1. A model of a car dealership. The number of customers is random and sales occur with a certain probability. New stock is delivered with a delay. At given model parameters the stock goes to zero at certain times. How to avoid this if:
a) the delivery time is no less than 10 days?
b) or the size of the warehouse is limited by 100 units?
c) or both of the above? Can you change the dealership operation in such a way, that the average number of units sold daily will be above 10, while the stock size will be maintained below 50?
|Stock (blue) and # deals/day (red) using either|
a moving average or 10, whichever is greater.
2. A simple population growth model, that has a certain growth rate and mortality rate. At this time it shows exponential growth with the population increasing to infinity. Modify the model in such a way that the population would grow to 400 and stabilize at about that size. Can this be achieved by changing parameter values only or the functions that describe flows need be modified?Booyah! #2 came much easier and more elegantly, if I may say so myself. All I did was modify the birthrate to be a function of the ratio of the population to the target population relative to the death rate. So my birthrate equation became
(Target_Population/Population)*Death_Rateand the model and output ended up looking like this. Notice that now we can stabilize the population at whatever target we want.
3. The system describes a human population, which growth also depends upon the level of "development". The larger the population, the higher level of development it can reach. The development in turn stimulates population growth. As development grows, more babies are born, therefore the higher the population. Is there anything wrong with this conceptual model? What modifications could make the system "sustainable"? (That is, keep both variables within certain limits, avoiding extinction and runaway growth to infinity)
4. A model of CO2 and biomass on this planet. CO2 is released into the atmosphere. It is consumed by the forests. The forests are cut at a certain rate. They also decompose adding to the pool of CO2. Currently the model displays infinite growth of forests. How to modify the system to stabilize it?
|Deforestation = Biomass / 100|
5. Model of body weight. You gain weight because you eat. You loose weight due to exercising. However the more you exercise, the higher your appetite. Also the more you weigh the higher the appetite. So the weight grows and grows. Can this pattern be changed?
6. A river is represented by 5 identical reaches. The water collected from the catchment area drains into the river reach N5 and then travels down the river. The precipitation pattern is random and results in sharp oscillations in the river stage in the reaches. What changes to parameters and/or functions are necessary to stabilize the system so that
1) there is no accumulation of water in the river;
2) the oscillations in the water depth are smoothed out.
Modify the model to describe the travel of one pulse of water through the river system.
Fun with modeling! I really love this stuff and am glad to have moved from the theoretical part of the course to the hands on part. Can't wait until we're building our own models. First, "Essential Math."