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How much would a species' birth rate have to go down in order for that species to go extinct within 100-200 years?

I've seen stuff about this... god, sorry I can't be very helpful, but I saw this somewhere in college in a math class, actually. But, I will go so far as to say that there's probably not an absolute answer like "5.7%". I would imagine it depends on other factors, particularly infant mortality rate.

Depends on how long the individuals live. A species with a human-like lifespan (less than 100 years) would have to have fewer births than deaths for several generations before complete extinction. To be completely extinct in only two or three generations, they would either have to be rendered sterile practically all in one go or lose all of their females. A technologically advanced species would just clone themselves and implant the cloned embryos if any females were left, even if all the males were sterile. If you propose an advanced enough technology, they might not even need the females if they invented artificial wombs.

James Tiptree, Jr.(Actually a woman named Alice Sheldon writing under a male pseudonym in the 70's, an ex-CIA agent) wrote an amazing story about what might happen in a situation like that. It's in the story collection "Her Smoke Rose Up Forever". It's called "Houston, Houston, Do you Read?" It's about a group of male astronauts who get pushed forward in time to a future Earth that's come up with a creative solution to a worldwide sterility plague. Very interesting story. The story of the author is even more interesting. She was a very talented writer, but one disturbed individual. Ended up killing herself in 1987. Most of her stuff is very feminist. You would think she was a man-hater, but she was married for many years to another SF author. She killed him before she suicided. He was bedbound and blind. She was in her 70's, he was 80 something. She shot him and then shot herself.

So in a species that has 200-300 year life-spans, how long would a 90-95% drop in birth rates take to make them no longer biologically viable to continue as a species?

Are we assuming there's no technology available to boost birth rates? (Hormonally stimulated multiple births, IVF, surrogate mothers, etc?) Why does this species have a 95% drop in birth rates and why aren't they doing anything about it? Before you do the calculations, you'd need to come up with a believable reason why this species is allowing itself to die off. Any efforts to improve the birthrate would slow the progression to extinction varying degrees. The calculations available for birth rate vs population viability in animal populations assume non-self aware populations incapable of changing the birth rate by artificial means. I don't think any intelligent species would just sit back and let it happen like that.

Ummm...if you REALLY want to get into the math, here's a link to get you started. I'm not even gonna go there. Maybe Elessar can explain it to you. Any time math equations have symbols in them which aren't numbers or recognizable letters in some language I can read, I bow out. http://darwin.eeb.uconn.edu/eeb310/lecture-notes/pva.pdf

Let's just say that all attempts at artificially boosting the birth rate have failed for these people, and it's a mystery as to why.