Suppose I am trying to estimate a future population that I believe to be log-normally distributed from a current value. But, every $n$ periods, I remove a fixed amount in the future.
For example, I may have a fish research pond where I have 1,000 fish. If I believe the population will grow 10%, plus or minus per period, for future years I will forecast the following years population (in each year) by mutiplying the the current year's log-normally-distributed estimate/projection by a log-normal where the $\mu$ of the underlying distribution would have the value $ln(1.10)$ (and some appropriate $\sigma$).
But, when one of those $n$ periods occurs, I would remove a fixed number of fish (say 100) for examination. After doing that, I seem to be left with what some people call a three-parameter log-normal, where the third parameter ('a') would convert every occurrence of $x$ in the pdf to $(x-a$) - in other words, the pdf would be shifted $a$ units to the left.
My question is this. If that is done, will multiplying by the annual growth log-normal multiplier continue to generate log-normal distributions?
I recognize that one issue would need to be some care and possibly re-scaling of parameters. But aside from this, does 'subtracting' a fixed amount from a log-normal distribution destroy the underlying property that the product of log-normals is, itself, log-normal?