MST210 Mathematical Methods Models and Modelling

1) [10 marks] Consider the “mortgage equation” from lecture 1(the slide is copied below)

2) [10 marks] Read the following texts

 

“How Populations Grow: The Exponential and Logistic Equations”

 

3) [10 marks] Read the following texts

 

Logistic function as a forecasting model: It’s application to business and economics

 

Describe briefly the problems the author had solved in this paper. What is the difference from the population dynamics?

 

Could you perform the same task for the British or World economy? Please formulate the problems and describe the data you need

 

4) [10 marks] Read the following text

 

“The Logistic Equation and Integration by Partial Fractions”

 

Invent and solve the logistic equation with hunting. What is the main difference from the logistic equation and its solution?

 

What is the main difference of the logistic equation with hunting and its solution from the logistic equation and its solution?

 

5) [10 marks] Find the explicit general solution of the Richardson model from the lecture 1 (the copy of this system is below):

When the solution is bounded for all initial states? When it is unbounded?

 

6) [10 marks] Let Y do not depend on X, Y=Y(t). In this case instead of the Richardson model we have one equation

where Y is a given function of t.

 

Please solve this equation for an arbitrary continuous bounded Y(t). Can this solution X(t) be unbounded? Why? Discuss the result: what could it mean for real life?

 

7) [20 marks] Smoking spreading. Assume that smoking causes specific illnesses. Also assume that smoking is virulent as infectious disease. Introduce several states, for example, NH (Non-smoking Healthy), NI (Non-smoking Ill), SH (Smoking healthy), SI (Smoking Ill), Q (Quit). Invent a scheme of transitions and write the kinetic equations. What is non-realistic in your model? How can we improve the model?

 

8) [20 marks] Dynamic of reputation and marking. A teacher expects good performance from students with good reputation and bad performance from students with bad reputation. Marking depends on expectations (the teacher is not perfect). Performance also depends on reputation (it is difficult for a student to perform excellent when he expects low marks). Reputation depends on marks. The process is not deterministic. Create a model of reputation dynamics. Discuss possible improvements of model and teaching.