Hey guys, I appreciate you tackling this problem! Let me give you a bit of background on where this is coming from, so you understand what I'm trying to accomplish.

I'm an author. I'm creating a science fiction universe and I have to have internal consistency while determining the rate of decline of the inhabitants of my extremely overpopulated nation-state as they go into the future with some very unethical measures of drastically culling the population. They exterminate a set number of individuals per year, with an aim to get down to a particular population number. I want to know how long it would take them, but I can't just take the current population, subtract the desired population, then divide that number by the constant number eliminated per year, because that doesn't take into account population gain and loss parameters of birth and mortality rates. Since these gains and losses are based on percentages, as the population changes each year by the constant extermination number, the resulting numbers of gains and losses must change dynamically as they reflect that updated total each and every year. Therefore, unless I can determine all three of these factors together, I can't get an accurate estimate on how long it will take my population to be culled to its desired size.

And of course the reason I am asking if you can show me how you did it is because I might have to do it again in the future, if aspects of the plot shift which affect the population of the state. (for example, to begin with, the nation occupied the whole continent of Australia = 8.5 million km^2, then I decided to devote 40% of that continent to wastelands, so the area dropped to 6.4 million km^2. To keep the population density proportionate, the entire starting population also has to drop by 40%. Therefore, I have to do all of these prospective population calculations again. And I'd like to know an equation I can just plug the numbers into, in the likely case that I have to shift things again.

I hope all that casts a little more light on the convoluted situation? Hahaha

Thanks! I really do appreciate the help. As is the case with many authors, math is not my strong suit. ;-)