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Population Ecology: Population Dynamics
Image from Wikimedia Commons
Global human population
United Nations projections (2004)
(red, orange, green)
U. S. Census Bureau modern (blue)
& historical (black) estimates
The demographic processes that can change population size:Birth, Immigration, Death, Emigration
B. I. D. E. (numbers of individuals in each category)
Population Dynamics
Nt+1 = Nt + B + I – D – E
For an open population, observed at discrete time steps:
For a closed population, observed through continuous time:
dN
dt = (b-d)N
dN = rN
(b-d) can be considered a proxy for average per capita fitness
dt
Population Dynamics
5 main categories of population growth trajectories:
Exponential growthLogistic growth
Population fluctuationsRegular population cycles
Chaos
Population Dynamics
Cain, Bowman & Hacker (2014), Fig. 11.5
Invariant density-dependent vital rates
Stable equilibrium carrying capacity
Deterministic logistic growth
r
dN
dt = rN
N
K 1 –
Population Dynamics
Cain, Bowman & Hacker (2014), Fig. 11.5
Deterministic vs. stochastic logistic growth
Invariant density-dependent vital rates
“Fuzzy” density-dependent vital rates
Stable equilibrium carrying capacity
Fluctuating abundance within a range of values for carrying
capacity
r ri
Population Dynamics
Cain, Bowman & Hacker (2014), Fig. 11.10
dN
dt = rN
N(t-)
K 1 –
Instead of growth tracking current population size (as in logistic), growth
tracks density at units back in time
Time lags can cause delayed density dependence,
which can result in population cyclesIf r is small,
logistic
If r is intermediate,damped oscillations
If r is large,stable limit cycle
Sir Robert May, Baron of Oxford
Population Dynamics
Time lags can cause delayed density dependence, which can result in population cycles or chaos
Photo from http://www.topbritishinnovations.org/PastInnovations/BiologicalChaos.aspx
Population Dynamics
Per capita rate of increase
Pop
ulat
ion
size
(s
cale
d to
max
. si
ze a
tta
inab
le)
Population cycles & chaos
Is the long-term expected per capita growth rate (r) of a population simply an average across years?
At t0, N0=100t1 is a bad year, so N1 = N0 + (rbad* N0) = 50t2 is a good year, so N2 = N1 + (rgood*N1) = 75
Consider this hypothetical example:rgood = 0.5; rbad = -0.5
If the numbers of good & bad years are equal, is the following true?rexpected = [rgood + rbad] / 2
Variation in r and population growth
The expected long-term r is clearly not 0 (the arithmetic mean of rgood & rbad)!
Variation in and population growth
Cain, Bowman & Hacker (2014), Analyzing Data 11.1, pg. 258
Nt+1 = Nt
=Nt
Nt+1
1.21
0.87
1.17
1.02
1.13
Arithmetic mean = 1.02
Geometric mean = 1.01
A fluctuating population
Variation in and population growth
Cain, Bowman & Hacker (2014), Analyzing Data 11.1, pg. 258
Nt+1 = Nt
=Nt
Nt+1
1.02
1.02
1.02
1.02
1.02
Arithmetic mean = 1.02
Geometric mean = 1.02
A steadily growingpopulation
1.02
1.02
1040
1061
1082
1104
1126
1020
1000
1148
Variation in and population growth
Cain, Bowman & Hacker (2014), Analyzing Data 11.1, pg. 258
Nt+1 = Nt
=Nt
Nt+1
1.01
1.01
1.01
1.01
1.01
Arithmetic mean = 1.01
Geometric mean = 1.01
A steadily growingpopulation
1.01
1.01
1020
1030
1040
1051
1061
1010
1000
1072
Which mean (arithmetic or geometric) best
captures the trajectory of the fluctuating population (the example given in the
textbook)?
Deterministic r < 0
Genetic stochasticity & inbreeding
Small populations are especially prone to extinction from both deterministic and stochastic causes
Population Size & Extinction Risk
Demographic stochasticity individual variability around r (e.g., variance at any given time)
Environmental stochasticity temporal fluctuations of r (e.g., change in mean with time)
Catastrophes
Each student is a sexually reproducing, hermaphroditic, out-crossing annual plant. Arrange the plants into small
sub-populations (2-3 plants/pop.).
In the first growing season (generation), each plant mates (if there is at least 1 other individual in the population)
and produces 2 offspring.
Offspring have a 50% chance of surviving to the next season. flip a coin for each offspring; “head” = lives, “tail” = dies.
Note that average r = 0; each parent adds 2 births to the population and on average subtracts 2 deaths [self & 1 offspring – since 50% of offspring live and
50% die] prior to the next generation.
Demographic stochasticity
Population Size & Extinction Risk
Environmental stochasticity
Population Size & Extinction Risk
How could the previous exercise be modified to illustrate environmental stochasticity?
Natural catastrophes
Population Size & Extinction Risk
What are the likely consequences to populations of sizes: 10; 100; 1000; 1,000,000
if 90% of individuals die in a flood?
Density (N)K
Zone of Allee Effects
Birth (b)
Death (d)
Rate
?
?
Population Size & Extinction Risk
Allee Effects occur when average per capita fitness declines as a population becomes smaller
Spatially-Structured Populations
Patchy population(High rates of inter-patch dispersal, i.e., patches are well-connected)
Spatially-Structured Populations
Mainland-island model(Unidirectional dispersal from mainland to islands)
Spatially-Structured Populations
Classic Levins-type metapopulation (collection of populations) model(Vacant patches are re-colonized from occupied patches
at low to intermediate rates of dispersal )
Original metapopulation idea from Levins (1969)
occupied
occupied
occupied
occupied
unoccupied
unoccupied
Assumptions of the basic model:
1. Infinite number of identical habitat patches
2. Patches have identical colonization probabilities
(spatial arrangement is irrelevant)
3. Patches have identical local extinction (extirpation)
probabilities
4. A colonized patch reaches K instantaneously (within-patch
population dynamics are ignored)
Spatially-Structured Populations
Classic Levins-type metapopulation (collection of populations) model(Vacant patches are re-colonized from occupied patches
at low to intermediate rates of dispersal )
Original metapopulation idea from Levins (1969)
occupied
occupied
occupied
occupied
unoccupied
unoccupied
dp
dt = cp(1 - p) - ep
c = patch colonization rate
e = patch extinction rate
p = proportion of patches occupied
Key result:metapopulation persistence
requires (e/c)<1
Habitats vary in habitat quality;occupied sink habitats broaden the realized niche
Source-Sink Population Dynamics
Original source-sink idea from Pulliam (1988)
source
source
source
sink
sink
sink