Transcript CHAPTER 15: TEMPORAL AND SPATIAL DYNAMICS OF POPULATIONS Robert E. Ricklefs

CHAPTER 15: TEMPORAL AND SPATIAL
DYNAMICS OF POPULATIONS
Robert E. Ricklefs
The Economy of Nature, Fifth Edition
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Some populations exhibit
regular fluctuations.
 Charles Elton first called attention to regular
population cycles in 1924:
 such cycles were known to earlier naturalists, but
Elton brought the matter more widely to the
attention of biologists
 Elton also called attention to parallel fluctuations
in populations of predators and their prey
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Evidence for Cycles in Natural
Populations
 Records of the Hudson’s Bay Company yield
important data on fluctuations of animals trapped
in northern Canada:
 data for the snowshoe hare (prey) and the lynx (predator)
have been particularly useful
 thousand-fold fluctuations are evident in these records
 Records of gyrfalcons exported from Iceland in the
mid-eighteenth century also provide evidence for
dramatic natural population fluctuations.
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Fluctuations in Populations
 Populations are driven by density-dependent
factors toward equilibrium numbers.
 However, populations also fluctuate about such
equilibria because:
 populations respond to changes in environmental
conditions:
 direct effects of temperature, moisture, etc.
 indirect environmental effects (on food supply, for example)
 populations may be inherently unstable
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Domestic sheep on Tasmania –
relatively stable population
after becoming established
(variation due to env. factors)
Phytoplankton pop.
Fluctuation is the rule for
natural populations.
 Tasmanian sheep and Lake Erie phytoplankton
both exhibit different degrees of variability in
population size:
 the sheep population is inherently stable:
 sheep are large and have greater capacity for homeostasis
 the sheep population consists of many overlapping generations
 phytoplankton populations are inherently unstable:
 phytoplankton have reduced capacity for homeostatic
regulation
 populations turn over rapidly
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Periodic cycles may or may not
coincide for many species.
 Periodic cycles: period between successive
highs or lows is remarkably regular
 Populations of similar species may not exhibit
synchrony in their fluctuations:
 four moth species feeding on the same plant
materials in a German forest showed little
synchrony in population fluctuations
 4-5 year population cycles of small mammals in
northern Finland were regular and synchronized
across species
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Temporal variation affects the
age structure of populations.
 Sizes of different age classes provide a history of
past population changes:
 a good year for spawning and recruitment may result
in a cohort that dominates progressively older classes
for years to come
 The age structure in stands of forest trees may
reflect differences in recruitment patterns:
 some species (such as pine) recruit well only after a
disturbance
 other species (such as beech) are shade-tolerant and
recruit almost continuously
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Commercial
whitefish:
excellent
spawning and
recruitment
in 1944
Population cycles result from
time delays.
 A paradox:
 environmental fluctuations occur randomly:
 frequencies of intervals between peaks in tree-ring
width are distributed randomly (they vary in direct
proportion to temperature and rainfall)
 populations of many species cycle in a non-
random fashion:
 frequencies of intervals between population peaks
in red fox are distributed non-randomly
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A Mechanism for Population
Cycles?
 Inherent dynamic properties associated with
density-dependent regulation of population size
 Populations acquire “momentum” when high birth
rates at low densities cause the populations to
overshoot their carrying capacities.
 Populations then overcompensate with low survival
rates and fall well below their carrying capacities.
 The main intrinsic causes of population cycling are
time delays in the responses of birth and death rates
to environmental change.
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Time Delays and Oscillations:
Discrete-Time Models
 Discrete-time models of population
dynamics have a built-in time delay:
 response of population to conditions at one time
is not expressed until the next time interval
 continuous readjustment to changing conditions
is not possible
 population will thus oscillate as it continually overand undershoots its carrying capacity
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Oscillation Patterns - Discrete
Models
 Populations with discrete growth can exhibit
one of three patterns:
 r0 small:
 population approaches K and stabilizes
 r0 exceeds 1 but is less than 2:
 population exhibits damped oscillations
 r0 exceeds 2:
 population may exhibit limit cycles or (for high r0)
chaos
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3 oscillation patterns
Time Delays and Oscillations:
Continuous-Time Models
 Continuous-time models have no built-in
time delays:
 time delays result from the developmental period
that separates reproductive episodes between
generations
 a population thus responds to its density at some
time in the past, rather than the present
 the explicit time delay term added to the logistic
equation is tau (t)
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Oscillation Patterns Continuous Models
 Populations with continuous growth can exhibit
one of three patterns, depending on the product
of r and τ:
 rτ < e-1 (about 0.37):
 population approaches K and stabilizes
 rτ < π/2 (about 1.6):
 population exhibits damped oscillations
 rτ > π/2:
 population exhibits limits cycles, with period 4τ - 5τ
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Cycles in Laboratory
Populations
 Water fleas, Daphnia, can be induced to cycle:
 at higher temperature (25oC), Daphnia magna exhibits
oscillations:
 period of oscillation is 60 days, suggesting a time delay of 12-15
days
 this is explained as follows: when the population approaches
high density, reproduction ceases; the population declines,
leaving mostly senescent individuals; a new cycle requires
recruitment of young, fecund individuals
 at lower temperature (18oC), the population fails to
cycle, because of little or no time delay of responses
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Storage can promote time
delays.
 The water flea Daphnia galeata stores lipid
droplets and can transfer these to offspring:
 stored energy introduces a delay in response to
reduced food supplies at high densities
 Daphnia galeata exhibits pronounced limit cycles with
a period of 15-20 days
 another water flea, Bosmina longirostris, stores smaller
amount of lipids and does not exhibit oscillations
under similar conditions
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Overview of Cyclic Behavior
 Density dependent effects may be delayed by
development time and by storage of nutrients.
 Density-dependent effects can act with little
delay when adults produce eggs quickly from
resources stored over short periods.
 Once displaced from an equilibrium at K,
behavior of any population will depend on the
nature of time delay in its response.
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Metapopulations are discrete
subpopulations.
 Some definitions:
 areas of habitat with necessary resources and
conditions for population persistence are called
habitat patches, or simply patches
 individuals living in a habitat patch constitute a
subpopulation
 a set of subpopulations interconnected by
occasional movement between them is called a
metapopulation
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Metapopulation models help
managers.
 As natural populations become increasingly
fragmented by human activities, ecologists have
turned increasingly to the metapopulation
concept.
 Two kinds of processes contribute to dynamics
of metapopulations:
 growth and regulation of subpopulations within
patches
 colonization to form new subpopulations and
extinction of existing subpopulations
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Connectivity determines
metapopulation dynamics.
 When individuals move frequently between
subpopulations, local fluctuations are damped out.
 At intermediate levels of movement:
 the metapopulation behaves as a shifting mosaic of
occupied and unoccupied patches
 At low levels of movement:
 the subpopulations behave independently
 as small subpopulations go extinct, they cannot be
reestablished, and the entire population eventually goes
extinct
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The Basic Model of
Metapopulation Dynamics
 The basic model of metapopulation dynamics predicts
the equilibrium proportion of occupied patches, ŝ:
ŝ = 1 - e/c
where e = probability of a subpopulation going extinct
c = rate constant for colonization
 The model predicts a stable equilibrium because when
p (proportion of patches occupied) is below the
equilibrium point, colonization exceeds extinction,
and vice versa.
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Is the metapopulation model
realistic?
 Several unrealistic assumptions are made:
 all patches are equal
 rates of colonization and extinction for all patches are
the same
 In natural settings:
 patches vary in size, habitat quality, and degree of
isolation
 larger subpopulations have lower probabilities of
extinction
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The Rescue Effect
 Immigration from a large, productive
subpopulation can keep a declining
subpopulation from going extinct:
 this is known as the rescue effect
 the rescue effect is incorporated into metapopulation
models by making the rate of extinction (e) decline as
the fraction of occupied patches increases
 the rescue effect can produce positive density
dependence, in which survival of subpopulations
increases with more numerous subpopulations
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Chance events may cause small
populations to go extinct.
 Deterministic models assume large populations
and no variation in the average values of birth
and death rates.
 Randomness may affect populations in the real
world, however:
 populations may be subjected to catastrophes
 other factors may exert continual influences on rates
of population growth and carrying capacity
 stochastic (random sampling) processes can also
result in variation, even in a constant environment
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Understanding Stochasticity
 Consider a coin-tossing experiment:
 on average, a coin tossed 10 times will turn up 5
heads and 5 tails, but other possibilities exist:
 a run with all heads occurs 1 in 1,024 trials
 if we equate a “tail” as a death in a population where
each individual has a 0.5 chance of dying, there is a 1 in
1,024 chance of the population going extinct
 for a population of 5 individuals, the probability of
going extinct is 1 in 32
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Stochastic Extinction of
Small Populations
 Theoretical models exist for predicting the
probability of extinction of populations because
of stochastic events.
 For a simple model in which birth and death
rates are equal, the probability of extinction
increases with:
 smaller population size
 larger b (and d)
 time
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Probability of extinction increases over time (t)
but decreases as a f of initial population size (N)
Stochastic Extinction with
Density Dependence
 Most stochastic models do not include
density-dependent changes in birth and
death rates. Is this reasonable?
 density-dependence of birth and death rates
would greatly improve the probability that a
population would persist
 however, density-independent stochastic models
may be realistic for several reasons...
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Density-independent stochastic
models are relevant.
 The more conservative density-independent
stochastic models are relevant to present-day
fragmented populations for several reasons:
 most subpopulations are now severely isolated
 changing environments are likely to reduce fecundity
 when populations are low, the individuals still compete
for resources with larger populations of other species
 small populations may exhibit positive densitydependence because of inbreeding effects and problems
in locating mates
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Size and Extinction of
Natural Populations
 Evidence for the relationship between
population size and the likelihood of extinction
comes from studies of avifauna on the California
Channel Islands:
 smaller islands lost a greater proportion of species
than larger islands over a 51-year period
 proportions of populations disappearing over this
interval were also related to population size
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Summary 1
 Populations of most species fluctuate over
time, although the degree of fluctuation
varies considerably by species. Some species
exhibit regular cyclic fluctuations.
 Both discrete and continuous population
models show how species populations may
oscillate.
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Summary 2
 Population oscillations predicted by models are
caused by time delays in the responses of
individuals to density. Such delays are also
responsible for oscillations in natural populations.
 Metapopulations are divided into discrete
subpopulations, whose dynamics depend in part on
migration of individuals between patches.
 The dynamics of small populations depend to a
large degree on stochastic events.
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