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I have found a formula to calculate the number of items y after a certain value x (e.g. time) given the size of items at the beginning of the series: $$ y_x = a^x y_0 $$ I would like to convert it to include a logistic term 1-x/k and I know that: $$ y_{x+1} = a^2 y_ x \left(1-\frac{y_x}{k} \right) $$ and $$ y_{x+1} = y_x \exp \left[a(1-\frac{y_x}{k}) \right] $$ How can I convert yₓ₊₁ ∝ yₓ to yₓ ∝ y₀ including the logistic term?

How can I solve for $y_0$?

Thank you

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  • $\begingroup$ If $y_x = a^x y_0$ then (a) $y_{x+1}=a y_x$, and (b) this is exponential, not logistic, and (c) if $a > 1$ then it faces unbounded growth while if $0 < a < 1$ then it faces exponential decay $\endgroup$
    – Henry
    Dec 22, 2020 at 12:37
  • $\begingroup$ Thank you, but it is possible to convert the last equation into a y₀-equivalent? (I have no background in math so I am a little navigating in the dark...) $\endgroup$
    – Gigiux
    Dec 22, 2020 at 12:43

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