Stochasticity You can download the notes from: http:www'helsinki'fijlaaksoteaching

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StochasticityYou can download the notes
fromhttp//www.helsinki.fi/jlaakso/teaching/
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Stochasticity
randomness, or we do not know how to
predict the next data point , noise
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White noise
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Red noise
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Blue noise
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How to generate stochasticity?
Can we find something that is truly random?
(http//www.fourmilab.ch/hotbits/)

• Pseudo-random processes random-number generators
• Deterministic process which is however
• sufficiently complex in dynamical terms
  (complexity it is difficult to predict
  successive data points prom the previous ones).

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Matlab example

• Built-in functions
• rand (uniform distribution between 01)
• randn (normal distribution)
• poissrnd (poisson distribution)

How to adjust statistical properties of random
numbers? Example adjust the mean and range X
rand(1,10) generate random numbers Y X a
b set mean, range
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Creating autocorrelated (i.e. coloured) noise
MATLAB example (AR1-process) for t 1 tmax
X(t1) X(t)a randnb end
Where a -1ltalt1 defines the autocorrelation
structure negative values of a produce blue
noise, positive values red noise when a0 we get
uncorrelated white noise. b defines the variance
- There are also other ways to produce correlated
variations -
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White AR noise,a0
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Red noise,agt0
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Blue noise, alt0
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The biological meaning of stochasticity !
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What is stochasticity in the context of
population models?

• Uncertainty in
• individual deaths and births ? demographic
  stochasticity
• uncertainty in environmental factors affecting
  population growth ? environmental stochasticity
• (weather, other species, )

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Demographic stochasticity individual-based
model of population growth process

• Matlab example random births
• Some assumptions
• No density dependence
• Only females, adults die after reproduction
• Fixed probabilities for producing n offsprings /
  female

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Environmental stochasticity and population level
models

• Example exponential growth model where to put
  the stochasticity?
• N(t) N(t)Rc (deterministic exp. growth)
• where parameter R is constant
• N(t) N(t)Rs (stochastic exp. growth)
• where Rs is stochastic (e.g., good year/bad
  year, or normally distributed)

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Environmental stochasticity and population level
models

• Example logistic growth model where to put the
  stochasticity?
• N(t) N(t) N(t)Rc(1-N(t)/Kc)
  (deterministic)
• where parameters Rc and Kc are constants
• N(t) N(t) N(t)R(1-N(t)/Ks) (stochastic K)
• where parameter K is stochastic (e.g., good
  year/bad year, or normally distributed)

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Stochasticity, models and data

• Uncertainty in the process itself
• demographic stochasticity
• environmental stochasticity
• Uncertainty due to measuring error
• Independent of the process
• Real data usually contains elements of both!

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Stochasticity, models and data

• The source (process vs. measurement error?) and
  type (statistical properties?) of uncertainty in
  data becomes important when
• decisions are made about how to model the system
• models are fitted to data

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GO, PRACTISE !1. demographic stochasticity2.
population level stochasticity