Population Growth and Regulation Chapter 14

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Population Growth and Regulation (Chapter 14)

• Population models
• Exponential and geometric population growth
• Survival and age distributions

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Two things that living things do
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• Population models are mathematical
  representations of how populations change over
  time.
• They are idealized
• capture general properties
• ignore random variation

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Population growth can be exponential
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• Exponential population growth
• Populations grow by multiplication, like interest
  from a bank account
• Describes populations where individuals are added
  continuously (bacteria, some insects)

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• Exponential population growth
• Equation N(t) N(0)ert
• N(0) population at time 0
• e base of the natural logarithm (2.72)
• r exponential growth rate
• if r gt 0, the population grows
• if r lt 0, the population shrinks
• if r 0, the population is stable
• t time

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r per capita birth rate per capita death rate
b - d
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• Geometric population growth describes
  population change over discrete intervals or
  generations

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• Geometric population growth
• Equation N(t 1) N(t)?
• ? geometric growth rate
• This is the proportional change in the population
• if ? 1.5, population will increase 50
• if ? 0.5, population will decrease 50

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• N(t 1) N(t)?
• N(1) N(0) ? , N(2) N(1)?
• so N(2) N(0) ?2
• and N(t) N(0)?t
• remember N(t) N(0)ert
• which means that er ? or ln ? r

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Exponential and geometric population growth can
describe the same data.
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• N number of individuals in the population
• N(t) N at time t
• N(t 1) N at time t 1
• ? N N(t 1) - N(t)

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• 4 ways that a population can change in size (BIDE
  model)
• birth (B)
• immigration (I)
• death (D)
• emigration (E)
• ? N B I - D - E
• closed populations have no immigration or
  emigration, so
• I 0 and E 0, and
• ? N B D

increase population size
decrease population size
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• Survival
• Survival rates change with age
• You can represent survival rates in a life table

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Survival rate is the proportion of organisms
surviving from one time period to the next.
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• Two ways to estimate survival
• Cohort life table
• start with a group of individuals all born at the
  same time
• record deaths as they happen
• used with organisms that can be easily marked and
  tracked
• plants
• sessile animals
• mobile animals on small islands

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• Cohort life table (contd)
• advantages
• easy to do
• disadvantages
• difficult for long-lived or mobile organisms
• can be complicated by environmental changes

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• Static life table
• measure lifespan of individuals of known age as
  they die during a single time period
• used with organisms that can be aged easily
• fish (otoliths)
• trees (tree rings)
• turtles/tortoises (carapace)
• some mammals (horns)

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• Static life table (contd)
• advantages
• dont have to follow every individual from a
  cohort
• can gather data at random
• disadvantages
• have to know the age of each individual at death

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Survivorship Curve
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• 3 types of survivorship curves
• Type I high survival for young individuals
• most mortality is among older individuals
• found in
• large vertebrates (humans, whales, ungulates)
• some annual plants
• some small invertebrates

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Type I Survivorship Curve
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• Type II constant survival throughout lifespan
• individuals die at the same rate regardless of
  age
• found in birds, turtles,
• small mammals

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Population Growth and Regulation (Chapter 14)

• Population growth curves
• Survival curves
• Age distributions
• Life tables
• Limiting populations

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Type I Type II
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• Type III high infant/juvenile mortality
• most mortality is among youngest individuals,
  early in life
• often very high mortality for eggs or seeds
• mortality is low after the juvenile period
• found in fish, perennial plants, marine
  invertebrates, sea turtles

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• Survivorship curves determine the age structure
  of the population the relative abundance of
  individuals of different ages

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• Birth rate number of young born per female in a
  given time period
• Birth rates often vary with age
• no breeding before age of maturity
• no breeding after certain age
• Therefore, the age structure of the population
  influences how the population grows.

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• Adding births to the life table...
• x age
• sx survival from time x to time x 1
• bx fecundity the birth rate of individuals of
  age x (females only!)

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t 0 t 1
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t 0 t 1
x
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t 0 t 1
x

x

x

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t 0 t 1
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If survival and reproduction are constant, the
population will reach a stable age distribution.
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• x time
• sx survival from time x to time x 1
• bx the birth rate (fecundity) of individuals of
  age x
• lx survivorship from birth to age x
• (the probability that an individual will
  survive to the beginning of age period x)

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• If you sum lx bx over all the age classes, you
  get R0, the net reproductive rate, which is the
  expected total number of daughters per female
  over a lifetime.

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R0 Slxbx 2.0 Each female produces 2
daughters on average over her lifetime.
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• T generation time, the average age at which an
  individual gives birth to offspring.
• T

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T Sxlxbx/ Slxbx 3.5/2 1.75 Average
generation time is 1.75
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• You can also determine r (the exponential growth
  rate) based on life tables.
• For our life table, r (ln 2)/1.75 0.39
• ? er 1.48 (48 growth each year)

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• Most populations have extraordinary potential for
  growth.
• Doubling time is the amount of time a population
  will take to double in size.

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Population Growth and Regulation (Chapter 14)

• Fun with life tables
• Population limitation
• Logistic growth
• Density dependence

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• LIFE TABLES CHEAT SHEET
• x time
• sx survival from time x to time x 1
• bx fecundity of individuals of age x
• lx survivorship from birth to age x
• R0 net reproductive rate Slxbx
• T generation time Sxlxbx/ Slxbx
• r exponential growth rate (ln R0)/T
• t2 doubling time (ln 2)/r

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R0 1.135 T 4.5 r
0.028 t2 24.7
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• Most populations have extraordinary potential for
  growth.
• Doubling time is the amount of time a population
  will take to double in size.

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• Elephant seal
• ? 1.096, t2 7.5 yrs
• if N(0) 100, N(100) 15,600
• Field vole
• ? 24, t2 80 days
• N(100) 1.05 x 10140
• Flour beetle
• ? 1010, t2 9.9 days
• N(100) 101102
• Water flea
• ? 1030, t2 3.5 days
• N(100) 103102

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Why are there so few plants and animals?
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• Environmental variation limits population growth.
  (remember optimum)

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• r b - d
• Limiting population growth requires a decrease in
  the birth rate and/or an increase in the death
  rate
• These factors act to slow and eventually halt
  population growth.

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• For exponential growth, r is constant
• In reality, r varies depending on N
• K carrying capacity of the environment, the
  maximum number of individuals that can be
  supported.
• r0 the maximum growth rate, when the population
  is very low

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• 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

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• Logistic growth has results in a characteristic
  sigmoid (s-shaped) curve.
• An inflection point at K/2 separates the
  accelerating and decelerating phases of
  population growth.

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• Why do populations stop growing as they approach
  K?
• Density-dependent factors death rates increase
  and/or birth rates to decrease with crowding
  (density N/area)

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• As density goes up, there is
• less space
• less food
• more social strife (fighting, infanticide,
  cannibalism)
• more disease
• more predation

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• In plants, density dependence can cause plant
  size to shrink with density
• Plants densities often decrease over time
• as density increases, plants die off
  (self-thinning)
• average plant size increases faster than the
  death rate

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• Density-independent factors may reduce population
  size but cannot regulate it.
• temperature
• precipitation
• catastrophic events
• storms
• volcanoes
• These factors operate regardless of the
  population density.

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Reading for Monday Chapter 15, p. 293-302
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Temporal Dynamics of Populations (Chapter 15)

• Geometric logistic growth
• Why populations fluctuate
• Population cycles

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Populations fluctuate over time
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• Why?
• Changing environmental conditions
• climate
• predators
• diseases
• Intrinsic dynamics of populations and
  density-dependence

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• Dynamics of populations with discrete generations
• The logistic equation for geometric growth
  (discrete generations)
• This equation can produce fluctuating populations.

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• At low densities, growth rates are high, so a
  small population grows quickly.

K
N
0 1
time
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• The population overshoots the carrying capacity,
  so the population decreases.

K
N
0 1 2
time
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• If the decrease brings the population back below
  K, the population will continue to fluctuate.

K
N
time
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• Large, long-lived animals have relatively small
  fluctuations
• can tolerate changing environment
• low reproductive rate -gt slow response
• Small, short-lived animals have large
  fluctuations
• populations turn over rapidly
• no defense against changing conditions

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Small organisms can have large fluctuations.
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Different species in the same environment can
fluctuate independently.
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Temporal variation can affect the population age
structure
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• Periodic cycles fluctuations with regular
  intervals between successive highs and lows

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• If r lt 1, the population will increase to K
  without oscillation

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• If 1lt r lt 2, the population will show damped
  oscillations, cycles that decrease in amplitude
  over time

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• If r gt 2, the population will show limit cycles
  (regular cycles).

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At very large r,the population may show chaotic
fluctuations.
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• Warm temps, high r lead to cycles
• Cooler temps, lower r, no cycles

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