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Exponential growth can be modeled as

$$ b (1+r)^N $$

For $b$ your starting quantity, $(1+r)$ your rate of growth, and $N$ the number of periods. But for $N \to \infty$, this formula can get out of control.

Is there a traditional way of controlling for this by factoring in some notion of a decay factor (so that for periods $N$ past some threshold, you stop growing asymptotically)?

user1770201
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1 Answers1

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The simplest extension of the exponential equation

$\frac{dx}{dt} = rx$

is the logistic equation

$\frac{dx}{dt} = rx(1-\frac{x}{C})$

where the rate of growth decreases as $x$ approaches $C$.

This differential equation has solution

$x(t)=\frac{Cx(0)e^{rt}}{C+x(0)(e^{rt}-1)}$

Interestingly, the discrete time step equivalent of the logistic equation (known as the logistic map) can exhibit chaotic behaviour.

gandalf61
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