Slow decreasing function that exhibits asymptotic behaviour

From you question, it is unclear exactly what shape you are looking for, and there are many functions that could describe the behaviour you're after. However, two possible options could be the negative exponential and a negative Gompertz function. Possible forms of these could be:

Negative exponential

$y(x)=e^{−ax+\ln{(1-b)}}+b,$

where $b=0.2$ and a is a rate parameter.

Negative Gompertz

$y(x) = 1-\alpha e^{-\beta e^{-\gamma x}},$

where $\alpha=0.8$ (describing the asymptote), $\beta$ in an inflection parameter (given a displacement along the $x$-axis), and $\gamma$ is a rate parameter.

These functions can give results such as:

In these examples, $a = 0.005$ for the negative exponential and $\beta = 150$, $\gamma = 0.008$ for the gompertz v1 and $\beta = 5$, $\gamma = 0.005$ for the gompertz v2.

An alternative parametrization of the Gompertz, which might be easier to understand, is:

$y(x) = 1-\alpha e^{-e^{-\gamma (x-\beta)}},$

where the inflection point ($\beta$) is directly related to the scale of the $x$-axis (so $\beta = 400$ would give an inflection at $x = 400$).

I hope these examples are interesting for you, and they can maybe provide a starting point.

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Note: This answer was originally posted at Biology-SE, where this question was first posted.