Modeling Populations

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Modeling Populations
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Population Dynamics

• Studies how populations change over time
• Involves knowledge about birth and death rates,
  food supplies, social behaviors, genetics,
  interaction of species with their environments
  and interaction among themselves.
• Models should reflect biological reality,
  yet be simple enough that insight may be
  gained into the population being studied.

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Types of Models

• We will study the development of some basic
  one- and two-species population models.
• Malthusian (exponential) growth human
  populations
• Logistics growth human populations and yeast
  cell growth
• Logistics growth with harvesting.
• Predator-Prey interaction two fish populations

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Malthus Model for Population Growth

• Thomas Malthus

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Brief Biography

• Thomas Robert Malthus (he went by Robert) was
  born on February 13, 1766. He was the second son
  and sixth child of Daniel Malthus, a country
  gentleman. Malthus was educated at Cambridge in
  mathematics. He became an ordained minister
  immediately after graduating in 1788, and became
  a curate near his family home in Surrey (Winch,
  1989).

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Malthus Basic Idea

• Thomas Malthus (1766 - 1834) was a British
  economist who startled early 19th century society
  with his pessimistic prediction that population
  growth would exceed food supply. Mankind would
  sink into deep misery. Thomas Malthus on the
  other hand focused on the interdependence between
  poverty and population growth.

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A Pessimistic View

• He  took the pessimistic view that population
  tends to expand faster than the food supply.
• Malthus was an English Anglican clergyman who
  devised and published his ideas on population in
  the 1790s.  Many had hoped that the French
  Revolution and the Enlightenment would lead to a
  new age of rational improvement in peoples'
  lives.  Malthus, an old Tory, argued instead that
  it was ultimately going to be impossible to bring
  about permanent improvement in the quality of
  life.

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Principle of Population

• In 1798 Malthus produced his famous essay,
  Principle of the Population
• A link to the above essay sis found here
  http//www.ac.wwu.edu/stephan/malthus/malthus.0.h
  tml
• It details a pessimistic view of times to come
  and offered as valuable reading material for such
  people as Karl Marx and Charles Darwin, both
  famous for their views on population

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Principle of Population

• In his famous Essay on the Principle of
  Population (1798), Malthus expressed the view
  that famine and poverty were natural outcomes of
  population growth.
• The increase of population will take place, if
  unchecked, in a geometrical progression, while
  the means of subsistence will increase in only an
  arithmetical progression.
• Geometrical 2-4-6-8--2k
• Arithmetical 2-3-4-5-1k
• Population will always expand to the limit of
  subsistence and will be held there by famine, war
  and ill health. Later he added moral restraint,
  as well.
• That pessimistic view was not popular among
  social reformers who were optimists, predicting
  an everlasting economic growth.

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The Malthus Model

• His model was based on the observation that the
  time required for human popu-lations to double
  was essentially constant (about 25 years at the
  time), regardless of the initial population size.

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US Population 1650-1800

• Data for U.S. population probably available to
  Malthus.
• The nearly-linear character of the right graph
  indicates good agreement after 1700 with the
  "uninhibited growth" model he produced.

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Governing Principle
To develop a mathematical model, we formulate
Malthus observation as the governing principle
for our model Populations appeared to increase
by a fixed proportion over a given period of
time, and that, in the absence of constraints,
this proportion is not affected by the size of
the population.
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Discrete-in-time Model

• t0, t1, t2, , tN equally-spaced times at which
  the population is determined ?t ti1 - ti
• P0, P1, P2, , PN corresponding populations at
  times t0, t1, t2, , tN
• b and d birth and death rates r b d, is the
  effective growth rate.
• P0 P1 P2
  PN
• ----------------------------------
  -----gt t
• t0 t1 t2
  tN

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The Malthus Model
Mathematical Equation (Pi 1 - Pi) / Pi
r ?t r b - d or Pi 1 Pi
r ?t Pi ti1 ti dt i 0, 1, ... The
initial population, P0, is given at the initial
time, t0.
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An Example
Example Let t0 1900, P0 76.2 million (US
population in 1900) and r 0.013 (1.3
per-capita growth rate per year). Determine the
population at the end of 1, 2, and 3 years,
assuming the time step ?t 1 year.
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Example Calculation
P0 76.2 t0 1900 ?t 1 r 0.013 P1 P0
r ?t P0 76.2 0.013176.2 77.3 t1
t0 ?t 1900 1 1901 P2 P1 r ?t P1
77.3 0.013177.3 78.3 t2 t1 ?t 1901
1 1902 ... P2000
277.3 (281.4), t2000 2000
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National Censuses, 19002000 Year
Population Area-sq mi Pop.Per sq mi 1900
75,994,575 2,969,834 25.6 1910 91,972,266
2,969,565 31.0 1920 105,710,620 2,969,451
35.6 1930 122,775,046 2,977,128 41.2 1940
131,669,275 2,977,128 44.2 1950 150,697,361
2,974,726 50.7 1960 179,323,175 3,540,911
50.6 1970 203,302,031 3,540,023 57.4 1980
226,545,805 3,539,289 64.0 1990 248,709,873
3,536,278 70.3 2000 281,421,906 3,537,441 79.6 1
. Beginning with 1960, figures include Alaska and
Hawaii. 2. Excludes armed forces overseas.