Wolfram Demonstrations Project

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.

Contributed by: Mark D. Normand and Micha Peleg

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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.