Population Models

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Population Models

• What is a population?
• Populations are dynamic
• What factors directly impact dynamics
• Birth, death, immigration and emigration
• in models we frequently simplify things in order
  to gain a better understanding of how the rest
  will work
• E.g. a closed vs. open population

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Population Models

• Start with treating time as a discrete
  (geometric population growth) unit rather than
  continuous (exponential growth)
• Is this realistic? Why or why not?

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Population Models
Nt Bt Dt It Et Nt1 Nt Bt -Dt

• Model development
• Consider using per capita rates (individuals)
• Rewrite the equation in terms of per capita
  rates
• With constant rates

bt Bt/Nt and dt Dt/Nt
Nt1 Nt btNt - dtNt
Nt1 Nt bNt - dNt
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Population Models

• Model is somewhat realistic, but still useful
• 1) provides a good starting point for more
  complex models (changes rates)
• 2) it is a good heuristic provides insight and
  learning despite its lack of realism
• 3) many populations do grown as predicted by such
  a simple model (for a limited period of time)

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Population Models

• Because this model does NOT change with
  population size, it is called density-independent
• Furthermore, (b-d) is extremely important
• ? is the finite rate of increase

Nt1 Nt (b d)Nt
Nt1 Nt RNt
Nt1 (1R)Nt
Nt1 ?Nt
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Population Models

• Doubling time
• Consider R0.1 ?1R (1.1)

Nt1 ?Nt
Nt double 2N0
2N0 ?t double N0
Divide both sides by N0 2 ?t double
Take the logarithm of both sides ln2 tdouble
ln?
Divide both sides by ln? ln2 / ln? tdouble
7.27 years
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Population Modelsexponential growth (continuous)

• Instantaneous rate of change
• Calculate the per capita rate of pop growth
• Calculate the size of the pop at any time

dN / dt rN
(dN / dt) / N r
Nt N0ert
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Population Modelsexponential growth (continuous)

• Doubling Time

Nt double N0ert double
Substitute 2N0 2N0 N0ert double
Divide by N0 2 ert double
Take natural log ln 2 rtdouble
Finally divide by r tdouble ln2 / r
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Logistic Population Models
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Logistic Population Models
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Logistic Population Models

• Similarly this population model will explicitly
  model birth and death rates
• Will also add in the concept of a carrying
  capacity (K), and one that is a continuous-time
  version

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Logistic Population Models

• Remember, the geometric model looked like this
• We can add two new terms to the model to
  represent changes in per capita rates of birth
  and death, where b and d the amount by which
  the per capita birth or death rate changes in
  response to the addition of one individual of the
  pop(n)

Nt1 Nt bNt - dNt
Nt1 Nt (bbNt)Nt (ddNt)Nt
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Logistic Population Models

• All four parameters (b, b, d, d) are assumed to
  remain constant through time (hence no bt)
• How and why should b and d vary with density?
• Logistic population models can be used to examine
  the potential impact of interspecific and
  intraspecific competition, as well as
  predator-prey relationships and harvesting
  populations

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Logistic Population Models

• We will explore the behavior of populations as
  numbers change
• There is an equilibrium population size

Neq b-d d-b
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Logistic Population Models

• However, is it realistic to think populations
  will grow exponentially continuously?

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Logistic Population Models

• This equilibrium defined is so important, it is
  called the carrying capacity
• This model gives us rate of change of population
  size

dN rN (K-N) /K)
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Logistic Population Models

• To derive the equation for population size
  requires us to use calculus

Nt K/ 1 (K-N0) / N0e-rt
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Logistic Population Models