What is the formula for exponential growth with a decay rate?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 → ∞, 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)?

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What is the formula for exponential growth with a decay rate?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  →  ∞, 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)?

Andrejkoxg · Accepted Answer

The simplest extension of the exponential equation            d      x              d      t        =  r  xis the logistic equation            d      x              d      t        =  r  x  (  1  −      x    C    )where the rate of growth decreases as x approaches C.This differential equation has solution  x  (  t  )  =            C      x      (      0      )              e                  r          t                            C      +      x      (      0      )      (              e                  r          t                    −      1      )      Interestingly, the discrete time step equivalent of the logistic equation (known as the logistic map) can exhibit chaotic behaviour.

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