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Each project includes a description of a base model and some possible extensions. Your group will need to fill in the details of the base model and analyze that, then choose ONE of the suggested extensions and modify your model to explore that extension. The exception to this is Project 4: Ontario’s Population which only has one suggested extension (since this project has you comparing results from two different places). Unless otherwise specified, you may construct either a discrete or continuous model.
Project 1: Arms Race
Imagine two economically competing countries: Purple and Green. Both countries desire peace and hope to avoid war. While neither country will go out of its way to launch an aggression, it also wouldn’t sit idly by if their country was attacked. Each country believes in self-defense and will fight to protect itself. Both nations feel that the maintenance of a large army and the stockpiling of weapons is ”defensive” when they do it but ”offensive” when the other nation does it.
In this project, you will investigate how arms spending varies over time in each nation, based on various assumptions about how each country’s arms buying behaviour influences the other.
Base model
Let p(t) and g(t) represent arms spending in year t by Purple and Green, respectively. The two nations are in competition, so there is an underlying sense of mutual fear: the more one nation arms, the more the other nation is incentivized to arm. To start, we could assume that each nation’s armament rate is directly proportional to the other’s arms spending (these should be linear terms in your model). We may also want to account for the fact that excessive armament expenditures come at a cost to each nation’s economy, i.e., the rate of change of each nation’s expenditures is directly and negatively proportional to its own expenditure (again, these should be linear terms in your model). Finally, we can additionally model any underlying grievances or feelings of goodwill of each country toward the other using constant terms. The sign of these constant terms determines whether the term models grievances, which increase armament spending, or feelings of good will, which decrease it.
Construct a continuous model to describe this scenario. What does your model predict for the long-term arms spending of each country?
Possible extensions
Logisitic mutual fear
For the mutual fear terms, assume that there is some inherent limit to the amount a nation can spend on armaments each year (call it Kp and Kg for Purple and Green, respectively). Then, you could replace the linear terms with logistic terms (a(1?x/Kp )y and b(1?y/Kg )x).
Three nation model
Imagine there is a third nation involved and build a model for armament spending between three mutually fearful nations. You could also consider the case where two of the countries are close allies who are not threatened by the arms build up of each other but are each threatened by the expenditures of the third nation.
Project 2: Carbon Cycle
Carbon is a key element in terrestrial ecosystems. It enters the soil when plants die, or shed leaves and branches (called “litterfall”, https://en.wikipedia.org/wiki/Plant_litt er); it leaves the system by being turned into carbon dioxide by bacterial metabolism or other chemical processes. In recent history, human industrial activity has begun to interfere substantially with the biological carbon cycle in various ways, leading to climate change (https://en.wikipedia.org/wiki/Climate_change).
In this project, you will model carbon cycling (https://en.wikipedia.org/wiki/Carbon _cycle) from the atmosphere and back through various components of an ecosystem (e.g., plants, litterfall, humus [https://en.wikipedia.org/wiki/Humus]) and examine the effects of human activity on this cycle.
Base model
Start simply by tracking carbon levels in the litter on a forest floor alone (litter being naturally-occurring debris like leaves, branches, and deadfalls, that decay over time, not human-made trash). Let c(t) be the density of carbon in the litter at time t (measured in grams of carbon per square metre, or gC/m2 ).
Assume carbon enters the litter through litterfall continuously at a constant rate, z, and leaves at a rate proportional to the amount of litter currently in the system via humification (the conversion of litter into humus). Set up a simple discrete-time (or continuous-time) model for this scenario and analyze it.
Now account for atmospheric carbon by assuming that humus and litter both respirate carbon dioxide in to the atmosphere at rates proportion to the amount of each substance in the system, and that atmospheric carbon gets converted back into litter via plant growth, which produce litter at some rate proportional to the number of trees. Set up a model for this scenario. You may find it easiest to set up one state variable for the density of carbon in each component of this system: litter, humus, atmosphere, and plants.
Finally, add some human activity to the model, and explore the repercussions. For instance, what if humans were to plant trees each year to draw out more atmospheric carbon? What if humans were to cut trees down to use this natural resource to produce goods? How might this affect model predictions?
Possible extensions
Expanded ecosystem
Model more components of the carbon cycle: plants (subdivided into leaves, branches, stems, and roots), litter, humus, and stable humus charcoal, denoted by x1 , x2 , ...x7 , respectively. Atmospheric carbon flows into plants via photosynthesis. Leaves, branches, and stems in- crease the carbon in litter, which then increases the carbon in the humus via humification. Roots increase the carbon directly in the humus (not via litterfall). Humus increases the carbon in the stable humus charcoal via carbonization. Litter, humus, and stable humus charcoal all increase carbon in the atmosphere via respiration.
As a simplification, assume that the atmosphere has a constant carbon content (un- changed either by giving carbon to plants or by absorbing carbon from the litter, humus, or stable humus charcoal), since it contains so much more carbon compared to the other components of the system. You can therefore imagine that the atmosphere is outside of the system and only model the other components. The atmosphere simply introduces a con- stant rate of carbon, z, into the system, and proportions p1 , ..., p4 indicate how much of this constant influx of carbon gets allocated to leaves, branches, stems, and roots, respectively. Parameters kij give the rate of carbon flow from xi to xj . Any carbon going back into the atmosphere can be thought of as simply leaving the system, which occurs at rate ki0 for carbon flowing from xi to the atmosphere.
A compartmental diagram of this model is as follows:
Model parameters for various ecosystems can be taken from the following table, which comes from ”A Simulation Study for the Global Carbon Cycle, Including Man’s Impact on the Biosphere” by Goudriaan and Ketner:
(The unit Gt is Gigatonnes, or a billion tonnes, where 1 tonne (metric ton) = 1000 kg =
1 Mg.)
Note that the “leaving litter” flow does not distinguish between carbon leaving litter by humification or respiration. The humification factor h gives the proportion of the “leaving litter” flow that goes into the humus, which leaves 1?h of the flow togo into the atmosphere via respiration. Similarly, the “leaving humus” flow must be divided into the proportion that goes into the stable humus charcoal via carbonization, c, and the proportion that goes into the atmosphere, 1 ? c.
You could consider one or several ecosystems based on the parameters in the above table, and compare the predicted carbon-cycling behaviour between ecosystems.
Seasonal parameters
Many parts of the world experience seasonal climate variation, which affects the growth and decay of plants. Consider some seasonal variation in plant-related flows (e.g. by assuming that some or all of per-capita plant-related flow rates are not constant but vary sinusoidally with a period of a year). How does this change your model predictions?
Project 3: Drugs in the Body
The way in which drugs are administered to individuals and then metabolized by the body is of great concern in pharmacology and medicine: drug dosage over time must be high enough for some period to have a medicinal effect on the patient, but not so high that they may overdose. There are various drug delivery methods available to those designing patient therapies. Metabolic pathways also vary depending on drug administration and type.
In this project, you will compare the effects of various drug delivery methods and metabolic pathways on a patient’s dosage over time.
Base model
The rate at which the body processes drugs depends on two factors: the rate at which the drug is administered and the drug processing rate. Let A(t) be the amount of a drug in the body (in milligrams) at time t. Then we can denote the dosing rate D(t), which is independent of the current amount of drug in the body A(t), and the processing rate P(A), which does depend on A. Then the model for A(t) can be cast as an “inflow minus outflow” relationship:
As a first pass, assume that a total of Dtot mg of the drug is given intravenously at a constant rate of r mg/hour for the first h hours, at which point drug administration ceases:
We also assume that drug processing occurs linearly, proportionate to the amount of drug currently in the system: P(A) = cA, where c is the clearance rate.
Note that, while D(t) is piecewise, you can split the domain into t ∈ [0, h] and t ∈ (h,∞), and then study what is happening in each piece of the domain to get a sense of the overall dynamics.
Assume the drug is given as a pill that is designed to dissolve slowly. As the pill dissolves, less and less of the medication is released. At any instant, the rate of release of the drug can be modelled by D(t) = Dmaxe?t/h. Calculate how much of the drug is released by the pill in the time interval [0, h]. Compare this dosage plan to the intravenous plans you previously explored.
Possible extensions
Logistic drug metabolism
Assume that instead of P(A) = cA (exponential drug clearance), the drug is metabolized logistically, i.e., P(A) = cA(1 ? A/K). Explain how one might interpret the new logistic parameter K in this context. How does this change the amount of drug in the bloodstream over time, compared to the intravenous treatment plans explored previously?
Intermediate absorption compartment
Consider a compartmental model where the drug needs to enter another compartment before becoming bio-available (e.g., it is injected in the blood but needs to diffuse into organs before it can be used, or it is swallowed into the digestive system and needs to diffuse into the blood stream). Model the amount of drug in each part of the body over time (blood, digestive system, organs—whichever apply in your context) as well as the diffusion processes between these body parts. Make a variety of assumptions for the diffusion mechanisms and explore how these affect model results.
Project 4: Ontario’s Population
In order to plan for the future, governments often use projections of population size to project demands on social services. Various factors affect population size: births, deaths, immigration, and emigration.
In this project, you will build a demographic model to project Ontario’s population growth or decay in the future.
Base model
The base model for this project will focus on population changes in California compared to the rest of the United States. You will then extend this to do something similar for Ontario’s population compared to the rest of Canada, and compare between Ontario and California.
Let (t) denote the populations of California and the US excluding California at time
t, respectively (so the first component is California’s population and the second is the rest of the US). The following table gives population values in 1955 and 1960, along with births, deaths, and net migrations (immigrations - emigrations) between these two regions:
Note that these population counts are in units of 1,000 people. Use this information to calculate the per-capita rates associated with each process by filling in the following table:
Two rates have already been computed for you so that you can verify your calculations are sound.
Set up a matrix model of the form P(t + 1) = (I + B - D + M)P(t), where the matrices
B , D , M are birth, death, and net migration matrices, respectively (which should all be diagonal), and I is the identity matrix. Explain why I is needed here. Using the 1955 data and the rates you computed, verify that your model accurately predicts the 1960 population.
Simulate the model to 2020 and verify your predictions against the https://www.cens us.gov/data.html). How close were you predictions to the actual outcomes? What may have caused discrepancies between you projection and the true outcomes?
What does your model predict in the long run for the population distribution between California and the rest of the US (as t → ∞)? You may find it easier to perform this analysis if you define a general ”growth” matrix G = I +B -D+M. What is the long-term rate of growth of the population of California? What proportion of US residents will live in California in the long run, according to your model?
Now work with immigration and emigration separately (instead of looking at net migra- tion). Migration between the two regions is summarized in the following table:
Ignoring births and deaths, construct a matrix model of transitions between regions with the form P(t + 1) = TP(t). Again, simulate the model to 2020 and compare it with US census estimates. Discuss any discrepancies. Determine the long-term behaviour of the model, including the long-run growth rate of California’s population, as well as the population distribution within the US. How do these results compare to the earlier model with births, deaths, and net migration?
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