2. The Gompertz equation
y'(t) = y[a-b*ln(y)]
is an important model for avascular tumor growth. In the avascular growth phase, tumor cells obtain nutrients directly from the surrounding tissue. (The transition from avascular to vascular growth is marked by the onset of angiogenesis, the formation of blood vessels, which are used to bring nutrients to the tumor.) In this model, y(t) represents the number of cells in the tumor and a and b are constants related to the tumor's growth rate.
Assume b>0. Determine the range of y over which the function y[a-b*ln(y)] is positive or negative (indicating increasing or decreasing tumor size).
Based on this information, what is the value of y(t) as t ---> infinity if we start with a positive number of tumor cells?
Repeat the qualitative analysis for b<0. Whet is the biological significance of y=exp(b/a)
c. Find the analytic solution to the equation subject to an initial condition y(0) = y0. To do this use z = ln(y) and solve for z
Here is the solution for this very interesting problem, ...
Th 5 page long solution shows how to analyze the equation in order to gain some insight into the behavior of the solution undere different conditions and then solves the equation completely.