Population Dynamics Part 1

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Population DynamicsPart 1

• Exponential Growth and Decay
• Logistic Model (Growth Cap)

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Population Dynamics

• References
• J.D. Murray Mathematical Biology, Second
  corrected Edition, Springer 1993
• Edward K. Yeargers, Ronald W. Shonkwiler James
  V. Herod An Introduction to the Mathematics of
  Biology (With Computer Algebra Models),
  Birkhauser 1996.

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Population Per-Capita Growth Rate
If rconst.gt0 ? Population increases
exponentially If rconst.lt0 ? Population decays
exponentially
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Malthus Model (1798)

• Model assumption Rate of change of a population
  is proportional to the population size
• Let ypopulation size and y0initial population
  size at t0.

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Mathematical definition of Population Per-Capita
Growth Rate using Malthus Model

• Let ypopulation size and y0initial population
  size at t0.

Units of r r s-1
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Exponential Growth (rgt0)

• T2Doubling time
• Sometimes we can estimate rgt0 from a measured
  doubling time

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Exponential Decay (rlt0)

• T1/2Half life
• Sometimes we can estimate rlt0 from a measured
  half life

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Local Exponential Growth

• Many populations exhibit exponential growth for a
  while
• ?? Local (rather than global) validity

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Environmentally Limited Population Growth

• In real populations there is no such a thing as a
  constant (per capita) growth rate r.
• Finite available resources eventually put an
  upper limit on population growth.
• How do we model it mathematically?

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Verhulsts Logistic Model (1845)

• Assumption As population increases, the
  per-capita growth rate gets smaller linearly.

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Logistic Model Carrying Capacity

• K the carrying capacity of the population
• At yK the growth rate r0

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Logistic Model Solution
It is a nonlinear differential equation. Analytic
solution happens to be available
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Behavior of Logistic Model Solution
As t?8 y?K, from any initial value
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Logistic Model Phase-Plane Solution
ye0 is an unstable equilibrium point. yeK is a
stable equilibrium point.