Math 4600, Final Exam (take home)
1. (From Modeling the Dynamics of Life by F. Adler) Consider an interaction between two mutually-inhibiting proteins with concentrations x and y , given by the differential equations dx dt
= f ( y )
− x dy dt
= g ( x )
− y.
Both f ( y ) and g ( x ) are positive decreasing functions.
a) Explain each of the terms in these equations. What is the influence of the proteins on each other?
Do part b) for the following two cases:
Case 1 : f ( y ) =
1
1 + αy
, g ( x ) =
1
1 + αx
;
Case 2 : f ( y ) = e −
αy , g ( x ) = e −
αx .
b) Vary value of α ( α > 0) and, convincingly, tell me what will be the dynamics of the system at these different values of α . Here are some points you may want to discuss: How many equilibria are there (your result from part b may be useful here)? Is it possible to have 3 equilibria? Draw nullclines and direction arrows in the representative cases. Are the eqilibria stable or instable? (Use analytical work, geometric analisis, sketches by hand, printouts of computer plots or simulations, as you see fit) c) How can a biological system with three equilibria act as a switch? Why may it be important?
2. There is evidence in some tumors that intermediate levels of nutrient are sufficient for the cells to survive, but not to proliferate, so that there is a layer of quiescent cells between the necrotic core and the outer proliferative layer, which consume nutrients at a lower rate than the proliferative cells. Set up a mathematical model for this situation. What conditions would you apply at the boundaries between the layers? (you do not need to solve this model)
3. Suggest a combined model of spherical cancer growth with both immune system interaction and growth inhibitor. This porblem does not have a unique solution. Make sure you explain and justify every term in your model.
4. Modify the model for disease spread in a family to describe the spread of a disease within and between two identical families of five each. Assume that a contact rate between families is different from the contact rate within them.
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