r ; (can also calculate from λ)
Species
r
Tdouble
T Phage
300
3.3 Minutes
„ E. Coli
59
17 Minutes
„ Paramecium 1.6
10.5 Hours
„ Brown Rat
0.012
57 Days
„ Milk Cow
0.001
1.9 Years
„ Beech Tree
0.0001 19 years
„ Humans
0.00005 38 Years
„
http://www.sdgfp.info/Wildlife/hunting/Pheasant/Pheasant11%20(Tourism).jpg
Logistic growth
„
„
„
„
„
„
„
Limits to growth
As population size increases
you’d think growth would
increase!
Does fig 11.15 apply to humans?
Do humans have a carrying
capacity? Births don’t
don t always
equal deaths.
Why do they level off?
Do invasive species fit the
logistic?
How do you determine K?
Don’t limiting factors vary, so the
plateau would wiggle?
Logistic growth
„
Pheasants on an Island with no predators,
lots of food
Extrapolation from first year’s growth
What really happened
Logistic Growth
dN/dt = rN
„
Rapid growth
(exponential or
geometric) when
population is small,
stabilizes at the
carrying capacity
www.hartnell.edu/faculty/nwheat/38.%20Animal%20Ecology.ppt
•The “Logistic” growth
equation was
developed by Pierre
Verhulst in 1844.
•“Logistic”
probably refers to
its ‘logical’
predictions, or to
the ‘logistics’ of
resource supply
http://en.wikipedia.org/wiki/Pierre_Fran%C3%A7ois_Verhulst
“Pastijn, Hugo 2006. Chaotic growth with the Logistic Model of P.-F Verhulst. Pp 3-11 in
“Understanding complex systems, - http://www.springerlink.com/content/w727x8h20018uh12
1
dN/dt = rN
Logistic Growth
requires a decrease
in ‘realized’ r when
populations get
large
Two ways to graph Logistic Growth
‘r’ with no subscript
is often meant to be
rrealized, not rmax
dN/dt = rmaxN (1-N/K)
(1/N)*(dN/dt) = rmax
(1-N/K)
(1/N)*(dN/dt) = dN/Ndt = rrealized
http://www.geo.arizona.edu/Antevs/nats104/00lect20.html
500000
What is K for this
population?
HIV Prevalence
(N people living with
HIV)
US
400000
(in 100,000s)
300000
„
„
„
„
„
Which is the best
term to describe
these data?
200000
„
100000
0
1985
A)) Exponential
p
B) Geometric
C) Logistic
D) Density Independent
E) Malthusian
1000000
HIV Prevalence (N
people living with HIV
„
1990
1995
Year
2000
2005
„
US
100000
10000
„
1000
100
dN/dt = Total #
individuals added to
the population
(1/N)*(dN/dt) = per
capita
p individuals
added to the
population = rrealized
rmax = maximum per
capita contribution to
population growth
0.6
US Aids Cases
0.5
Re
ealized r (dN/Ndt)
HIV prevalence in US
http://www.utm.edu/departments/cens/biology/rirwin/441_442/441loggrowth.htm
0.4
0.3
0.2
01
0.1
0
-0.1
-
200,000
400,000
600,000
HIV Prevalence
(N Living with AIDS)
10
1
1985
1990
1995
Year
2000
2005
http://www.cdc.gov/hiv/topics/surveillance/resources/slides/index.htm
„
„
„
„
„
What is the best
estimate of “K” for
this population?
A) 0.3
B) 200,000
C) 400,000
D) 600,000
E) Not enough
information
0.6
Why does K/2 give the highest population
growth rate?
US Aids Cases
0.5
0.4
0.3
0.2
01
0.1
0
-0.1
-
200,000
400,000
600,000
dN/dt
„
Re
ealized r (dN/Ndt)
What is K for this
population?
HIV Prevalence
(N Living with AIDS)
N
2
Spotted Salamander (Ambystoma maculatum) –
UA Field Station, Bath Nature Preserve.
Grizzlies in Yellowstone
Data From Dr. Peter
Niewiarowski, UA
Biology
The Logistic Model and Real Populations
Carrying capacity can vary over time
„
„
Some populations
fluctuate greatly
around K.
K is the population size supportable by
available resources and conditions
‰
‰
„
Not necessarily the ideal conditions!
Births = Deaths
Usually
y set byy the limiting
g resource or
condition
‰
Liebig’s law
www.laredo.edu/science/rviswanath/SSI%202005/lecture39.pp
Reindeer of St. Matthew Island
„
„
„
„
Eat lichens
(slow growing)
No predators
Ate so much
lichen that it
couldn’t
regenerate
“overshoot” K
Density Dependence
„
„
Why would abiotic factors be density
independent (later claimed not)
Which are which?
Klein 1968. Iournal of Wildlife
Management, Vol. 32, No. 2,
April 1968
http://www.druidry.org/obod/lore/animal/stag/reindeer_mig.jpg
3
Density-Dependent vs. Density-Independent?
„
The book oversimplifies this
Density Dependent:
„ Density Independent :
‰
‰
‰
Growth rate is higher at
low density
Growth rate is less at
high density
Often biotic factors
„
„
„
‰
Resource limitation
(competition)
Predation
Behavior (movement away or
towards)
Also some biotic factors
„
Hiding places….
‰
‰
„
„
„
‰
How does the environment limit population
growth by changing birth and death rates?
„
Growth rate does not vary
predictably with density
Often abiotic factors
„
Drought
Freeze
Hurricane
Floods….
Births and Deaths
balance each other
at “K”
Birth Rate
Rate
Also some biotic factors
„
„
Realized r
„
Diseases that kill all
Variation in food supply due to
variation in weather
Death Rate
Population Size
K
Population Size
How does the environment limit population
growth by changing birth and death rates?
„
Births and Deaths
balance each other
at “K”
Rate
Daphnia
Birth Rate
Death
Rate
Birth Rate
Population
Size
Rate
K
Death Rate
Rate
Population Size
K
Birth Rate
Death Rate
Population Size
The many meanings of ‘r’
„
Terms related to per capita population growth rate
‰
‰
‰
‰
‰
r = Intrinsic rate of increase = per capita rate of increase = per
capita rate of population growth
One more thing: Two similar
but very different terms
„
rmax = maximum r in the absence of constraints
Logarithmic-scale on a graph
‰
r realized = realized r = r actually occurring now = dN/Ndt =
1/N*dN/dt
Ro = Net Reproductive rate= number of offspring per individual
λ = finite rate or increase = geometric rate of increase =
population multiplication factor
‰
∆N = change in population size
‰
dN/dt = Rate of population growth
„
„
K
(vs. r above; note that dN/dt = rN)
‰
„
Axis goes up by a multiple with
each step (e.g., 1, 10, 100,
1000…)
Helps determine if the pattern is
multiplicative
Logistic growth
‰
S-shaped curve in population
growth, caused by density
dependence
They are all at least partly distinctive and different
4
Human Population Growth
„
„
„
Human Population Growth
Interesting that we inhabit most of earth
How can projections be made when
technology (food production, housing)
changes?
See next presentation
http://www.ecorazzi.com/2009/04/13/sir-david-attenborough-joins-org-working-to
population-growth/
Doubling times across the globe
r
Doubling time
Africa
0.029
24 yr
South America/Caribbean
0.020
35 yr
y
North America
0.007
98 yr
Asia
0.017
41 yr
Europe
0.001
1025 yr
Oceania
0.012
57 yr
Continent
http://en.wikipedia.org/wiki/Overpopulation
Human Population Growth
How long can this rate of growth be sustained?
5
How can human populations be limited?
6