EV222 Homework Assignments

#1 - Due Tuesday 10/02

1. An invasive population is growing at a rate of 4% per year.  How long will it take the population to double?  To triple?  What key assumption are you making?

2.  A pollutant is released into a lake.  Immediately after its release, its concentration is 250 ppm.  Two years later the concentration is 160 ppm.  Assuming exponential decay, what is the decay rate?  How long will it take until the concentration is 10 ppm?

3.  For each of the situations below, write down a differential equation and initial condition that satisfy the description:
          a) Water flows into a million cubic foot reservoir at 25 cubic feet/minute and flows out at 19 cubic feet/minute.

          b) A glacier gains 500 cubic meters of new ice per year, and evaporates at a rate of 15% per decade. 
                In 1950 the glacier contained 56000 cubic meters of ice.

          c)  A prairie dog colony has a per capita birthrate of 1.6 per year, and a per capita death rate of 1.3 per year. 
                It also gains 200 individuals per year through migration.  The current population is approximately 10,000. 

4.  Implement each of the models in question 3 in a single Stella file.  Run each model over an appropriate time interval,
        and plot the results.


#2 - Due Thursday 10/04

1.  Aspen trees are efficient colonizers of areas opened up by disturbances like forest fires, avalanches, and logging.  The shade from aspens protects seedlings of coniferous trees like spruce which eventually crowd out the aspens.  High density monocultures of either aspen or spruce are susceptible to insect and disease outbreaks, and spruce forests are susceptible to fire.  Draw a causal loop diagram of the aspen-spruce system.   Write a few sentences explaining how the positive and negative feedback loops in your diagram affect the long-term coexistence of these species.  Be clear about the spatial and temporal scales over which you are describing coexistence.

2.  Spruce budworm is a significant pest in many North American coniferous forests.  It is characterized by occasional massive outbreaks, which often have no obvious cause.  It has been subject to extensive mathematical modeling to try to explain this phenomenon and evaluate control strategies.  We will consider a simple, classic model proposed in the 1970s by Morris et al.
Let S be the population of spruce budworms, expressed in units of individuals per tree.  In the absence of predation, we assume that the budworms will exhibit logistic growth.  Birds feed on the budworms, representing an important cause of budworm mortality.  However, birds also have other prey, so they are not reliant on budworms - they tend to switch their prey preference depending on the abundance of various prey species.  We will assume for simplicity that the bird population is constant.
    a)  Let B be the (constant) bird population.  We assume that the rate of predation by birds is: f(S) = a*B*S2/(C2 + S2).
         Graph f(S) over the range (0, 5000) for B = 10, a = 20 and C = 1000.  Then change a and C to determine what effect they have.
        What is the biological meaning of each of these parameters?  Think about what they represent in terms of bird behavior.
     
    b) Write down the differential equation that includes logistic growth of the budworms and the predation function above.  What are         the units of each parameter value?

    c)  Draw the phase line and determine the stability of any equilibria for the following parameter values:
           i) r = 0.1, K = 4000, B = 10, a = 20, C = 1000
           ii) r = 0.1, K = 8000, B = 10, a = 20, C = 1000

    Discuss how this analysis helps shed light on the outbreaks of spruce budworm.

3.  Implement the spruce budworm model in Stella.  Use the second set of parameter values, and run the model with low and high initial budworm populations to illustrate both equilibria.



#3 - Due Monday 10/08  - Do in groups

Choose one of the following questions about the chronic wasting disease model.  After a thorough investigation, turn in your modified Stella file and a brief (< 2 pages plus figures) paper explaining what you did and summarizing your findings.

1.  Carry out a complete sensitivity analysis on all of the parameters in the model.  Vary each parameter over a reasonable range, and look at the effect on the number of infected deer.  Categorize the parameters into high, medium, or low sensitivity.  What parameter(s) should biologists focus their efforts on measuring?

2.  Incorporate a carrying capacity for the deer.  How does the value of the carrying capacity affect the epidemic?  How does it affect the ability of hunting or culling to control the disease?

3.  Use reservoirs instead of conveyors for the exposed and infectious deer.  Be sure to adjust parameter values to keep the same average length of time spent in each stage.  How does this affect the epidemic?  How does it affect the ability of hunting or culling to control the disease?

4.  We used "frequency-dependent" transmission, in which the exposure rate was proportional to the fraction of deer that are infectious.  Replace this with "density-dependent" transmission, in which the exposure rate is proportional to the number of deer that are infectious.  You will need to adjust the "contacts" parameter downward - it now represents the fraction of the population that an individual comes into contact with per unit time.  Compare the behavior of the new model with the old one.  What do you conclude about the ability of hunting or culling to control the disease?

#4 - Due Tuesday 10/09 - Do individually or in pairs

Consider a predator-prey model that includes logistic growth of the prey, and a Type 2 functional response for predation.
Note: Type 2 functional response is the one that includes a handling time, but is _not_ s-shaped.  In class I mis-stated the names.  The correct nomenclature is: Type 1 = Lotka-Voterra; Type 2 = handling time (aka saturating); Type 3 = s-shaped

1.  Write down the differential equations.

2.  Sketch the phase plane (isoclines and direction arrows) for two cases.  In both cases, let r = 0.1, g = 0.005, h = 0.5, b = 0.33, and m = 0.1.  In the first case, let K = 100.  In the second case, let K = 600.  Use your phase plane to try to predict the dynamics of the model (stable equilibrium?  sustained oscillations?...).

3.  Run the Stella model "paradoxofenrichment.stm" in the class folder on the course drive.  Do both of the cases from question 2; be sure to try several different initial conditions.  Print out or sketch a graph of the dynamics for each case.  Were your predictions in #2 right?

4.  Why is this called the paradox of enrichment?  Discuss whether "enriching" a system by increasing K is stabilizing or destabilizing, and whether this was obvious or surprising.  What are some of the ecological implications of this for human-impacted systems?

#5 - Due Friday 10/12 - Do in pairs

Implement the river pollution model in Stella.

1.  Assume no effluent.  Let c = 0.1 mg/(L*min), k1 = 0.03 /min, k2 = 0.04 /min, and Osat = 11 mg/L.  Run the model long enough to estimate the equilibrium.

2.  Assume that upstream from the effluent source the  flow rate of the river  is 300,000 L/min and  the oxygen and BOD levels are at the equilibrium values from above.  Assume effluent flows into the river at 100,000 L/min with oxygen at 2 mg/L and BOD at 40 mg/L.
Plot oxygen and BOD levels for 200 minutes after the effluent input.

3.  Assume that the river is flowing at a rate of 20 meters/min.  Oxygen values below 6 are typically considered to indicate significant pollution.  How far (in meters) downstream from the effluent source is the river polluted?

4.  Suppose that there is a popular fishing area 2 km downstream from the eflluent source.  If we want to make sure that oxygen levels at the fishing area are at least 8, how large can the effluent flow be?  Show a graph to support your answer.

#6 - Due Tuesday 10/16 - Do in pairs

Read each question carefully; most have several parts.  Write up your answers in text boxes within Stella, and include supporting graphics.  When you are done, copy your Stella file to the course drive HW folder.  Note that you will be modifying the model and running different versions withing the same file.  It is a good idea to keep the model flexible by using converters rather than "hardwiring" new constants into the formulas.

1. Run the air pollution model to predict emissions over the next 30 years.  Plot the total emissions along with the total number of cars and the average age of cars.  Explain the pattern in emissions.  According to the model,  what percent of total emissions 30 years from now will be contributed by each of the 4 vehicle age categories?

2.  Suppose that the average number of miles driven per year by each car goes up by 100 miles per year for the next 30 years.  Without changing the model, use a simple calculation to predict what percentage increase in emissions this will cause.  Then modify the model and run it.  How close is the agreement with the simple calculation? 

3.  Suppose that new technology allows the emissions rate of new vehicles to decrease by 2% per year for the next 30 years.  Assume that the rest of the vehicle fleet has emission rates 0.5 g/year higher than the age group immediately younger.  (These will also change over time.)  Modify the model and plot the total emissions over the next 30 years.  (Assume that miles driven per year doesn't change.)

4.  Suppose the state is considering a buyback program, in which the government will subsidize owners of old (>11 years) cars to allow them to buy new cars.  Assume that funding is available to replace 5000 old cars with new cars each year.  Show that in the original model (no mileage or efficiency changes), the buyback program causes emissions to increase.  Can you explain why?  Do you expect a buyback program to perform better if efficiency increases over time as in problem 3?    Run the model and check.  Finally, plot the total number of cars with and without the buyback program.  Does this explain your results?  Do you think it is realistic?  How might the model be modified to make this more realistic?  (You don't need to carry out the modifications!)

#7 - Due Friday, 10/19 - Do in pairs

1.  In the Stella model globalwarming1, carry out sensitivity analysis on the parameters "albedo" and "ra".  Study how these parameter affect the long-term surface temperature.  Summarize your results and explain why the model responds the way it does to these changes.

2.  In the Stella model globalwarming2, change the shape of the graphical function relating ra to CO2.  Compare linear, concave up, and concave down cases.  In all three cases the value of ra should increase with the concentration of CO2.  Look at the effect on the pattern of surface temperature over time.  Summarize and explain your results.

3.  In the model globalwarming2, change the CO2 dynamics so that they go up 5 ppm each year for 20 years, then up 2 ppm for 20 years, then down 3 ppm per year for 60 years.  (Be sure to set the ra graphical function back to its original value.)  Compare the temperature dynamics  in this scenario with the dynamics from the original CO2 pattern (before you changed it). 

4. Suppose that CO2 will go up 5 ppm each year for X years, then stay constant.  If we want to keep the surface temperature from exceeding 291.0 K, how soon do we need CO2 to level off?

5.  There are a number of possible feedbacks between greenhouse gasses and temperature: energy consumption, melting ice sheets, economic changes, permafrost thawing, CO2 dissolved in the ocean, plant growth, cloud formation,.....
Choose one positive feedback and one negative feedback and modify the globalwarming2 model to include them. Run the model with only one feedback loop at a time, then with both present.  Summarize and explain how they affect the CO2 and temperature dynamics.  Keep in mind that complex models that include these processes predict temperature changes on the order of a few degrees.  If your model predicts that temperatures will go up or down by a hundred degrees, you probably need to adjust some parameter values.
Note:  These are extremely complex processes that are still poorly understood!  Your model will be necessarily simplistic - include these factors in the simplest possible ways. 

Be prepared to discuss your results from question 5 in class on Friday.  Write up your explanations within Stella, and turn in your files to the HW drive by 4:30 on Friday.