A comment on Bochynek & Burd (2024) ‘Pollination efficiency and the pollen–ovule ratio’

There is a long-debated question in pollination biology regarding the effect of selection on pollen production. Intuitively, one would expect selection for increased pollen production if pollen is a limiting resource (i.e. only a fraction of ovules is successfully fertilised). Yet, classical sex allocation theory suggests that under panmixia, mortality of offspring (or analogously, loss of pollen) does not influence the evolutionarily stable level of sex allocation (Fisher, 1930; Leigh, 1970; West, 2009).

Bochynek & Burd (2024) address this question quantitatively with a new mathematical model of sex allocation in a simultaneous hermaphrodite. The model is presented in the context of plant reproduction, although it could apply equally well to, for example,broadcast spawning animals. An interesting aspect of their model is that it begins with a set of dynamical equations describing the production of pollen and ovules and their fertilisation and mortality, similar to equations used to model broadcast spawner fertilisation dynamics in the evolution of anisogamy (e.g. the dynamical equations describing the resident population in Lehtonen & Kokko (2011) are essentially identical, but the evolving trait is gamete size instead of sex allocation). The authors then build an evolutionary model of allocation into pollen and ovules around this underlying set of equations. Surprisingly, their panmictic model predicts an effect of population size, pollen mortality and fertilisation success on sex allocation, in contrast with the classical results mentioned above. Bochynek & Burd argue that the classical Fisherian sex allocation of exactly equal male and female allocation under panmixia is a consequence of the ‘atemporal simplification’ of classical models, whereas their model explicitly accounts for the dynamics of fertilisation over time. However, no further justification is given for why atemporal simplification should yield a different result than the dynamic model of Bochynek & Burd.

Here, I re-analyse the model of Bochynek & Burd and show that it confirms the classical result of an exact 50 : 50 sex allocation rather than challenges it. I then discuss a different set of biological assumptions the mathematical model of Bochynek & Burd does match, relate their model to earlier theory and then generalise it. The reinterpreted model matches previous models in the relevant parts of parameter space but also has something new to say.

中文翻译：

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性别分配：种群大小和结构、受精成功和繁殖体二态性的影响

对Bochynek和Burd （2024） '授粉效率和花粉-胚珠比例'

授粉生物学中有一个长期争论的问题，关于选择对花粉产生的影响。直觉上，如果花粉是一种限制性资源（即只有一小部分胚珠成功受精），人们会期望选择增加花粉产量。然而，经典的性别分配理论表明，在 panmixia 下，后代的死亡（或类似的，花粉的损失）不会影响进化上稳定的性别分配水平（Fisher， 1930;Leigh， 1970;West，2009 年）。

Bochynek和Burd（2024）通过一种新的数学模型定量地解决了这个问题，即在同时雌雄同体中进行性别分配。该模型是在植物繁殖的背景下呈现的，尽管它同样适用于广播产卵动物等。他们模型的一个有趣之处在于，它以一组描述花粉和胚珠的产生及其受精和死亡率的动力学方程开始，类似于在异性进化中用于模拟广播产卵器受精动力学的方程式（例如，描述Lehtonen和Kokko的常住人口的动力学方程式（2011） 基本相同，但进化的性状是配子大小而不是性别分配）。然后，作者围绕这组基本方程构建了一个分配给花粉和胚珠的进化模型。令人惊讶的是，他们的泛米模型预测了种群规模、花粉死亡率和受精成功率对性别分配的影响，这与上述经典结果形成鲜明对比。Bochynek和Burd认为，在panmixia下，男性和女性完全相等的经典费舍尔性别分配是经典模型“非时间简化”的结果，而他们的模型明确解释了受精随时间的动态变化。然而，没有给出进一步的理由来解释为什么非时间简化应该产生与Bochynek & Burd的动态模型不同的结果。

在这里，我重新分析了Bochynek和Burd的模型，并表明它证实了精确的50：50性别分配的经典结果，而不是挑战它。然后，我讨论了Bochynek和Burd的数学模型确实匹配的一组不同的生物假设，将他们的模型与早期的理论联系起来，然后将其推广出来。重新解释的模型与参数空间相关部分的先前模型相匹配，但也有一些新的东西要说。