Chapter 9.4, Problem 17E

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

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

(a) Find the solution of this equation that satisfies the initial condition P(0) = P₀.

(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?