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BIOL 4120: Principles of Ecology
Lecture 10: Temporal And
Spatial Dynamics of
Populations
Dafeng Hui
Office: Harned Hall 320
Phone: 963-5777
Email: [email protected]
Temporal and Spatial Dynamics
of populations
Numbers of
gyrfalcons
exported from
Iceland to
Denmark during
1731 and 1770
reflected
population cycles.
Topics (Chapter 12)
10. 1 Fluctuation is the rule for natural
populations
10.2 Temporal variation affects the age
structure of populations
10.3 Population cycles result from time
delays in the response of populations to
their own density
10.4 Metapopulations are discrete
subpopulations linked by movements of
individuals
10.5 Chance event may cause small
population to go extinct
10.1 Fluctuation is the rule for
natural populations
Domestic sheep on the island of Tasmania of Australia. Introduced in
early 1800s and reached the K in about 30 years and varied slightly
What are the causes?
Sensitivity to environmental change and response
time of the population
Sheep: large, greater capacity for homeostasis, and
better resist physical changes
long life, generation overlap, even out the
short-term fluctuation in birth rate
Algae and diatom: short life span (a few days),
rapid turn over, high mortality, population size
depends on continued reproduction, which is
sensitivity to food availability, predation, and
physical conditions. Phytoplankton populations
are intrinsically unstable.
Periodic cycles of some species
Population cycles of grouse in Finland are synchronized across species and
areas (Three species in two areas). (6-7 year cycles)
Populations of four moth species in the
same habitat fluctuate independently
10.2 Temporal variation affects the age
structure of populations
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Variation in population size
over time often leaves its
mark on age structure
Age structure influences the
rate of population growth
Commercial whitefish in
Lake Erie
1944 resulted a large
population
Age distributions of forest trees show the effects of disturbances
on seedling establishment (tree ring count)
Survey in Pennsylvania, 1928 over the past 400 years
10.3 Population cycles result from time
delays in the response of populations to
their own density
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Cycling of populations has been observed
E.g: hare cycles, 11-yr cycle in sunspot
Oscillation and time delays
Oscillation may reflect intrinsic dynamic qualities
of biological systems (some oscillate even with
small environmental fluctuation)
Time delays in responses: high birth rate ->
overshoot population -> high death rate -> low
population.
Time delays and oscillations in discretetime models
ΔN(t)=N(t+1)-N(t)
=RN(t)
R is proportional increase or
decrease in N per unit of time
Let’s make R densitydependent,
ΔN(t)=RN(t) (1-N(t)/K)
=RN(t)/K (K-N(t))
K-N(t) is the difference
between the size of the
population and its carrying
capacity at time t
Damped oscillation, limit cycle
or chaos
Time delays and oscillations in
continuous-time models
Logistic model:
dN/dt=r N(1-N/K) or
dN(t)/dt=rN(t)(1-N(t)/K)
(1-N(t)/K) is a component that
show density dependent
influence by population size
If not by current, but by a
population size at time (t-τ)
dN(t)/dt=rN(t)(1-N(t-τ)/K)
Oscillation depends on (rτ):
r τ=0, no oscillation
r τ=1, damped oscillation,
r τ=2, limit cycles
Cycles in laboratory populations
Water flea experiment
25oC, large oscillation
period of cycle: 60 days
time delays: 12-15 days
(average age give birth:
12-15 days at 25oC)
18oC, no oscillation
reproduction fell off
quickly with increasing density,
and life span was longer than at
25oC. Deaths were more evenly
distributed over all ages and
some individuals gave birth even
at high population densities.
Generation overlap broadly, No
time delay.
Introduction of time delays results in regular
population cycles
Blowfly experiment
AJ Nicholson, Australian
Control population by
provide limited food
supplies to larvae and
unlimited food to adults.
Adults population cycles
from 0 to 4000.
Period 30-40 days
Cause: a time delay in
the responses of
fecundity and mortality
to the density of adults.
A time delayed logistic model
(rt=2.1) provides a good fit
for the blowfly data.
What happens if you do not provide enough food for
adults?
Limited food supplies to adults limited the time delay and results in
the elimination of population cycles
10.4 Metapopulations are discrete
subpopulation linked by movements of
individuals
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Habitat Patches, subpopulations and
metapopulation
Processes contribute to dynamics of
metapopulations
• Growth and regulation of subpopulations within patches
• Colonization of empty patches by migrating individuals
to form new subpopulations
• and the extinction of established subpopulations.
Basic model of metapopulation dynamics
One population is divided into discrete subpopulations, each
subpopulation has a probability of going extinct (e).
If (p) is the fraction of suitable habitat patches occupied by
subpopulations, then subpopulations go extinct at the rate
(ep).
The rate of colonization of empty patches depends on the
fraction of patches that are empty (1-p) and the fraction of
patches sending out potential colonists (p). The rate of
colonization within the metapopulation as a whole as a single
rate constant (c) times the product (p(1-p)).
The rate of change in patch occupancy:
dp/dt = cp(1-p)
–
colonization
ep
extinction
Basic model of metapopulation dynamics
The rate of change in patch occupancy:
dp/dt = cp(1-p)
–
colonization
ep
extinction
A metapopulation attains equilibrium size when
colonization equals extinction
cp(1-p) = ep
Thus, p^= 1 – e/c
This is the proportion of occupied patches at
metapopulation equilibrium
Recap
Population cycles: very common
Mechanisms: result from time delays in the
response of populations to their own
density
Two models: how R or rt influences the
population size change.
Metapopulations and subpopulations
Basic model of metapopulation dynamics
Basic model of metapopulation dynamics
dp/dt = cp(1-p)
–
colonization
ep
extinction
p^= 1 – e/c
Rate of e/c is very important
If e = 0, then p^=1, all patches occupied, none
disappears
If e>=c, then p^ =0, then metapopulation heads
toward extinction
0< e < c, intermediate value, results in a shifting
mosaic of occupied and unoccupied patches.
Basic model of metapopulation dynamics
In this model, there are many assumptions
1.All patches are equal
2.(e) and (c) for each patch are the same
3.Each occupied patch contribute equally to dispersal
4.Colonization and extinction in each patch occur
independently of other patches
5.Colonization rate is proportional to the fraction of
occupied patches (p)
In reality:
Patches vary in size, habitat quality, degree of isolation
from other patches. Larger patch support large
subpopulation, lower probability of extinction. Smaller,
more isolated patches are less likely to be occupied.
Larger, less isolated patches are more likely
to be occupied
Shrew on
islands in
two lakes
in Finland
Few
occupied
patch
area <1
Isolation?
Larger, less isolated patches are more likely
to be occupied
Butterfly on patches of
calcareous grassland in
England (Hanski et al. 1991)
Patch size and isolation are
important
Glanville fritillary butterfly study by Illka Hanski,
Finland
A survey for p^: Occupied patches of dry meadows, Aland
Islands, Finland
Exp. One: Total 1600 suitable patches, only 30% were
occupied at any given time.
Exp. Two: Introduced populations to 10 of the 20 suitable
habitat patches on the smaller, isolated island of Scottungia, in
August 1991
Observed over next 10 years
Number of extinctions varies between 0 to 12 per year
Number of colonization between 0 and 9
Subpopulations: started at 10, dropped to as few as 2, and
increased as high as 14, ended at 11.
None of the original 10 survived the decade, the
metapopulation as a whole persisted.
The rescue effect
Immigration from large, productive subpopulations can keep
declining subpopulations (small ones) from dwindling to small
numbers and eventually becoming extinct. This phenomenon is
known as rescue effect.
Dispersal is critical for colonizing empty patches, as well as
maintaining established populations.
Model: modify the model and add rate of extinction (e)
decrease as the fraction of occupied patches (p) increases
(more rescuers)
dP/dt = cp(1-p) - ep(1-p)

p=0, 1 if c><e.
This model predicts that p^ will either equal to 0, otherwise, it
will increase to 1, as when p<1 but close to 1, 1-p is small,
reduce the extinction rate. With rescue effect, all patches will
be occupied.
10.5 Chance event may cause small
populations to go extinction
Deterministic model and stochastic model:
Population models we described before are based on
average values of birth rate and death rate, and
assume no difference among individuals. Such
models, whose outcomes can be predicted with
certainty, are called deterministic models.
Models built in with chance factors (randomness),
such as birth and death rate vary from each
individual to another and from one time to another
(with mean over certain time or of all individuals is
fixed). The result of each model run varies and can’t
be predicted, stochastic model.
Three types of randomness
1. Unpredictable catastrophe, such as appearance of
a predator, disease, fire etc (birth and death)
2. Environmental variation (some rules, small
variation not predictable). Physical and
environmental factors (influence birth and death)
3. Stochastic processes such as death of an
individual, number of offspring produced by an
individual. Even under constant environment, these
values could change for an individual.
(Overall, there is a probability distribution)
(coin tossing is an example of a stochastic process)
Stochastic population processes produce a
probability distribution of population size
N(0)=10,
lambda=1.5
No stochastic
process
involved,
what’s the
population
size at t=1?
Chart on left:
N=10, pure
birth, b=0.5
and
stochastic
process
involved,
All give birth,
N1=10+10=
20.
Chance events exert their
influence more forcefully in small
population than in large ones
Coin tossing: a set of 5 pennies, 5 heads in a
trial is 1/32; 10 pennies, is 1/1024. If each
individual in a population is a coin, and
heads mean death, a population of 5 has a
higher probability of extinction, just by
chance.
Stochastic extinction of small populations
Random walk: A population subject to stochastic birth and
death process is said to take a random walk, meaning that its
numbers may increase or decrease strictly by chance.
When the size of such a population does not respond to changes
in density, its ultimate fate is extinction, regardless of how its
size might increase in the meantime.
Mathematicians have calculated the probability of extinction. For
simplicity, given same birth rate and death rate, the probability
of a population will die out within a time interval t is
p(t)=[bt/(1+bt)]^N.
Stochastic extinction of small populations
The probability of
stochastic
extinction increase
over time, but
decrease as a
function of initial
population size N.
b=0.5,
p0(t)=[bt/(1+bt)]
^N
N=10, t=10,
p=0.16
t=1,000, p=0.98
Stochastic extinction with density dependence
Stochastic extinction models usually do not include densitydependent changes in birth and death rates.
If density-dependent birth and death rate includes, it rarely
goes extinction (unless the population size is very small),
as a population drops below K, the birth rate will increase
and death rate will decrease.
Whether the density-independent stochastic models are
relevant to natural populations?
They are.
1)Fragmentation by human beings creates many small
subpopulation, often so isolated that eventual demise can’t
prevented by immigration from other populations
2)Changing environmental conditions reduce fecundity
3)Endangered species can’t compete with other species
4)Small populations sometimes exhibit positive density
dependence (Allee effect), their number may decline more
rapidly.
Size and extinction of natural populations
Small size populations become more susceptible
to extinction, particularly on small islands
An Example: species lists for birds in 1917 and
1968 on two islands.
Over 51 years, 10 species disappeared from
Santa Barbara Island (3 km^2), only 6 of 36
disappeared from the large Santa Cruz Island
(249 km^2)
Extinction rate: 1.7% and 0.1% per year.
The End
10.9 Population extinction
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If r becomes negative
(birth rate < death
rate), population
declines and will go
extinction.
Factors: Extreme
environmental events
(droughts, floods, cold
or heat etc), loss of
Overgraze, only 8 in 1950
habitat (human).
Allee effect, genetic drift,
Small populations are
inbreeding (mating
susceptible to extinction between relatives)
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Small population size may result in
the breakdown of social structures
that are integral to successful
cooperative behaviors (mating,
foraging, defense)
The Allee effect is the decline in
reproduction or survival under
conditions of low population density
There is less genetic variation in a
small population and this may affect
the population’s ability to adapt to
environmental change
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Hackney and McGraw (West Virginia
University) examined the reproductive
limitations by small population size on
American ginseng (Panax
quinquefolius)
• Fruit production per plant declined with
decreasing population size due to
reduced visitation by pollination
Recap
Population cycles: very common
Mechanisms: result from time delays in the
response of populations to their own
density
Two models: how R or rt influences the
population size change.
Metapopulations and subpopulations
Basic model of metapopulation dynamics