Chapter 15: Temporal and Spatial Dynamics of Populations

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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 Hudsons 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)
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Phytoplankton pop.
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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
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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
  over- and 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
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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 t
• rt lt e-1 (about 0.37)
• population approaches K and stabilizes
• rt lt p/2 (about 1.6)
• population exhibits damped oscillations
• rt gt p/2
• population exhibits limits cycles, with period 4t
  - 5t

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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, s
• s 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)
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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
  density-dependence 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.