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# Figure 1 the various cases for the competing species

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Figure 1: The various cases for the competing species system (d) For the case that the tumor-free equlibrium (1 /b 1 ,0) is to be stable, linearize about this equilib- rium point ( X = 1 /b 1 + ǫu and Y = 0 - ǫv , where ǫ is small compared to 1 /b 1 ) to obtain, parenleftbigg u v parenrightbigg = parenleftbigg - a 1 - c 1 b 1 0 a 2 (1 - c 2 b 1 a 2 ) parenrightbigg parenleftbigg u v parenrightbigg . It is now seen that the tumor cannot recur if the eigenvalue 1 - c 2 /b 1 a 2 is negative, and thus the condition for the tumor-free equilibrium to be stable is c 2 b 1 a 2 > 1 . 2

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2. Purpose: To explore another model equation using a more complicated tumor growth function, in an attempt to include the effects of angiogenesis. The comparison of tumor growth curves is, continued in the Projects, using published data. See also Exercise 3. Research on tumor angiogenesis is a hot topic, and a literature search on current theories and proposed mechanisms would be an interesting research project. Exercise: It has been observed both in vivo and in vitro that solid tumors experience an initial period of quick growth,followed by a period when growth slows or stops, follwed by another period of growth. It has been suggested that the first period of growth is during the ‘avascular’ phase, when the tumor has not yet developed any internal vasculature, so that the cells must acquire nutrients through diffusion from outside the tumor. Once the tumor reaches a certain size, the tumor cells release ‘angiogenic growth factors’, which stimulate the growth of blood vessels towards the tumor, and finally reaching into the interior of the tumor. After the tumor has been ‘vascularized’, another period of growth occurs. In this exercise, the angiogenic process will be modelled by a drastic slowing in the growth rate when the tumor reaches a certain size, denoted by T a . (a) Sketch a graph of a possible growth function, F ( T ) = dT dt , which is positive for 0 < T < T MAX , and is very small for T near T a < T MAX . In the spirit of generating tractable models, the function should be as simple as possible, within the prescribed constraints. (b) Write a differfential equation for T ( t ), after writing an equation for the function F ( T ) you graphed in 2a. (c) Solve the differential equation in part 2b, and compare the result to logisitc tumor growth by plotting solution curves for the two types of growth, using the same initial conditions. (De- pending on the form of F ( T ), you may or may not be able to find an explicit solution for the differential equation. If an explicit solution is not available, use a numeric solver.) The compar- ison will be most meaningful if the same initial growth rates (when T is close to zero), and the same carrying capacity are used. Comment on the effect of varying the tumor size during vascularization, i.e. T MAX .
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