Bateman gradients from first principles

Abstract

In 1948, Angus Bateman presented experiments and concepts that remain influential and debated in sexual selection. The Bateman gradient relates reproductive success to mate number, and Bateman presented this as the cause of intra-masculine selection. A deeper causal level was subsequently asserted: that the ultimate cause of sex differences in Bateman gradients is the sex difference in gamete numbers, an argument that remains controversial and without mathematical backup. Here I develop models showing how asymmetry in gamete numbers alone can generate steeper Bateman gradients in males. This conclusion remains when the further asymmetry of internal fertilisation is added to the model and fertilisation is efficient. Strong gamete limitation can push Bateman gradients towards equality under external fertilisation and reverse them under internal fertilisation. Thus, this study provides a mathematical formalisation of Bateman's brief verbal claim, while demonstrating that the link between gamete number and Bateman gradients is not inevitable nor trivial.

Fig. 1. The Bateman function of Eq. (1).

This figure shows how the basic Bateman gradient asymmetry arises from simple biophysics and mathematics of fertilisation. The population is monogamous aside from a mutant individual, whose number of fertilisation partners (‘matings’) varies on the horizontal axes within panels. a–d show the effect of variation in sex-specific gamete numbers under efficient fertilisation, while e–h show the effect of variation in sex-specific gamete numbers under inefficient fertilisation. Parameter values used are shown in the figure. Females (gamete number n_x) are indicated by blue crosses and connecting lines, while males (gamete number n_y) are indicated by black dots and connecting lines. Under isogamy, females and males are undefined, and the two colours overlap. The typical sex-specific shapes of Bateman gradients arise from a single equation (which itself is not sex-specific) when a difference in gamete numbers is assigned to n_x and n_y, confirming Bateman’s intuition that the primary cause of the difference in selection is that females produce fewer gametes than males. The entire range of gamete number ratios presented in the figure is observed in nature, from equal gamete size in many unicellular organisms to vertebrates, where sperm count per ejaculate can commonly exceed 10⁹ (see ref. and Supplementary Information therein).

Fig. 2. The Bateman function of Eq. (2) for an externally fertilising population with potential for population-wide polygamy and gamete competition.

Results are shown for two values of resident matings (m = 1 and m = 2). a–h show the effect of variation in sex-specific gamete numbers and in fertilisation efficiency with m = 1, while i–p show the same with m = 2. Parameter values used are shown in the figure. The value m = 2 is used here because it is comparable to the mean number of matings in Bateman’s work (see Fig. 3 for corresponding results with internal fertilisation, but note that the aim of the models is not to quantitatively reproduce Bateman’s results). Females (gamete number n_x) are indicated by blue crosses and connecting lines, while males (gamete number n_y) are indicated by black dots and connecting lines. Under isogamy, females and males are undefined, and the two colours overlap. Further variation in m is examined in Fig. 4.

Fig. 3. The Bateman functions of Eqs. (3) and (4) for internal fertilisation.

Where Figs. 1 and 2 show that the sex-specific shapes of Bateman functions are ultimately caused by differences in gamete number, Fig. 3 shows that internal fertilisation does not invalidate this outcome when fertilisation is efficient. As in Fig. 2, results are shown for two values of resident matings (1 and 2), and the value m = 2 is used because it is comparable to the mean number of matings in Bateman’s work. a–h show the effect of variation in sex-specific gamete numbers and in fertilisation efficiency with m = 1, while i–p show the same with m = 2. Parameter values used are shown in the figure. Inefficient fertilisation combined with relatively low asymmetry in gamete numbers and the added asymmetry of internal fertilisation can in principle reverse the Bateman gradients (second and fourth row). Females (gamete number n_x) are indicated by blue crosses and connecting lines, while males (gamete number n_y) are indicated by black dots and connecting lines.

Fig. 4. The Bateman functions of Eqs. (2)–(4) when the resident number of matings varies.

The gametic system is anisogamy with n_x = 100 (female, indicated by blue crosses and connecting lines), n_y = 100,000,000 (male, indicated by black dots and connecting lines). The number of resident matings m varies between a – d as indicated above the panels. Results are visually indistinguishable for Models 2-3 and with fertilisation efficiency parameters a = 0.001/a = 1. Increased number of resident matings (i.e., increased gamete competition) decreases the steepness of the male Bateman gradient but does not eliminate the asymmetry between female and male gradients, in line with earlier theoretical results by Parker and Birkhead.