Question

Assume that the population of a town obeys Malthusian growth. Assume that the size of the population was 2000 in year 1900, and 50 000 in year 1950.

(a) Find the value of the growth constant k.
(b) How long does it take for the population to grow by 20%?
(c) How big was the population in year 2000?
(d) What was the rate of change of the population in year 2000?
(e) Calculate the size of the population in year 2001, and from that the actual change in population during the one year, from 2000 to 2001.
(f) Explain why the two values calculated in (d) and (e) above do not necessarily have to be the same.

Answer 1

Malthusian growth model is a simple exponetial growth model.

where population at any given time t is

P = P₀ e^kt

where k is the growth constant.

so here if t = Year - 1900

P(1900) = 2000 = P₀

P(1950) = 50000 = P₀ e^50k

so diviing both equations

50000/2000 = e^{50 k}

25 = e^{50 k}

ln 25 = 50 k

k = 0.0644

so here the population growth equation is

P = 2000 e^{0.0644 t}

(b) Here let say at time t, the poulation is P and at time (t + t₁), it get increased to 1.2P

so here

1.2P = 2000 e^{0.0644 (t + t₁ )}

P = 2000 e^{0.0644 t}

1.2P/P = e^{0.0644 t₁}

ln (1.2) = 0.0644 t₁

t₁ = 2.83 year

(c) at year 2000, t = 100

P = 2000 * e^{0.0644 * 100}

P = 1252813

(d) Here rate of change of the population in the year 2000 = kP = 0.0644 * 1252813 = 80681 person per year

(e) For year 2001, t = 101

P(2001) = 2000* e^{0.0644 * 101} = 1336249

actual change in the population = 1336249 - 1252813 = 83436

(f) Here the value we get in the problem (d), is the instantenous change at year 2000, but as the population is continously increasing at every moment, so these two values of (d) and (e) can be differed.