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I have found lots of problems that I am working on that mimic this pattern.% of population with a given age in university

The pattern as I see is (using my question as an example): age is always positive integer

Right tail after certain point is all 0 (for this example, any number before 14 is always zero)

Short exponential on the right(a quick growth period until...)

A period of plateau of a large jump (this is the 4 or 5 year cohort in my example)

A long dropoff with an eventual extinction (small decline until extinction around 70)

So I see a few parameters here: MinThreshold,Rate of exponential incline, Peak, MaxThreshold, Rate of exponential decline

I can multiple the whole equation by (using excel for prototype)

=IF(ageOfUser>peakAge,0,IF(ageOfUsermaxThreshold,0,userAge))) * slow exponential equation or Fibianci with loss

added to

=IF(ageOfUser<=maximumThreshold,0,IF(ageofUser > maximumThreshold,0,ageofUser)) * fast exponential decrease equation

Depending on the data set, there might be a plateau period which I would need to define. I would think this type of equation would explain population over time for species also.

So two questions, 1> How the heck would I represent this in mathematical equation?

2> Has anyone already done something in biology for an equation like this? (I realize this might not be the exact forum but I am hoping for a mathematical biologist might see this)

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    Without looking too closely at the data, I;d say that insisting that age was an integer was a bad idea. Whatever you use to model the data will have to model that rounding process, and to what end? beyond that, I assume the curve is just a normal distribution with the usual sampling noise. Or, if the third moment is high, model it as a [log-normal](https://en.wikipedia.org/wiki/Log-normal_distribution) – 2017-02-15

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Lulus answer to use log-normal can give me the flavor of what I need. Setting mu =0, and sd =1 gives the curve.

So I have in excel:

=LOGNORM.DIST(YearsPastMin,0,1,FALSE)

I then use a modifier to get it to peak at the correct place (In this case 18%)

=LOGNORM.DIST(YearsPastMin,0,1,FALSE)* modifier(3.4) * n enter image description here

Interesting enough, the squared error is actually the same curve with certain params but it still has too much error.

So while I still have another question, this is the equation I was looking for. I need to figure out how to manipulate the slope for both sides but that will have to wait until another question.