While working through this, keep in mind that the sheet is live so you can modify anything you want to check out how it is done.
Create a vector 'pop' with a
For calculations to come later, create an index variable 'i', with integer values from zero up to the number of estimates of
Be careful with your index values:
If you later create a vector variable using an
algebraic function of pop, that vector will begin with row zero, rather than row one.
You can use a regression approach to estimate the mean growth rate
Use the slope function to determine the slope of regression of ln(pop) on year,
as described by Mills.
Here's some code to add a ln-linear regression fit
to a plot of the raw data:
And then check the value of
First calculate the vector of
And look at them.
If you click on the table, you will get a slider bar
to move through the rest of the values.
And to be careful, plot
Plotting your data is ALWAYS a good idea, because it often reveals outliers, data entry errors, logical flaws, algebraic errors and so forth.
Calculate mean
Note that ln(mean
Assign values 1 to 50 to t, to project dynamics for 50 years.
Start with a population size of 100.
Draw 50 growth rates from a ln-normal distribution with mean
Look at the 50 random values of
And confirm that they are a reasonable simulation of the original data, by calculating and displaying the stochastic mean growth rate.
If you put the cursor in the rlnorm function (above) and press F9, a new iteration of the random sampling from the ln-normal distribution will be calculated.
Make a stochastic exponential growth projection model using the random draws from the empirically estimated distribution of population growth rates.
And confirm that the output is a reasonable simulation of random-walk dynamics from an initial
population of 100 individuals, by display the vector of N values...
...and by plotting the population growth projection of N[t vs t.
First use the
Assign some values to K and
... and modify the stochastic exponential growth equation so that it is theta-logistic.
Plot the results as a trajectory of ppopulation sizes, recognizing that this a model of population growth with a reflecting upper boundary (K) and an absorbing lower boundary, zero.
Scroll up to look at the outcome under exponential growth, with the same set of random draws for growth rates.
Modify the parameters describing density
dependence and recalculate the sheet several times to investigate the impact on projected dynamics.
A second density-dependent model, also commonly used, was developed by Robert May.
Like K and
Compare this model's predictions to the theta-logistic model of density-dependent growth.
Modify the parameter values and replot several times to explore this model's behavior.