Hcfyk

I think you'd have to mess with telomere length. I'm not sure exactly how you would do this and have it pass down to their offspring, but if the species loses a specific amount of telomere length per generation, eventually things stop working. (This might result in severe medical problems in the last few generations.)

Note that within 50 generations there's a chance that this change mutates away, and the larger the number of offspring, the more likely one of them rolls the dice just right. Have a backup plan. (But then that's a problem with almost anything biological.)

Alternate: make them extremely susceptible to a certain chemical, not naturally found in the environment. Release the chemical once generation 50 rolls around. (Does require you to stick around.)

Make fifty generations of eggs (or whatever). Put them in incubators that only trigger the hatching criteria at the appropriate time. This will be much easier to manage than the biological alternative.

Biologically, what you would do is change the gene copying mechanism to remove a sequence from the DNA. For the first fifty generations, it removes a meaningless sequence inserted for that purpose. On the fifty-first generation, it removes something important. It's also possible that it might remove something reproduction related in the fiftieth generation so that it's sterile.

The problem with the biological alternative is that it will tend to have a high failure rate. Think of the things that can go wrong:

1. It might start at the wrong place and end early.


2. It might continue too long, removing two sequences (or more) instead of one.


3. It might fail, allowing the animal to keep breeding perpetually.


4. It might start at the wrong place and introduce a problematic mutation in that generation.

And variants of those.

Another easier alternative would be to make the animals long-lived. Then you could have just one sterile generation that lives for fifty normal generations.

`Grandchildless' gene in the fruit fly (Drosophila Melanogaster) exists. It increases the fly's average number of offspring, but the offspring are sterile. You may look into whether the gene has been modified for a larger number of generations.

If it's OK that not exact but expected number of generations is limited then you could add mutation which makes child sterile. With 10% for such mutation in single generation you get about 0.5% that 51th generation will exists.

The drawback you have decreasing population during whole period.

UPDATE

It works only for animals that grow up few childs in a moment. Most big enough animals select this strategy. If your animal produces numerous offspring (like mouse or fish) then we should change mutation: It should reduce every next generation by 10%. And somehow you need to make sure it will appear in each generation.

You can't do it reliably.

If it can breed for multiple generations it can evolve and change whatever you did to it. More importantly if it can breed for multiple generations it already has everything it needs to keep replicating forever so such a mutation is easy. The best you can do is limit their diversity so detrimental genes kill off most of each generation and does not leave enough diversity to survive catastrophic events, but that is not going to be precise or 100% reliable.

Make part of the childbearing process rely upon a specific material, such as a mineral or a chemical, passed down from mother to children (maybe while the child is developing in the womb). There is only a certain amount that is necessary, and they start out with far more than they all need, but it is non-replenishable and slowly degrades or is lost. Eventually, around the 50th generation (or whatever other point you wish, just adjust the variables) there won't be enough to catalyze a successful pregnancy, and any pregnancies will fail.

The weakness of this is threefold: one, it's imprecise: if a significant number of the breeding population dies, then the end may come that much faster (as the catalyst now has much less to go around). It's also somewhat reliant upon guesswork, for some individuals may run out faster than others, leading to a few stragglers still having successful pregnancies in the 51st generation, or a sudden lack of viable births in the 49th, etc. Finally, there's the lesson we learned from Jurassic Park—life finds a way. It's entirely possible that (if this race is sapient and has access to science of a sufficient level) they may diagnose this issue and correct it via synthesis of the catalyst, or at least delay their extinction. But I think all these are easily overcomable with a bit of handwaving or proper setup of the scenario.

The encoding would need to be in the individuals phenotype and control the expression of individuals gametes on entering sexual maturity.

That way when two generational limited animals mate and their offspring carry their genomes, the “decrement to sterility” could be part of the genetic instructions that generate the animals eggs or sperm.

And with each generation, barring mutation, the number of generations that genome could sire would be one less than the previous generation.

There might be tricks to work around the ‘decrement to sterility’ genetic encoding. Take the case were a 1st generation male impregnates a 4th generation female. The offsprings genotype would contain two different expressions of the generational limit. If the trait is dominant, then the child would be 2nd generation, but if it was recessive the child would be 5th generation.

The telomere approach of user3757614's approach is close but it ticks as the creature grows so it won't be reliable.

Thus lets take a slightly different approach:

1) There is a loose gene in the sperm, along with a chemical that replaces a target pattern in the egg with the gene. This gene is vital for survival.

2) Include 51 copies of the target pattern in the creature's DNA. This must likewise be vital for survival.

After your 50 generations are up the creature is missing the vital gene, it quickly dies in utero.

Mutation is going to be a serious problem here, you probably want something like a quad-DNA approach (instead of our two-copy approach) with very good checking chemicals to make mutation very unlikely. Note that such a creature inherently must be from the lab, the mutation protection needed to keep the death switch working over 50 generations makes it evolve very slowly--which means it won't have time to evolve in the first place.

Of the existing answers I like this one by Aos Sidhe best, but I would probably do it slightly differently.

Make your animal susceptible to a poison/virus which causes infertility. The poison should be airborne. Set 3 or 4 (for redundancy) meteorites moving towards you plant so they arrive in 50 generations, they carry the poison/virus, and will air burst as they enter in the atmosphere.

There is less risk of genetic drift devaluing the poison/virus as it does not have a counterpart on the planet. Winds should delivery the poison/virus world wide in a year or so, even if only a single meteorite arrives. Faster if the animal has developed intelligence and travel.

Tandem repeat mutation disease and genetic anticipation.

Some genetic diseases occur earlier and earlier with each generation. Most of these diseases are caused by mutations in genes with tandem repeats. The number of repeats gradually expand, and affected children manifest the disease younger than their parents did.

Expandable DNA repeats and human disease. Mirkin SM.

One of the central principles of classical (mendelian) genetics is
that mutations are stably transmitted between generations. As long ago
as 1918, however, a different type of inheritance was described for a
human neurological disorder, myotonic dystrophy1. This type of
inheritance was characterized by increased expressivity: that is, a
decreased age of onset and increased severity in individuals of
subsequent generations. A similar hereditary pattern was later
observed for other neurological diseases: for example, Huntington’s
disease, spinal and bulbar muscular atrophy, and several ataxias. The
penetrance — that is, the probability that a given mutation results in
disease — can also increase in successive generations, as was first
demonstrated for fragile X syndrome2. This unusual type of inheritance
— characterized by a progressive increase in the expressivity and,
sometimes, the penetrance of a mutation as it passes through
generations — was called genetic anticipation.

The gradual expansion of the mutant gene is your timer counting down to 50 generations. With each generation is gets slightly longer. A (probably neurologic) disease which initially would not manifest within an individual's lifetime begins to show up in the very aged. With each generation, younger individuals develop the disease. At the 50th generation the disease occurs at the time of puberty, and these individuals do not live to bear children.

You can do it by having the first generation of females have a set number of eggs, all of which are transferred to their first offspring (except for the one that generates the offspring). This way each child has one fewer egg available than their parent, and when the eggs run out, the species dies.

How it happens biologically: I would say take your pick between the gestation process engulfing all the remaining eggs at the start (and maybe this is necessary in order for gestation to be successful), or at some stage of the growth of the embryo the eggs are transferred in (and again, this is necessary for successful growth). The first might be easier to work up as a biological process without (too much) handwaving.

This also gives you some options for twins taking half the egg supply each (or some other percentage) and so having some of the animals unable to produce a full number of generations) and an option to have (mad?) scientists interested in taking eggs from 'weaker' animals in order to extend the generational life of 'stronger' animals.

Express mtDNA-specific cytidine deaminase in oocyte mitochondria.

Mitochondria are cellular organelles which, among other things, convert hydrocarbons like glucose into fuel that the cell can use, typically ATP. Mitochondria actually contain their own DNA, called mtDNA, which is separate from the cell's nuclear DNA. Because mitochondria originate from bacteria, they lack the sophisticated DNA protection and repair mechanisms provided to nuclear DNA. Because mitochondria are only passed on by the mother (the ones in sperm do not make it to the oocyte), mutations can build up over generations. This is especially likely given the fact that the internal environment of the mitochondrion is highly toxic to DNA due to high concentrations of free radicles. To prevent this, oocyte mitochondria are kept in a highly inactive state, limiting DNA damage.

So... what if you could speed up the mutation rate of mtDNA? There is a class of enzymes called cytidine deaminases which mutate nucleic acids, including DNA and RNA. It converts a nucleic base called cytidine to uridine, causing a mutation. This mutation is quickly corrected in nuclear DNA, but not in mtDNA. If you were to express a slow-acting cytidine deaminase in the mitochondria of maternal oocytes (egg cells), then over time, sufficient mutations would accumulate in the mitochondria of offspring that the embryos would eventually become non-viable. Because these mutations would be passed along in each generation, there would be a gradual reduction in viability.

Polyploidy.

I don't think the telomere approach will work at all. If it did, we would have died out. Or rather, would never have existed. Short telomeres are a normal thing, happens to everybody (except the ones who die in an accident), yet nature "just makes it" so gametes are good to go anyway. Sperm from a 100 year old is, well, not precisely as good as sperm from a 17 year old in terms of numbers and mobility and such, but... whatever. It still works, and whatever comes out is a perfectly good individual with perfectly good telomers.

Now, polyploidy might be a better strategy. While it can be somewhat troublesome on higher animals, it's a well-established thing for plants, some frogs and amphibian stuff, a couple of marine vertebrae, and some worms.

If you have a population of, say, 2n males and 4n females, the offspring will be 3n. For a 3n individual, undergoing meiosis is, uh, troublesome. Because 3 doesn't divide by 2 so well.

So, that's that. No gametes, no offspring. Unless you do it like wild dandelion, which despite being unable to reproduce just tells you "Yeah, you know what, f... off!" and simply goes agamosperm, reproducing anyway. But for an animal, that's reasonably unlikely to happen.

The real challenge is that you want not 2 or 3, but 50 generations which are OK, so you would need to find a number which only results in an odd pairing after 50 generations. That's probably a tough one.

It's fiction though, so... got some leeway, I'd say. Might just say "poly" and not mention how many.