Discrete-Time Malthus Suppose yn represents the population of the world the n th year afte...

Discrete-Time Malthus Suppose yn represents the population of the world the n th year after 1800. That is, y0 is the population in 1800, y1 is the population in 1801, and so on. From the definition of the derivative as the limit of a difference quotient, we can write

so Malthus’s differential equation model dy/dt = 0.03y can be approximated by y(t+1) − y(t) = 0.03y, that is, y(t + 1) = y(t) + 0.03y(t). Thus writing yn for y(n), we obtain7

(a) Use this discrete model (5) to estimate the population in the years 1801, 1802, 1803,..., 1810.

(b) Estimate the world’s population in the year 1900 using this discrete model. You might do this with a spreadsheet, using equation (5), or you might develop an algebraic formula by observing a pattern.

(c) Comment on the difference in results comparing this discrete process with Malthus’s continuous model in Table 1.1.1. Explain.