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====== Density-independent population dynamics with demographic stochasticity - Tutorial for Ecovirtual ======

{{section>en:ecovirt:roteiro:den_ind:di_ed_base#density-independent_population_dynamics_with_demographic_stochasticity}}

====The binomial distribution====

{{section>en:ecovirt:roteiro:avisos#roteiro_ecovirtual}}


Let us get acquainted with the idea of a probability distribution, by calculating values from the binomial distribution. On the **R Commander** menu, click //Distributions// -> //Discrete distributions// ->  //Binomial distribution// -> //Binomial distribution graph//. A window like this will open:

{{ :ecovirt:roteiro:den_ind:binomial_plot_dialog.png |}}

With the options:

^Option ^ Effect^
^''Binomial trials'' | number of trials, in our case $N_0$|
^''Probability of success''| probability of success at each trial, in our case the probability that an individual survives|
^''Mass function graph'' / ''Cumulative function graph''| should the graph be made for each value or for the cumulative probability until this value? |

Make the graph for our example:
- Set the ''Binomial trials'' to $2$
- Set the ''Probability of success'' to $0.5$
- Select the ''Mass function graph'' option
- Click ''Apply''

{{section>en:ecovirt:roteiro:den_ind:di_ed_base#the_binomial_distribution}}


==== Computer simulation ====

{{section>en:ecovirt:roteiro:den_ind:di_ed_base#computer_simulation}}

In the **EcoVirtual** menu, click //Ecovirtual// -> //One Population// -> //Demographic Stochasticity//. A window like this will open:

{{ :ecovirt:roteiro:math:onepop_demo_stoc_dialog.png |}}

{{section>en:ecovirt:roteiro:den_ind:di_ed_base#parametros}}

====Population size distribution====

Let's inspect the distribution of the population sizes in the time $t=2$. To do this, we need to store the simulations in an R object. Run more simulations after setting the following options:

* ''Enter name for last simulation data set'': sim1
* ''Maximum time'': 2
* ''Number of simulations'': 1000
* ''Initial size'' : 20
* ''birth rate'' :  0
* ''death rate'': 0.693

And click ''OK''. The results will be stored as a list of 1000 tables in R, called ''sim1''. Each table contains the times in which the population lost an individual and the size of the population after that time, up to the maximum time set in the options window.

To see the first table, close the simulation window, copy the following command to the ''Script'' window of the Rcmdr and click ''Submit''

<code>
sim1[[1]]
</code>

The table will be written on the window ''Output''. Inspect the other tables: run the same command again, now changing the numeric index inside the double brackets to any other value between one and one thousand. 

{{section>en:ecovirt:roteiro:den_ind:di_ed_base#population_size_distribution}}


===== Births and deaths =====
{{section>en:ecovirt:roteiro:den_ind:di_ed_base#births_and_deaths}}

===== To learn more =====
 {{section>en:ecovirt:roteiro:den_ind:di_ed_base#to_learn_more}}


{{tag>Rcmdr uma_população crescimento_exponencial tempo_discreto tempo_contínuo estocasticidade_demografica}}