Developing The Logistic

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Developing The Logistic
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Lecture Goals

• Early concepts of population growth
• Fibonacci series
• Geometric Growth
• Discrete Continuous models
• Difference and Differential
• Exponential Growth (continuous)
• Adding limits
• Changing the model
• The Logistic Population Model

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Lets get Medieval
1202 C.E. - Arabic numbers become popular in
Europe, quantitative work becomes much
easierLiber Abaci published

• How many rabbits produced from a pair in a year
  if
• Two months to maturity
• Each mature pair produces a new pair each month
• No deaths occur

- Fibonacci (1180 1250)
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Nt Nt-1 Nt-2

• Limiting assumptions
• Not really empirical

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1798 Essay on the Principle of Population
I said that population, when unchecked,
increased in a geometrical ratio
- Thomas Malthus (1766-1834)
But what does this mean
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The ratio between the current population and the
previous one (last year) is constant
Let this ratio be represented by l R
l 2
Nt
l Nt-1
1, 2, 4, 8,
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Though it cant always apply, when might this
model provide a reasonable approximation?
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This reasoning leads to the observation that

• Most organisms have a high potential for
  reproduction (even elephants and people)
• No species has overgrown the world
• Thus, only a small proportion of individuals will
  reproduce successfully to their full potential
• Early death or poor reproductive success

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Sidebar Importance to Evolutionary Theory
"In October 1838, I happened to read for
amusement Malthus on Population it at once
struck me that under these circumstances
favourable variations would tend to be preserved,
and unfavourable ones to be destroyed. The
results of this would be the formation of a new
species. Here, then I had at last got a theory by
which to work". - Charles Darwin, from his
autobiography (1876)
See http//www.ucmp.berkeley.edu/history/evolution
.html
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