# What causes exponential growth of a population?

Aug 7, 2014

The rate of growth of a population in people per year is the births minus the deaths for that year. But what determines those numbers?

Births are caused by women having babies, and the number of pregnant women is proportional to the size of the population. The deaths are also proportional to the population P(t) at time t, so the rate of change of P(t) is about P'(t) = (births - deaths) in year t.

The equation $P ' \left(t\right) = k P \left(t\right)$ has the family of solutions
$P \left(t\right) = A {e}^{k t}$, where A is the initial population (when t=0) and k is the growth rate as a decimal. (For example $P \left(t\right) = 200 {e}^{0.03 t}$ would model a population initially 200 people growing at a rate of 3% per year.)

You can prove that P'(t) = k P(t) by taking the derivative of
$P \left(t\right) = A {e}^{k t}$ getting $P ' \left(t\right) = A k {e}^{k t} = k P \left(t\right)$.

To prove these are the only solutions, let g(t) = $\frac{P \left(t\right)}{e} ^ \left(k t\right)$, then you can use the quotient rule to show that g'(t) = 0 meaning
g(t) = constant, say A, so $P \left(t\right) = A {e}^{k t} .$

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