Chapter 2.1, Problem 30P

A tumor may be regarded as a population of multiplying cells. It is found empirically that the “birth rate” of the cells in a tumor decreases exponentially with time, so that $\beta \left(t\right)={\beta }_0{e}^{-\alpha t}$ (where $\alpha$ and ${\beta }_0$ are positive constants), and hence $\frac{dP}{dt}={\beta }_0{e}^{-\alpha t}P,P\left(0\right)=P_0$ .

Solve this initial value problem for $P\left(t\right)=P_0\exp \left(\frac{{\beta }_0}{\alpha }\left(1-e^{-\alpha t}\right)\right)$ .

Observe that P (t ) approaches the finite limiting population $P_0\exp \left({\beta }_0/\alpha \right)\text{as}t\to +\infty$ .