I was trying to find a function that would follow this model:
The Y axis is population; the x axis is years. Each point is a local max or min point. Is it possible? I have no idea how to make a sinusoidal wave that decreases over time.. I need a clear formula that will be able to help me predict populations in 62 years and in 117 years.
Say that the upper envelope curve (the line with negative slope passing through your upper points) is given by $y = u(x)$ and the lower envelope is given by $y = \ell(x)$.
The midline curve for your sinusoid will be given by their average $$ m(x) = \frac{u(x) + \ell(x)}{2}, $$ and their amplitude is given by $$ a(x) = \frac{u(x) - \ell(x)}{2}. $$ Notice that $m + a = u$ and $m - a = \ell$, as functions.
Now, your function is $$ f(x) = m(x) + a(x) \cdot \cos( \omega x ), $$ where $\frac{2 \pi}{\omega}$ is the period. I have ignored phase shift, as you seem interested in a wave that achieves its maximum at $x = 0$.