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=(b−d)N(t)
and as Δt→0
dNdt=(b−d)N.
With an initial population size of N0, and with r=b−d 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.