Martin Burd

Reproductive efficiency reflects the gap between an organism's potential success and its actual achievement. Exploring how theoretical sex allocation models deal with reproductive efficiency reveals shortcomings in two fundamental elements of the theory, fitness gain curves and the Shaw‐Mohler equation. Gain curves depict success but make no reference to the production of the entities that accrue fitness, so they imply nothing about efficiency in its common sense, the fraction of successful reproductive entities in relation to the number produced. Because the number of successful entities cannot exceed the number produced, a gain curve implies a minimum allowable production, but the implication itself exposes new problems. Gain curves are used to replace terms in the Shaw‐Mohler equation that were meant to represent an individual's fair competitive share of total population fitness. If gain curves themselves represent individual fitness outcomes, the theoretical solutions can imply unequal aggregate success of male and female mating agents, a biological impossibility in a sexually reproducing population. If gain curves represent inputs to an arena of mating interactions, the theoretical solutions can imply misleading or biologically impossible patterns of reproductive efficiency. The relationship between pollination efficiency and pollen‐ovule ratios in flowering plants is used as a window into the shortcomings of gain curves and the Shaw‐Mohler equation. There are new modeling approaches to sex allocation that are more explicit about the processes that lead from production of male and female reproductive entities to their eventual fitness outcomes.

Gain curves have been a staple of sex allocation theory for decades. They represent patterns in which fitness is obtained from resource investments in reproductive functions. The monotonic forms that have been used for gain curves can represent fitness accrual by individuals, but only on the assumption that sufficient mates are always available to allow the stipulated monotonic pattern of reproductive success to occur. However, sexual populations do not have external banks of mating opportunities that lie outside the dynamics of the population (such opportunities would, by definition, be part of the population). Thus, the reproductive behaviour of whole populations cannot be simple scaled‐up versions of individual gain curves. As sex allocation evolves within a breeding population, frequency‐dependent selection creates a shifting advantage for the rarer sex. Individual gain curves cannot then remain stable possibilities at the population level. Evolutionary models based on fixed gain curves can predict evolutionary outcomes with unequal total fitness for male and for female function, an outcome that the biology of syngamy does not allow. Such biologically impossible outcomes are easily demonstrated. Gain curves have also been widely used as a framework for interpretation of interspecific empirical patterns, such as low male allocation in monogamously mating hermaphroditic animals or self‐pollinating plants, and higher male allocation in wind‐pollinated than in animal‐pollinated plants. However, if gain curves incorrectly characterize whole populations or species, interspecific differences in gain curves cannot explain these patterns. Even if they superficially appear to predict the empirical pattern, other processes must be operating. The selective effects of local mating competition and sex‐specific dispersal patterns have long been known. They are likely replacements for gain curves as explanations of many broad interspecific patterns, but the predominance of gain‐curve explanations has distracted attention from these alternatives. A revision of our understanding of gain curves seems needed.

The success of invasive species relies heavily on the production, dispersal and genetic composition of propagules. For range expanding species, breeding strategy and level of reproductive investment will strongly influence their capacity to establish and invade new areas. A hermaphroditic lifestyle provides the advantage of increasing the number of seed bearing individuals within a population while a dioecious habit may enable more rapid adaptation to new environments, improve resource use efficiency, fecundity and dispersal. Pittosporum undulatum, a tree native to coastal areas of southeastern Australia, has many characteristics of an invasive species within and beyond its native range. A previous study detected a male bias within invasive populations, with a high proportion of fruit deriving from female-only trees, leading to recommendations for the removal of ‘matriarch’ trees as a simple management technique. We expanded that study and investigated breeding systems of different populations of P. undulatum by assessing tree density, gender, resource availability and fruit load of 871 individuals in seven native and seven invasive populations. All populations comprised either females (47%) or hermaphrodites. No male-only trees were observed within the study. More females produced more fruit than hermaphrodites, especially in the native site. This could not be attributed to environmental differences between sites. These data support the current management practices of targeting the removal of females as a simple method for containing invasions given the benefits of reducing the workload and spreading limited management resource. Our work highlights the value in understanding the breeding strategy employed by focal invasive species as a means of developing improved and more targeted control methods.

Summary: Pollination presents a risky journey for pollen grains. Pollen loss is sometimes thought to favour greater pollen investment to compensate for the inefficiency of transport. Sex allocation theory, to the contrary, has consistently concluded that postdispersal loss should have no selective effect on investment in either sex function. But the intuitively appealing compensation idea continues to be raised despite the lack of theoretical endorsement. We address the theoretical issue with a model that directly represents pollen loss (and ovule loss through floral demise or loss of receptivity) as rate‐dependent dynamical processes. These loss rates can be varied to examine the effect of pollination efficiency on optimal sex allocation. Pollen–ovule ratios follow from the sex allocation based on the resource costs of pollen and ovule production. This model confirms conventional findings that pollen loss should have essentially no effect on sexual resource allocation in large, panmictic populations. Pollen limitation of seed set does not alter this conclusion. These results force us to rethink the empirical association of pollination efficiency with low pollen–ovule ratios. This pattern could arise if efficient pollen transport commonly results in stigmatic deposition of cohorts of related pollen. Empirical evidence of correlated paternity supports this explanation.

This article is a Commentary on Duncan et al. (2024), 244: 307–317.

Fitness gain curves were introduced into the framework of the Shaw–Mohler equation, the foundation of sex allocation theory. I return to the Shaw–Mohler equation to consider how it embodies the rare-sex advantage underlying frequency-dependent selection on the sex ratio. The Shaw–Mohler formulation is based on the numbers of males and females randomly mating in a panmictic population. Gain curves are meant to describe reproductive success through male and female functions in hermaphrodites and have been inserted in place of male and female numbers in the Shaw–Mohler equation. In doing so, gain curves bypass consideration of the implicit mating process in the Shaw–Mohler argument and can lead to anomalies like unequal total male and female fitness in a population. If gain curves truly represent fitness gain, equality of total male and female fitness requires a constant sex allocation of equal resource investment into male and female functions. The blurring of input with fitness outcome has led to misinterpretation of what gain curves mean in reproductive ecology. They can describe a particular reproductive ecology, such as diminishing fitness returns on resource investment, but lack causal efficacy with respect to sex allocation.

Fitness gain curves were introduced into the framework of the Shaw-Mohler equation, the foundation of sex allocation theory. I return to this equation to consider how it embodies the rare-sex advantage underlying frequency-dependent selection on the sex ratio. The Shaw-Mohler formulation is based on the numbers of males and females randomly mating in a panmictic population. Gain curves were meant to describe reproductive success through male and female functions in hermaphrodites and were inserted in place of male and female numbers in the Shaw-Mohler equation. In doing so, gain curves bypass the implicit mating process in the Shaw-Mohler argument that creates frequency-dependent selection. Gain curves can then lead to anomalies like unequal total male and female fitness in a population. If gain curves truly represent fitness gain, they must have a population-level effect on both sexes, a basic necessity of sexual reproduction. The blurring of inputs with fitness outcome has lead to misinterpretation of what gain curves mean in reproductive ecology. They lack causal efficacy with respect to sex allocation. Because gain curves have been a staple of evolutionary ecology for decades, the implication is that much of our understanding of sexual allocation in hermaphrodites needs to be revisited. I outline some directions such an effort might take.

Gain curves were introduced to explain how hermaphrodites could displace a dioecious population, and to account for sexual allocation in hermaphrodites. Terms for gamete production employed for the first purpose were transformed for the second into male and female gain curves that ostensibly defined fitness outcomes. These gain curves pose a conceptual challenge if they are specified separately because fitness at the population level cannot occur through one sex function independently of success through the other. If gain curves truly represent fitness outcomes, anomalies can arise, such as inequality of total male and female fitness in a population. Gain curves were originally used in a mathematical framework that treated the ostensible gain functions as inputs of male and female actors to a mating arena rather than as mating outcomes from that arena. I present a model of sex allocation that incorporates power functions to describe both gamete production and fitness gain in a manner that explicitly separates these two roles. In this formulation, the gamete production functions have the identical effect on optimal sex allocation originally attributed to gain curves while the true fitness gain curves lose nearly all effect on the optimum. Thus, despite the label, gain curves were implicitly describing inputs rather than outcomes. Because gain curves have been a staple of evolutionary ecology for decades, the implication is that much of our understanding of sexual allocation in hermaphrodites needs to be revisited. I outline some directions such an effort might take.