The Gompertz differential equation was suggested' as a model for tumor growth: =r.X. In dt (+) Here, X denotes the tumor size (note that X > 0), r and K are positive constants; K is called the carrying capacity. We may write (+) concisely as = F(X). (a) Find lim F(X) and lim F(X). Also show that the so-called proliferation rate F(X)/X becomes infinite as X0*. For which value of X is the growth rate X maximal? What is the value of this maximal growth rate? X-+00 X-0+ lim i In (4) might be helpful The rule lim z In(x) = I40+ (b) Find and classify all equilibria of (*). (c) Let a E R. Find S adn du using an appropriate substitution. (d) Solve the Gompertz differential equation with initial value X(0) = Xo. (e) Explicitly calculate lim X(t). Does this line up with the result you got in part (b)? (f) Chemotherapy can be used to treat a tumor. The "fractional kill hypothesis" states that a defined chemotherapy concentration will kill a constant fraction of the cells in a population; thus taking chemotherapy into acount, the model (+) changes to =r.X - In ()- - d. X, with some constant d, the mortality rate of the tumor cells the treatment achieves. Find and classify all equilibria of this model. Argue that the tumor cannot be remove no matter what rate d the treatment achieves. (g) We have seen in part (a) that the proliferation rate in the Gompertz model is unbounded for small tumor sizes which is not realistic for biological systems. The "Gomp-ez" model combines an exponential model with the Gompertz model as follows: dx (-- In (4)) x if 0s X < Xe, if X 2 Xei here, X, is a critical size threshold at which exponential growth of the tumor changes to the Gompertz model (e.g., Xe 10 tumor cells for human tumors). Argue that in the Gomp-ex model chemotherapy can be used to remove the tumor. What is the minimal value of the mortality rate d the treatment has to achieve to be sure that the tumor will eventually be removed?