Download Population Growth and Interactions

10/7/2010
Does exponential growth occur in
nature?
Population Growth and
Interactions
YES!
1. Monk Parakeet
Van Bael and Pruett-Jones 1996. Wilson
Bulletin 108:584-588.
1960’s – started appearing
1970’s – so widespread USFWS began control
measures
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10/7/2010
Used CBC data to estimate population growth
Chicago population has tripled in 3 years
1975 – population reduced ½ so control ceased
Why successful??
population
p
ggrowingg at exponential
p
Since then p
rate
r = 0.146 (population doubling time 4.8 years)
Nest is unique – only
species doesn’t nest in
cavities
Nests can contain upto
200 nest chambers
Exponential growth is
unrealistic
2. White-tailed Deer
Introduced into George Reserve in Michigan
1928 – 2 bucks, 4 does (480 ha enclosure)
1934 – 164 deer!
„
„
Growth without limits is physically impossible
For short periods populations may follow
exponential growth
laboratory populations (E.
(E. coli in test tube)
Introduced species (while spreading)
„ Human populations at various times
„
„
Natural populations may grow at exponential rate
but for only short periods of time.
When?
May be important in populations
-During process of establishment
-During exploitation of favorable resources
But growth doesn’t last forever
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Populations in nature
„
„
„
Do not explode routinely. Why?
Hypotheses
dN / dt varies over time or space, averages 0
„
„
Density dependent population growth
„
„
„
resource depletion
direct interference
„ predation, parasitism
„ emigration
Density independent population
pop lation gro
growth
th
„
„
dN / dt declines as N increases
„
Density dependent population growth
Resource depletion
„
„
„
„
Resource -- something necessary and capable of
being depleted (e.g., food, water, space)
As N increases resource per capita decreases
May reduce b
May increase d
Direct interference
„
„
„
„
Predation & parasitism
„
„
„
„
Predators learn to attack more common prey
As N increases attacks or pathogen
transmission may increase
May increase d (obvious)
May reduce b (time spent hiding)
As N goes up, dN / N dt goes down
Stabilizes population
Why might dN / N dt decrease with N ?
Interference - direct harm of one individual by
another
As N increases encounters and aggression
increase
May reduce b … harm, waste of time
May increase d … harm, cannibalism
Emigration
„
„
Leave when it gets crowded
As N increases proportion leaving increases
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10/7/2010
Alternative ways of expressing
logistic growth
„
Logistic population growth
K
dN / N dt = r0 [ (K
(K - N ) / K ]
„
per capita population growth
„
dN / dt = r0N [ (K
(K - N ) / K ]
„
Nt = K / [ 1 +[(K
+[(K-N0 ) / N0 ] e-r0 t ]
„
„
Nt
totall population
l i growth
h
at any point
slope of curve = dN / dt
population size
t
Logistic population growth
„
Logistic population growth
When N = K , dN / N dt = 0, no population growth
•
•
•
K = carrying capacity
density at which population growth ceases
r0 = intrinsic rate of increase
dN / dt
K/2
Logistic population growth
„
dN / N dt = r0 [(
[(K
K - N) / K ]
Interpreting (K
(K - N ) / K
Nt
Assumptions of logistic growth model
„
„
K
K is constant over time
„
does not vary year to year etc.
„
dN / Ndt declines linearly with N
„
Effect of density N on dN / Ndt is instantaneous
… no delays
„
„
„
alternative … nonlinear decline
alternative … density now affects dN / Ndt some time
in the future (time
(time lag)
lag)
Continuous overlapping generations
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10/7/2010
Logistic growth: Real data
Laboratory populations of
Paramecium aurelia &
Paramecium caudatum
Laboratory population of
Drosophila melanogaster
Filling carrying capacity
t=0
Filling carrying capacity
t=0
t=1
ΔN / NΔt
=4/1
=4
Filling carrying capacity
t=0
t=1
t=2
ΔN / NΔt
= 4 /1
=4
ΔN / NΔt
=8 / 5
= 1.6
Filling carrying capacity
t=0
t=1
t=2
ΔN / NΔt
= 4 /1
=4
ΔN / NΔt
=8 / 5
= 1.6
t=3
ΔN / NΔt
= 8 / 13
= 0.62
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10/7/2010
Logistic growth: Numerical Example
Logistic growth: Numerical Example
(let r0 = 0.10, K = 100)
(let r = 0.10, K = 100)
N
1
10
50
90
99
100
110
(K-N)/K
99/100 = 0.99
90/100 = 0.90
50/100 = 0.50
0 50
10/100 = 0.10
1/100 = 0.01
0/100 = 0
-10/100 = -0.10
dN / dt
0.099
0.900
2 500
2.500
0.900
0.099
0
-1.100
2.5
dN / N dt
0.099
0.090
0 050
0.050
0.010
0.001
0
-0.010
dN / dt
d
Nt
0
Logistic growth: Numerical Example
K / 2=50
K=100
Logistic population decline
(let r0 = 0.10, K = 100)
r0 = 0.10
K
dN /Ndt
Nt
Nt
0
K / 2=50
K=100
t
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