Chapter 9.4, Problem 17E

Consider a population $P=P\left(t\right)$ with constant relative birth and death rates $\alpha$ and $\beta$, respectively, and a constant emigration rate m, where $\alpha ,\beta$, and m are positive constants. Assume that $\alpha >\beta$ Then the rate of change of the population at time t is modeled by the differential equation

$\frac{dP}{dt}=kP-m\text{ where k=}\alpha \text{-}\beta$

(a) Find the solution of this equation that satisfies the initial condition $P\left(0\right)=P_{0.}$

(b) What condition on m will lead to an exponential expansion of the population?

(c) What condition on m will result in a constant population? A population decline?

(d) In 1847, the population of Ireland was about 8 million and the difference between the relative birth and death rates was 1.6% of the population. Because of the potato famine in the 1840s and 1850s, about 210, 000 inhabitants per year emigrated from Ireland. Was the population expanding or declining at that time?