This will be a short installment.

Skyman pointed out that the last section of the first gynogen post was a bit tricky to understand. I’ll try to redo that section here, and hope that it will be more intelligible.

Let us assume:

1. that the amount of sperm that is produced by males in a population is not a limiting factor, but that the population is limited by resources or shelter, for instance. This means that a superabundance of fry are born every year in the autumn, but only a more or less constant amount of individuals survive the winter to be able to mate the following summer. At the start of the breeding season, the size of the population is therefore more or less constant over time.

2. that males cannot tell gynogens and sexual females apart.

3. that fertilization rates in both sexual and asexual females are the same, so that there is no bias in fry production of one female over another. In essence, this assumes that all females, whether gynogens or sexuals, produce the same amount of eggs, and the same proportion of eggs in each subpopulation becomes fertilized. We can, for simplicity, assume that all eggs in all females and all gynogens are fertilized.

4. that survival to reproductive age is the same in gynogens and sexuals (of both sexes), so that there is no bias in which kind of fish survives. The proportion of individuals of a certain kind in any given year is therefore the same as the proportion of fry of that kind produced the previous year.

5. that the system is closed, so there is no migration in to or out of, say, the pond.

6. that sex rations in the sexual populations are equal, so that in any given breeding season, 50% males and 50% females are produced.

7. that all individuals die after they have mated, so that there is more or less complete replacement of generations, and no individuals survive to reproduce a second breeding season.

This would mean that the ratio of sexuals and gynogens in any given year is dependent only on the ratio of gynogens and sexual females the previous years (as males do not replace themselves in the population independently).

Now, let’s say that we have a pond in which the proportions of gynogens (red), males (green), and females (blue) are the same (33% each):

In the next breeding season, the females and gynogens will produce the same amount of eggs, and all of these will be fertilized by the males. However, while the females will produce 50% males and 50% females, the gynogens will produce 100% gynogens. The males will not replace themselves independently, so the rations between the three sexes after that breeding season will be as in the right column:

If we expand that to the same scale as before, we see that the actual ratio between the three sexes has changed from the previous 33%/33%/33%:

Now, roughly half of the population is gynogens, and a quarter each are males and females. Over the winter, large amounts of fry die, but the proportions remain, and when the next breeding season come around, we see the same thing: gynogens replace themselves according to the proportion in the population, whereas females produce 50% males and 50% females:

And if we scale that up as before, we see that the proportion of gynogens has increased from one third to two thirds in only two generations, while the proportion of either of the sexuals has decreased dramatically:

The system is thus highly unstable, and over the next few generations, following the same pattern as above, we can see that the proportion of males and females diminishes quite rapidly (*P. formosae* here represents the gynogens):

At some point, the proportion of males becomes so low that we can no longer assume that the amount of available sperm is unlimited, and even if we did, it would be obvious that given a limited time for mating, the probability that all males only encounter gynogens increases rapidly. Even disregarding that, there will be some point in this progression when the proportion of males is lower than 1 individual, and this, if not before, is when the system breaks down. The year where no sexual females are fertilized by any male is the year when the sexual species becomes extinct. The gynogen may survive for another year, provided at least one of them has been fertilized, but that is the upper limit. As there is no migration in to or out of the system, the maximum survival expectancy of the gynogens is exactly one year after the last breeding season in which at least one gynogen was fertilized by a male.

I asserted last time that this is the case regardless of the starting proportions, but didn’t show that, and likely do not have the mathematical know-how to show that anyway. Luckily, then, my friend Skyman is so bored of his PhD project that he will do *anything* rather than work on that. He produced this nifty pamphlet which explains the math behind this sort of calculation, and as I don’t understand half of it, I will just post it here (with his permission) and assume that my readers (if any) are more savvy than I am:

I can, however, understand graphs (or at least some graphs), so Skyman made me some of those as well. He made a graph for the situation detailed above (for the terminology used in the graphs, see the paper):

However, these starting proportions are not very natural, of course. If a new pond was settled by exactly one male, one female, and one gynogen, as in an experimental population, this may very well be the case, but under natural conditions, the most likely scenario is that in a population consisting of more or less equal numbers of males and females, a gynogen suddenly appears, through hybridisation, mutation, or some other process. This scenario would look like this:

Here, we start with almost 50% males and almost 50% females, but with a single gynogen. We expect the whole system to crash after approximately 16 generations, as compared to less than ten in the example above, so it is obvious that while the starting conditions ultimately do not matter, they may effect the time period over which we may observe the system at all. So Skyman varied the input variables a bit. It should come as no surprise that if the proportion of males is higher than the proportion of females in the starting population, the system crashes earlier (after less than 10 generations), as there are already fewer females that can produce the next sexual generation:

Conversely, if we increase the starting proportion of females, we get a system that may survive for longer, in this case maybe up to 20 generations:

If we drastically increase the proportion of starting females, we may prolong the population even further (up to perhaps 90 generations), but even this will not save the system:

So, the question remains: if some of these systems have persisted for 100,000 years or more (some estimates for one of these systems reaches 280,000 years), one or more of the assumptions must be unjustified, but which? I will (hopefully) be abel to return to this, and start answering this question tomorrow.

Note that the file Gyn.pdf has been replaced by gyn_copy.pdf, which I hope is the more recent version in which some spelling errors and other things have been corrected. Thank you, Skyman, for pointing this out.