Population Growth

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Population Growth

• Chapter 11

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Populations grow by multiplication.

• A population increases in proportion to its size,
  in a manner analogous to a savings account
  earning interest on principal
• at a 10 annual rate of increase
• a population of 100 adds 10 individuals in 1 year
• a population of 1000 adds 100 individuals in 1
  year
• allowed to grow unchecked, a population growing
  at a constant rate would rapidly climb toward
  infinity

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Two Models of Population Growth

• Because of differences in life histories among
  different kinds of organisms, there is a need for
  two different models (mathematical expressions)
  for population growth
• exponential growth appropriate when young
  individuals are added to the population
  continuously
• geometric growth appropriate when young
  individuals are added to the population at one
  particular time of the year or some other
  discrete interval

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Exponential Population Growth 1

• A population exhibiting exponential growth has a
  smooth curve of population increase as a function
  of time.
• The equation describing such growth is
• N(t) N(0)ert
• where N(t) number of individuals after t time
  units
• N(0) initial population size
• r exponential growth rate
• e base of the natural logarithms (about 2.72)

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Exponential Growth
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Exponential Population Growth 2

• Exponential growth results in a continuously
  accelerating curve of increase (or continuously
  decelerating curve of decrease).
• The rate at which individuals are added to the
  population is
• dN/dt rN
• This equation encompasses two principles
• the exponential growth rate (r) expresses
  population increase on a per individual basis
• the rate of increase (dN/dt) varies in direct
  proportion to N

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Geometric Population Growth 1

• Geometric growth results in seasonal patterns of
  population increase and decrease.
• The equation describing such growth is
• N(t 1) N(t) ?
• where N(t 1) number of individuals after 1
  time unit
• N(t) initial population size
• ? ratio of population at any time to that 1
  time
• unit earlier, such that ? N(t
  1)/N(t)

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Geometric Population Growth 2

• To calculate the growth of a population over many
  time intervals, we multiply the original
  population size by the geometric growth rate for
  the appropriate number of intervals t
• N(t) N(0) ? t
• For a population growing at a geometric rate of
  50 per year (? 1.50), an initial population of
  N(0) 100 would grow to N(10) N(0) ? 10
  5,767 in 10 years.

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Exponential and geometric growth are related.

• Exponential and geometric growth equations
  describe the same data equally well.
• These models are related by
• ? er
• and
• loge ? r

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Varied Patterns of Population Change

• A population is
• growing when ? gt 1 or r gt 0
• constant when ? 1 or r 0
• declining when ? lt 1 (but gt 0) or r lt 0

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Geometric growth rates (and conversion to
exponential rate)
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Per Individual Population Growth Rates

• The per individual or per capita growth rates of
  a population are functions of component birth (b
  or B) and death (d or D) rates
• r b - d
• and
• ? B - D
• While these per individual or per capita rates
  are not meaningful on an individual basis, they
  take on meaning at the population level.

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Intrinsic rate of increase is balanced by
extrinsic factors.

• Despite potential for exponential increase, most
  populations remain at relatively stable levels -
  why?
• this paradox was noted by both Malthus and Darwin
• for population growth to be checked requires a
  decrease in the birth rate, an increase in the
  death rate, or both

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Consequences of Crowding for Population Growth

• Crowding
• results in less food for individuals and their
  offspring
• aggravates social strife
• promotes the spread of disease
• attracts the attention of predators
• These factors act to slow and eventually halt
  population growth.

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The Logistic Equation

• In 1910, Raymond Pearl and L.J. Reed analyzed
  data on the population of the United States since
  1790, and attempted to project the populations
  future growth.
• Census data showing a decline in the exponential
  rate of population growth suggested that r should
  decrease as a function of increasing N.

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Behavior of the Logistic Equation

• The logistic equation describes a population that
  stabilizes at its carrying capacity, K
• populations below K grow
• populations above K decrease
• a population at K remains constant
• A small population growing according to the
  logistic equation exhibits sigmoid growth.
• An inflection point at K/2 separates the
  accelerating and decelerating phases of
  population growth.

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Integration of Logistic Model
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Three views of Logistic Growth Model
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Human population growth has not declined in the
US as predicted!
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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
• 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?

• 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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Discrete-time model and cycles
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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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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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Stochasticity can affect births.

• Consider a population in which only births occur,
  such that N(t) N(0)ebt.
• On average we expect the population to grow by a
  factor of 1.65 (e0.5) in one time interval.
• For a small population of 5 individuals
• the average size after one time interval would be
  5 x 1.65 8.24, but this could vary
  from as few as 5 to as many as 20, just by chance

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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 is affected by time and
population size