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