9464

A Probabilistic Model for Population Extinction

A fluctuating organismic population can become extinct if its size drops below a certain threshold. This Demonstration simulates hypothetical life histories in which the population's random fluctuations have lognormal distributions, characterized by and , which terminate when the population size falls below this chosen critical number. The program records the extinction times and plots a histogram with a superimposed exponential distribution function having the same mean as the recorded extinction times. It also shows the quantile-quantile (q-q) plot used for testing the exponential distribution as a model.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Snapshot 1: relatively small fluctuations and low threshold
Snapshot 2: relatively large fluctuations and low threshold
Snapshot 3: relatively large fluctuations and high threshold
Snapshot 4: very small fluctuations and low threshold
There are many stochastic models of populations. This Demonstration addresses a special case of organismic populations whose observed size fluctuates following the model , where is the number of individuals after (arbitrary) time units, and and are the logarithmic mean and standard deviation that you can choose with the sliders, and is a random number between zero and one () drawn from a uniform distribution.
If the population size hits a low , being the threshold you chose, the population becomes extinct, and all successive counts equal zero [1]. The program generates a number of random records using the chosen parameters, retains the extinction times according to the above criterion, and plots their histogram. It then superimposes on the histogram the PDF of the exponential distribution having the same mean as that of the extinction times and draws the corresponding quantile-quantile (q-q) plot.
The model parameters , , and , the generation parameters, the number of runs, and the run length , as well as the seed value, are entered with sliders. You can also view any individual run by selecting its index with a slider.
Reference
[1] M. Peleg, Advanced Quantitative Microbiology for Foods and Biosystems: Models for Predicting Growth and Inactivation, Boca Raton, FL: CRC Press, 2006.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+