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I was thinking about the following question: why do most creatures on earth reproduce asexually or bisexually, but not trisexually?

Looking on the internet, I read an interesting perspective https://www.zhihu.com/question/303528094 that attempts to answer this question via the following mathematical model, which I call the reproduction model.

  • A species has $m$ sexes, and $n$ different types of sexual chromosome, which we call $\{1,2,\cdots, n\}$.

  • Each individual of the species has a genotype of $m$ sexual chromosomes, which can be described as an $m$-element multiset with each element in $\{1,2,\cdots, n\}$. There are disjoint families $F_1, F_2, \cdots, F_m$ of genotypes, such that an individual exhibit sex $i$ iff their sexual chromosome lies in family $F_i$.

  • There is a distribution $\mu$ on $F_1 \sqcup \cdots \sqcup F_m$ that describes the distribution of each possible genotype across the entire species. We require equal number of individual of each sex: we must have $\mu(F_1) = \mu(F_2) = \cdots = \mu(F_m) = 1/m$.

  • The main point of the argument is that the distribution $\mu$ must be stable. Specifically, consider the following reproduction process: for each $i$, we sample a random individual from sex $i$ according to $\mu$. We then form an offspring by selecting a uniformly random chromosome from each individual. Then the genotype of this offspring must also be distributed according to $\mu$.

Let me give two stable examples.

  1. Asexual reproduction: $m = n = 1$, and $F_1 = \{(1)\}$. This is clearly stable.
  2. Bisexual reproduction: $m = 2, n = 2$, $F_1 = \{(1, 1)\}$, $F_2 = \{(1, 2)\}$, and $\mu$ is uniform. This is also stable, since during reproduction there is an equal chance that an offspring of type $(1,1)$ or $(1,2)$ are born.

The author of this models argue that trisexual reproduction is not stable, by considering the following model:

$m = 3, n = 2$, $F_1 = \{(1,1,1)\}, F_2 = \{(1,1,2)\}, F_3 = \{(1,2,2)\}$, $\mu$ is uniform. Then the probability that the offspring is of sex $1,2,3$ is $2/9,5/9,2/9$ respectively. For example, the offspring is of sex $1$ only if all three parents donate type $1$ chromosome, which happens w.p. $1 \cdot 2/3 \cdot 1/3 = 2/9$. So this model is unstable.

However, the author said that they cannot rule out models with more than $2$ chromosome types i.e. $n \geq 3$.

So my question is

Does there exist a stable reproduction model with more than $2$ sexes?

abacaba
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  • I made a significant change to this model, since the problem I stated before is trivial. – abacaba Nov 27 '23 at 08:31
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    I believe a pair of genetic loci with two alleles each gives a stable 4-sex situation. Take two unlinked loci and label the alleles X/Y and Z/W. There are four possible genotypes (sexes): XYZZ, XYZW, XXZZ, XXZW. Assume that, as in human sex differentiation, YY and WW are untenable pairs--hence two XY individuals cannot mate, nor two ZW individuals. Hence types 1 and 4 mate, producing all four genotypes will equal probability, and types 2 and 3 mate, also producing all four genotypes with equal probability. If off-type mating cannot produce offspring, this would be stable. – Eric Snyder Nov 27 '23 at 09:23
  • I guess what I have in mind is a type of organism where all sexes engage in mating to produce a single individual, like in Asimov's novel "The Gods Themselves". – abacaba Nov 27 '23 at 18:49
  • Back when all gametes were the same, natural selection favoured some becoming larger, which favoured others becoming smaller but combining with the larger ones, which led to a bimodal distribution that also drove further differences between the sexes. – J.G. Nov 27 '23 at 18:57
  • @abacaba In that case, I imagine any such system for $m>2$ will be unstable by the binomial theorem. I can't be certain though--perhaps a three-locus setup would allow for three separate sexes a la The Gods Themselves? – Eric Snyder Nov 28 '23 at 00:34

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