Problem Set #3

This problem set has three parts and is worth 10 points total.
Part A deals with using Excel to fit sets of points.
Part B deals with pulling parameters for a set equation off of a graph.
Part C deals with choosing a appropriate mechanism and equations that underlay the problem.
Part D deals with analyzing STELLA models
Part E deals with isolating the parts of a equation

Part A: Use Excel to choose an appropriate equation for this graph.

Instructions:

• enter the values
• chart them
• click on the series of points to select them
• go to the pull down menu Chart:add trendline to get the dialog box
• select which type of trend line or regression you want - or try different ones
• but before you close -- choose options and have the equation displayed on the chart

Part B: fitting equations to graphs (3 points)

The following three graphs are built on the following equations.

sigmoidal	= maximum1 * X^n/(X^n+20^n)
sigmoidal + 20	= 20 + maximum2 * X^m/(X^m + 25^m)
integrated form of the logistic equation (see below*)	=K/(1 + (1/Nzero - 1/K)*K*EXP(-rmax*X))
	where: maximum1 and maximum2 are constants n and m are exponents > 1 K = 200 rmax = 0.1 Nzero = 20

5 Assignment: replicate this above figure in Excel, finding the right parameters in each of the 3 equations given above. Comment on the suitability of these equations to fit a population growth. What if you were doing a fit for a chemical reaction?

* This is the integrated form of the logistic growth equation. Instead of iteratively calculating the change in population, this form of the equation calculates the population for any time entered. For more information on the integrated form of the logistic equation you can check out the Wikipedia entry on Verhulst at http://en.wikipedia.org/wiki/Pierre_François_Verhulst

Part C: Plotting and transforms

6. Replot the data on this figure and plot it. Then fit the points to an exponential function. Why is this plotted as a log-transform rather than simple arithmetic scale?

7. Create a similar set of data for x and y that when plotted on a log(X) vs. log(Y) gives a straight line. For example, from this figure you might use (100, 30), (1000, 60) and (10000, 120). Plot this data with on a linear plot and on a log-log plot. What is the underlying mechanism that leads to these observations, and why does it make sense to plot this as a log-log plot, i.e. what is the meaning of a ratio for each axis.

Part D: Analyzing STELLA models

8. Population growth

The following population model holds as the birth and death rate change as shown in the graph below.

a. Sketch what you think the relative total population would look like over this period of changes in the birth and death rates. (Start with 1000 people).

new_births = birth_rate*population
loss_to_death = death_rate*population

 

9. This model of the logistic growth and harvest from a population has simulated variation and noise.

a. Describe all the components of this model, piece by piece. In particular, describe the equations that cause variation in the growth rate. How do they operate?

b. Link the variation simulated in the model equations to the graphical output shown below (i.e. use the graph to illustrate how the variation works and at what scale.)

accumulated_harvest(t) = accumulated_harvest(t - dt) + (harvest_flow) * dt
INIT accumulated_harvest = 0

INFLOWS:
harvest_flow = population*harvest_proportion
population(t) = population(t - dt) + (new_animals - harvest_flow) * dt
INIT population = 50

INFLOWS:
new_animals = population*growth_rate
OUTFLOWS:
harvest_flow = population*harvest_proportion
carrying_capacity = 800 + 200*sin(time/6)
growth_rate = short_variability*max_growth_rate*(carrying_capacity-population)/carrying_capacity
harvest_proportion = 0
max_growth_rate = .1
short_variability = 1-.2*random(0,1)

Part E: Analyzing pieces out of an equation

10. The carbon dioxide in the atmoshpere has been changing with a general trend over the last decades and it changes over the season. This can be modeled by a combination of an exponential function and a sin function. If you want to see some real data please see http://cdiac.ornl.gov/trends/co2/csiro/csiro-mlo.html. The figure below, looks very similar and was generated by the follwoing function:

CO2 =355*EXP(.005*(year-1990))+3*SIN(year*2*PI())

a. In your own words describe the pattern shown by this figure and all the parts of the function.