Supplement to my Substack post replicating Maire et. al (2001), which established that anisogamy can evolve without requiring either pre-existing mating types (that is, each gamete must be paired with another of the right type in order to form a zygote) or very large mutations. Novel discoveries of this replication are (i) that the robustness of the most promising of their models is far worse than claimed (likely a typo or mathematical error), bringing it beyond biological plausibility; but (ii) that a reparameterisation of this model is far more robust and realistic than even claimed.
We have a population in which each individual has two copies of a gamete-gene, one of which is randomly passed (possibly with some small mutation) to each gamete that the individual produces. The size of each individual's gametes is given as the average of the values of their two gamete-gene copies. Each individual produces a number of gametes in inverse proportion to the size of their gametes (modeling the trade-off between quantity and quality). Of course, there is a very small minimum gamete size (one cannot produce subatomic gametes, for instance).
Gametes pair randomly with one another, fusing into zygotes which each survive to maturity according to its size \(z\) (which is the sum of the sizes of the two gametes that formed it). In particular, the survival function is a sigmoid curve that is zero below some positive size, and approaches this cutoff fairly gradually. One such survival function is as follows.
\[S(z)=\left(1\ –\ 1.05\cdot e^{–z^2}\right)\cdot95\%\]
The shape we needed (graphed in the Substack post) is determined by the \(1.05\) parameter, and the qualitative results are robust to this being strictly between \(1.0\) and \(1.1\); meanwhile, the percentage is a ceiling on survival probability, and the qualitative results are robust to this being arbitrarily low or high (although of course in the latter case there will be an effective ceiling of \(100\%\)). By contrast, Maire et al. effectively combine these two parameters, and so claim to just require a ceiling survival probability of \(50\%\) or higher; but what is actually needed, for their survival function, is a ceiling survival probability of about \(50%\) or higher. This appears much less robust, since biologically realistic ceiling survival probabilities may be much lower than these values. However, one can separate out the shape (how one approaches the cutoff size) from the ceiling value, and so be robust to arbitrary ceiling values.
In this visualisation, the horizontal axis is gamete size on a logarithmic scale (so that it is evenly spaced in terms of number of gametes produced), with color on a roughly linear scale. We assume a fixed population size of \(256\), and assume that every individual lives for just one generation. The gamete-gene values ranges between \(0.0001\) and \(4\), and mutations go either up or down by one step on the log scale, which has \(512\) steps so that each mutation is makes the value about \(2\%\) larger or smaller. The probability of such a mutation occurring for each gamete can be adjusted using the slider (which has a default value of \(10\%\)). Each copy of the gamete-gene in the population has a value of \(0.5\) initially. The mean value of the gamete-genes decreases to an attractor point, but (given the set-up) this attractor point is a fitness minimum, such that it can be invaded both by mutants with slightly higher values and with slightly lower values; so, the population splits in two. For a (gentle, visually-explained) mathematical explanation and some intuition about these dynamics, see the Substack post.
In the end, the males are haploid (two small-value copies of the gamete-gene) and the females are diploid (one small-value copy of the gamete gene, and one large-value copy of the gamete-gene). The maximum possible gamete-gene value is \(4.00\) in the simulation, but (as the female population never reaches a \(1.50\) gamete size) this choice empirically is large enough not to interefere with things. However, the minimum gamete size is important: if it is within a few orders of magnitude of the "middle" of the survival sigmoid, then the males only produce a few orders of magnitude more gametes than the females. In that case, two gametes produced by females, each with large-value copies of the gamete-gene, may merge into a "super-female" zygote that produces massive gametes (roughly twice as big as the female gametes). Indeed, there may then stably be three sexes (with genotypes small-small, small-large, and large-large). Recall that, by contrast, human sex determination is such that our females are haploid and our males are diploid, rather than vice versa.