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Mathematics LibreTexts

1.1: The Malthusian Growth Model

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Let N(t) be the number of individuals in a population at time t, and let b and d be the average per capita birth rate and death rate, respectively. In a short time Δt, the number of births in the population is bΔtN, and the number of deaths is dΔtN. An equation for N at time t+Δt is then determined to be

N(t+Δt)=N(t)+bΔtN(t)dΔtN(t)

which can be rearranged to

N(t+Δt)N(t)Δt=(bd)N(t)

and as Δt0

dNdt=(bd)N

With an initial population size of N0, and with r=bd positive, the solution for N=N(t) grows exponentially:

N(t)=N0ert

With population size replaced by the amount of money in a bank, the exponential growth law also describes the growth of an account under continuous compounding With interest rate r.


This page titled 1.1: The Malthusian Growth Model is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Jeffrey R. Chasnov via source content that was edited to the style and standards of the LibreTexts platform.

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