Inleiding in de Biomathematica - PowerPoint PPT Presentation

Title: Inleiding in de Biomathematica


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Inleiding in de Biomathematica

• Franjo Weissing
• Holger Waalkens

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Some ODEs can be solved by separation of
variables
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Properties of the solution of an ODE
Qualitative behaviour
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Example
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Example
Conclusion the sign of f(z) determines the
qualitative behaviour of z(t)
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Example
Conclusion the sign of f(z) determines the
qualitative behaviour of z(t)
The conclusion holds for all autonomous ODEs
autonomous ODE
non-autonomous ODE
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Example

• Theorem for autonomous ODEs
• every solution is either monotonically increasing
  or decreasing,
• or stationary
• every solution converges monotonically to a
  stable equilibrium
• or to ?

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Qualitative analysis of an autonomous ODE

• the equilibria determine the qualitative
  behaviour of z(t)

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Qualitative analysis of an autonomous ODE

• the equilibria determine the qualitative
  behaviour of z(t)

• equilibria

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Qualitative analysis of an autonomous ODE

• the equilibria determine the qualitative
  behaviour of z(t)

• equilibria

• stability

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Qualitative analysis of an autonomous ODE

• the equilibria determine the qualitative
  behaviour of z(t)

• equilibria

• stability

thus
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1stExample
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1stExample

• equilibria

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1stExample

• equilibria

• stability

• the quantitative analysis shows that
• is a stationary solution and
  for all initial conditions

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1stExample

• equilibria

• stability

• the quantitative analysis shows that
• is a stationary solution and
  for all initial conditions

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Maple
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2ndExample
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2ndExample

• equilibria

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2ndExample

• equilibria

• stability

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2ndExample

• equilibria

• stability

Thus
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2ndExample

• equilibria

• stability

Thus
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3rd Example
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3rd Example

• quantitative analysis

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3rd Example

• quantitative analysis

• qualitative analysis

• equilibria

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3rd Example

• quantitative analysis

• qualitative analysis

• equilibria

• stability

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3rd Example

• quantitative analysis

• qualitative analysis

• equilibria

• stability

thus
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3rd Example

• quantitative analysis

• qualitative analysis

• equilibria

• stability

thus

• conclusions for monotonic
  convergence to
• for
  monotonic convergence to

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3rd Example
?
?
unstable
?
stable
?
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Exponential and logistic growth
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Exponential and logistic growth
?
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Exponential and logistic growth
?
Model 1
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Exponential and logistic growth
?
Model 1
hence exponential growth
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Exponential and logistic growth
?
Model 1
hence exponential growth
but this is not realistic -- due to
intraspecific competition the per capita
speed of growth decreases when the
population density increases
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per capita speed of growth decreases linearly
with N
Model 2
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per capita speed of growth decreases linearly
with N
Model 2
(cf. feiten en formules)
This is called logistic growth
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Qualitative analysis of logistic growth

• per capita speed of growth

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Qualitative analysis of logistic growth

• per capita speed of growth

• growth equation

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Qualitative analysis of logistic growth

• per capita speed of growth

• growth equation

• equilibria (evenwichten)

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Qualitative analysis of logistic growth

• per capita speed of growth

• growth equation

• equilibria (evenwichten)

• stability

hence
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Qualitative analysis of logistic growth

• per capita speed of growth

• growth equation

• equilibria (evenwichten)

• stability

N(t) for N0gtK
N(t) for N0ltK
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the logistic growth equation can be solved
explicitly
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the logistic growth equation can be solved
explicitly
N0ert
exponential growth
N(t)
logistic growth
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the logistic growth equation can be solved
explicitly
N0ert
exponential growth
N(t)
logistic growth
Properties

• at low density

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the logistic growth equation can be solved
explicitly
N0ert
exponential growth
N(t)
logistic growth
Properties

• at low density
• monotonic convergence to stable equilibrium K

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the logistic growth equation can be solved
explicitly
N0ert
exponential growth
N(t)
logistic growth
Properties

• at low density
• monotonic convergence to stable equilibrium K
• NK/2 ? speed of growth is maximal, i.e. N(t) has
  point of
• inflection when density is equal to K/2

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the logistic growth equation can be solved
explicitly
N0ert
exponential growth
N(t)
logistic growth
Properties
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Example

• Suppose a population is growing according to the
• following logistic growth equation specified by
  the following
• properties
• At low densities the population doubles every 4
  hours.
• The maximal speed of growth is reached at the
  density
• 5000 ml-1. How much time does it take the
  population to
• grow from a density of 1000 ml-1 to a density of
  9000 ml-1?

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Example

• Suppose a population is growing according to the
• following logistic growth equation specified by
  the following
• properties
• At low densities the population doubles every 4
  hours.
• The maximal speed of growth is reached at the
  density
• 5000 ml-1. How much time does it take the
  population to
• grow from a density of 1000 ml-1 to a density of
  9000 ml-1?

Solution
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Example

• Suppose a population is growing according to the
• following logistic growth equation specified by
  the following
• properties
• At low densities the population doubles every 4
  hours.
• The maximal speed of growth is reached at the
  density
• 5000 ml-1. How much time does it take the
  population to
• grow from a density of 1000 ml-1 to a density of
  9000 ml-1?

Solution
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Maple