bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

The evolution of anisogamy does not always lead to male competition

Mattias Siljestam1 and Ivain Martinossi-Allibert1,2,*

1Department of Ecology and Genetics, Animal Ecology, Uppsala University, Norbyvägen 18D, 752 36 Uppsala, Sweden; 2Department of Organismal Biology, Systematics, Uppsala University, Norbyvägen 18D, 752 36 Uppsala, Sweden

This manuscript was compiled on December 18, 2020

1 Anisogamy has evolved in a large proportion of sexually reproduc- tion and parental care, but it does not prove very insightful 21 2 ing multicellular organisms allowing the definition of the female when it comes to revealing the influence of anisogamy on these 22 3 and male sexes, producing large and small gametes, respectively. traits because of the multitude of confounding factors that are 23 4 Anisogamy is the initial sexual dimorphism: it has lead the sexes not entirely understood. On the other hand, the theoretical 24 5 to experience selection differently, which makes it a good starting approach on which we will focus relies on thought experiments 25 6 point to understand the evolution of further sexual dimorphisms. For and mathematical models that aim at understanding the evolu- 26 7 instance, it is generally accepted that anisogamy sets the stage for tionary origins of the male and female sexes, i.e. the evolution 27 8 more intense intrasexual competition in the male sex than in the fe- of anisogamy. This line of research has also endeavoured to gen- 28 9 male sex. However, we argue that this idea may rely on assump- erate predictions on further evolutionary changes that would 29 10 tions on the conditions under which anisogamy has evolved in the result from the state of anisogamy, sometimes referred to as 30

11 first place. We consider here two widely accepted scenarios for the the sexual cascade (Parker, 2014), where anisogamy ultimately 31

12 evolution of anisogamy: gamete competition or gamete limitation. leads to sexual dimorphisms in intrasexual competition and 32

13 We present a mechanistic mathematical model in which both gamete may also influence dimorphism in parental care. We believe 33

14 size and an intrasexual competition trait for fertilisation can coevolve this approach to be most useful to specifically understand the 34

15 in a population starting without dimorphism between its two mating influence of anisogamy on further sexual dimorphisms, because 35

16 types. Two different intrasexual competition traits are investigated, it starts in a context where anisogamy is the only difference 36

17 gamete motility and the ability of gametes to capture gametes of the between males and females, therefore automatically removing 37

18 opposite mating type. We show that gamete competition and ga- the influence of potentially confounding factors. 38

19 mete limitation can lead to greatly different outcomes in terms of Starting in 1948, Bateman (Bateman, 1948) conducted an 39 20 which sex competes most for fertilisation. Our results suggest that influential and later questioned and debated (Sutherland, 1985; 40 21 gamete competition is most likely to lead to stronger competition in Tang-Martinez and Ryder, 2005; Gowaty et al., 2012) study in 41 22 males. On the other hand, under gamete limitation, competition in Drosophila melanogaster in which he measured higher variance 42 23 form of motility can evolve in either sex while gamete capture mainly in reproductive success in males than in females. He suggested 43 24 evolves in females. This study suggests that anisogamy does not that this result may be explained by the state of anisogamy 44 25 per se lead to more intense male competition. The conditions un- in the following way: males produce small, cheap, numerous 45 26 der which anisogamy evolves matter, as well as the competition trait gametes and females fewer, larger, more energetically costly 46 27 considered. ones, therefore the number of gametes produced by males 47 would never be a limiting factor of their reproductive success, 48 anisogamy | mating types | motility | intrasexual competition | sexual while it is the case for females. This would allow males to 49 cascade | gamete limitation | gamete competition | adaptive dynamics reach higher potential reproductive success than females. As 50 a consequence, because each offspring has a male and female 51 1 ost sexually reproducing multicellular organisms have parent and the population sex-ratio is assumed to be balanced, 52 2 Mevolved mating types and anisogamy (dimorphism in the fact that some males reach a high reproductive success 53 3 gamete size)(Lessells et al., 2009). In systems with two mating implies that other males will have low or null reproductive 54 4 types, the one producing larger gametes is denoted the female success. At the same time, most females are expected to have a 55 5 sex while the one producing smallPREPRINT gametes is the male sex. Be- reproductive success that is close to average. This results in a 56 6 cause it is the earliest possible sexual dimorphism, researchers higher variance in reproductive success for males. In the early 57 7 interested in sexual selection have tried to understand how 1970s, Trivers (Trivers et al., 1972) followed up on Bateman’s 58 8 anisogamy may have influenced the evolution of sexual dimor- idea and suggested an explanation for the causal relationship 59 9 phisms or sex-biases that evolved later on, most notably in between anisogamy, intrasexual competition and parental in- 60 10 intrasexual competition for mating and in parental care. Two vestment. Trivers argued that initial energy investment in 61 11 main approaches have been used to this end: first the study of offspring should determine which sex competes and which sex 62 12 sex-biases in intrasexual competition and parental care in con- cares for offspring. The larger initial energy investment in the 63 13 temporary species through empirical observation, supported zygote by females should automatically trigger competition for 64 14 by meta-analysis and review (Clutton-Brock, 1991; Cox and mating opportunities in males and create a stronger incentive 65 15 Calsbeek, 2009; Janicke et al., 2016; Singh and Punzalan, 2018) 16 and second mathematical modelling (e.g. Kokko and Rankin, Both authors contributed equally to this work. Conceptualization: I.M.; Methodology: M.S.; Formal 17 2006; Kokko et al., 2012; Jennions and Fromhage, 2017). The analysis: M.S.; Visualization: M.S.; Writing: I.M and M.S.; Funding Acquisition: I.M.

18 empirical approach is most powerful to inform us on how eco- Authors declare no competing interests. 19 logical settings, evolutionary history, and particularities of the * 20 mating system may influence sexual dimorphisms in competi- To whom correspondence should be addressed. E-mail: [email protected]

December 18, 2020 | vol. XXX | no. XX | 1–20 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

66 in females to provide further care in order to increase off- reproductive success is limited by female fecundity, while male 127 67 spring survival. Trivers’s ideas (Trivers et al., 1972) regarding reproductive success is limited by male competitive ability. We 128 68 parental investment have been critically examined early on and therefore predict, in agreement with previous work (Schärer 129 69 resulted in the production of a wealth of theoretical work (e.g. et al., 2012; Lehtonen et al., 2016), that if anisogamy evolves 130 70 Dawkins and Carlisle, 1976; Queller, 1997; Kokko and Jen- in a gamete competition context, it should favour the evolution 131 71 nions, 2008; Jennions and Fromhage, 2017), which have shown of competition traits in the male sex. 132

72 that although there may be a causal link between anisogamy In the gamete limitation framework, which originated with 133 73 and parental investment, it ought to be more complex than Kalmus (Kalmus, 1932) the evolution of anisogamy arises as a 134 74 Trivers had initially predicted (Knight, 2002). In contrast, the strategy to increase gamete encounter rates, in an environment 135 75 mechanism through which anisogamy should result in more where gamete density is very low. The evolution of anisogamy 136 76 intense intrasexual competition in males has been relatively in this framework is favoured by sperm limitation, i.e. the fact 137 77 neglected by theoreticians. Although initially a verbal model, that not all the larger gametes get fertilised and this suggests 138 78 it has not received much critical attention until recently (see that both sexes experience selection to increase fertilisation 139 79 Lehtonen et al., 2016), and instead has been supported by success. For that reason, we predict that the gamete limitation 140 80 verbal arguments (e.g. Schärer et al., 2012; Parker, 2014) or context should allow the evolution of competition traits in 141 81 correlational empirical data (Janicke et al., 2016). both sexes. 142

As demonstrated by Lehtonen and Kokko (Lehtonen and 143 82 Here, we re-examine the causal relationship between Kokko, 2011), gamete competition and gamete limitation 144 83 anisogamy and the evolution of intrasexual competition. Al- are not two mutually exclusive theories on the evolution of 145 84 though much work has been done on the evolution of anisogamy anisogamy, but rather two complementary ideas that speculate 146 85 since Trivers developed his ideas (to cite a few: Parker et al., on how anisogamy may have evolved in populations of high or 147 86 1972; Parker, 1978; Schuster and Sigmund, 1982; Smith, 1982; low gamete densities. It is therefore relevant to include both 148 87 Hoekstra et al., 1984; Hoekstra, 1984; Bulmer, 1994; Dusen- scenarios in the same model, which we do here by including 149 88 bery, 2000; Bulmer and Parker, 2002; Dusenbery, 2006; Les- gamete density as a variable parameter of our model (regulated 150 89 sells et al., 2009; Lehtonen and Kokko, 2011), the interaction by population density or the per individual gamete energy 151 90 between the evolution of gamete size and the evolution of allocation, or both). Furthermore, gamete limitation may be 152 91 intersexual competition traits is still not well understood. We a particularly relevant scenario since the organisms that are 153 92 argue that considering the coevolution of anisogamy and in- thought to have evolved anisogamy in the first place are sessile 154 93 trasexual competition (as opposed to their independent or broadcast spawning marine invertebrate (Lehtonen and Parker, 155 94 sequential evolution) is important, because the selection pres- 2014; Parker, 2014), and this type of organism can easily be 156 95 sures that are responsible for the evolution of anisogamy may subjected to sperm limitation (Levitan and Petersen, 1995; 157 96 still be at play when intrasexual competition evolves. For Levitan, 1998a,b; Yund, 2000; Crean and Marshall, 2008). 158 97 this reason we develop a model where both gamete size and a In the present study, we develop and analyse a mathemat- 159 98 competition trait are evolving simultaneously, starting from a ical model of the coevolution of gamete size together with 160 99 state of isogamy (two mating types with equal gamete size). an intrasexual competition trait, starting from a population 161 100 Classically, there are two main theoretical frameworks for the without sexual dimorphism. By varying gamete density, we 162 101 evolution of anisogamy (Lehtonen and Parker, 2014), gamete explore several scenarios ranging from extreme gamete limita- 163 102 competition and gamete limitation (detailed below) and we tion (low density) to intense gamete competition (high density). 164 103 want to examine whether these should favour different sexual Intrasexual competition traits in this model are gamete-level 165 104 dimorphisms in intrasexual competition. traits that increase the fertilisation success of an individual 166 105 Most models of the evolution of anisogamy assume that at the expense of the success of other individuals of the same 167 106 there is a limited amount of energy an individual may allocate mating type. We investigate independently the evolution of 168 107 to gamete production resulting in a trade-off between gamete two different competition traits. The first one is gamete motil- 169 108 number and individual gamete size (for example: Parker et al., ity, which in nature is often found prominently in the smaller 170 109 1972; Smith, 1982; Bulmer and Parker, 2002). In the gamete gametes. The second trait is the ability of gametes to capture 171 110 competition framework, initially developed by Parker (Parker nearby gametes of the opposite mating type in order to achieve 172 111 et al., 1972), the evolution of anisogamy is mainly driven by a PREPRINTfertilisation by increasing their apparent size, a strategy ob- 173 112 constraint imposed on the survival of the zygote (Bulmer and served in nature under the form of a jelly coat surrounding 174 113 Parker, 2002), which depends on gamete size. Larger gametes the larger gametes of several marine invertebrates (e.g. Farley 175 114 harbour more resources and provide zygotes with higher sur- and Levitan, 2001; Podolsky, 2002). We explore a wide range 176 115 vival chances. Once one mating type starts to evolve larger of possible scenarios and our model provides novel insights 177 116 gametes, the other mating type is somewhat released from on the relationship between the evolution of anisogamy and 178 117 the constraint of providing energy to increase zygote survival sex-biases in intrasexual competition. 179 118 and can therefore reduce energy investment into individual 119 gametes in order to increase the number of gametes produced, Model 180 120 which is a good strategy in this context to increase fertilisa- 121 tion success. In this framework, there is no sperm limitation Here, we present a mechanistic mathematical model of the 181 122 because gamete density is high enough and gametes encounter evolution of anisogamy, where we consider the coevolution 182 123 each other easily, which means that all larger gametes are of gamete size and an intrasexual gamete competition trait 183 124 automatically fertilised while some proportion of the smaller for fertilisation. We consider a community of sessile marine 184 125 gametes are in excess. This situation is thus reminiscent of animals reproducing through external fertilisation. This is 185 126 the view that Bateman and Trivers held, which is that female inspired by the conditions under which anisogamy is thought 186

2 | Siljestam and Martinossi-Allibert bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

187 to have first evolved in animals (Parker, 2014). Below, we first where Kg gives the half-saturation constant of gamete sur- 246 188 give a verbal presentation of the structure and analysis of the vival, i.e. the excess mass required for a 50% probability of 247 189 model before going into the mathematical details. survival (Figure 1). We will refer to Kg as the gamete survival 248 190 The life cycle of the organism is modelled in two steps and constraint. 249

191 assumes discrete non-overlapping generations. In the first step Hence, the number of gametes per zygote of mating 250 192 of the life cycle gametes are produced. A fixed number of type i entering the mating pool, ni,0, equals the num- 251 193 brooding spots on the sea floor are occupied by individuals ber of gametes produces times their survival probability: 252 194 (zygotes) that produce gametes. The number of gametes ni,0(mi, ri) = ni(mi, ri)sg(mi). 253 195 produced per zygote is traded off against both gamete size

196 and the proportion of energy spent on an intrasexual gamete Gamete fertilisation. In the mating pool, the second and last 254 197 competition trait, with larger gametes having a higher chance step of the life cycle, gamete fertilisation occurs, which pro- 255 198 of surviving to next step of the life cycle (Figure 1). In the duces the zygotes of the next generation. Here, the gametes 256 199 second step of the life cycle gametes fertilisation occurs. The that survived enter the mating pool where they move around 257 200 surviving gametes are synchronously released into a common randomly and can collide with each other. Each collision be- 258 201 mating pool, and fertilisation is the result of collisions of tween gametes of the two mating types x and y can result 259 202 gametes of the two mating types, generating the zygotes of the in a fertilisation with probability p. A fertilisation event re- 260 203 next generation (Figure 2). These newly formed zygotes have moves both gametes from the mating pool and produces a 261 204 to survive to enter the next generation with larger zygotes juvenile zygote offspring. To determine the collision frequency 262 205 having a higher chance of survival (Figure1). between gametes of the two mating types we use collision the- 263 206 In our analysis, we investigate the evolution of anisogamy by ory (originally used to model chemical reaction rates between 264 207 allowing for coevolution of two gametic traits: gamete size and gas particles, McNaught et al., 2014). The collision frequency 265 208 the relative investment into a competition trait. We assume depends on three factors: the sizes of the gametes mx and my, 266 209 evolution to start under the constraint of isogamy, meaning the gamete speeds vx and vy (see details further down) and 267 210 equal trait values for both mating types. This constraint can the density of gametes, where all three factors increases the 268 211 be removed by disruptive selection. We first solve for isogamic collision frequency (See in Appendix1, Eq. A1). 269 212 attractors under constrained isogamy (i.e. gamete trait val- The initial gamete density of mating type i is given by 270 213 ues to which evolution leads). Then, we determine if these ni,0(mi, ri)d, i.e. the number of surviving gametes per zygote 271 214 isogamic attractors are endpoints of evolution or if disruptive entering the mating pool (at t = 0) multiplied by the zygote 272 215 selection appears leading to the evolution of anisogamy. When density d. The number of gametes then declines over time as 273 216 anisogamy arises, we then use simulations to determine how they collide and get fertilised (Figure 2, Eq. A6). 274 217 the trait values of gamete size and investment in competition Finally, the probability that a newly formed juvenile zygote 275 218 for the two mating types evolve. By varying gamete density survives until the next generation is an increasing function of 276 219 in the model we transition from analysing the evolution of its size mz (which is given by the combined sizes of the two 277 220 anisogamy in a gamete limitation context at low density to a gametes that fused to produce it mz = mx + my) accord- 278 221 gamete competition context at high density. ing to sz(mx, my) = (mx + my − 1)/(Kz + mx + my − 1), 279 where Kz gives the half-saturation constant of zygote sur- 280 222 Gamete production. In the first step of the life cycle, zygote vival (Figure 1), and we will refer to Kz as the zygote survival 281 223 individuals are randomly chosen to occupy brooding spots constraint. 282 224 occurring with a density of 2d, where the zygotes then can 225 grow and produce gametes. The zygotes are assumed to be in The intrasexual competition traits. We investigate separately 283 226 abundance such that all brooding spots are occupied. The two two alternative intrasexual competition traits: gamete motility 284 227 mating types x and y are of equal frequency such that zygotes or fusion partner capture. 285 228 of each mating type occupies spots with a density of d. (i) Competition through motility: In this scenario we 286 229 Each zygote possesses a fixed amount of energy e allocated assume that gametes are in still water resulting in no basal 287 230 to gamete production. A proportion of the energy ri is spent on motility of the gametes. The competition trait is then a 288 231 a gamete competition trait (see details further down), where i 232 denotes the mating type and canPREPRINT be either x or y. We refer to 233 the trait ri as relative competition investment. The remaining 1 234 energy e(1 − ri) is converted into gamete mass with gametes 235 of size mi. The number of gametes ni produced per zygote is 236 thereby a function of both the gamete size mi and the relative 237 competition investment ri and is given by 0.5

e(1 − ri) 238 ni(mi, ri) = . [1]

mi Survival probability(s) 0 239 The probability that a gamete survives until it enters the K 5 K 10 K 240 mating pool (the second step of the life cycle) is an increas- Excess mass(m-1) 241 ing function of its size mi. We assume gamete cells must 242 have a minimum size of 1 to perform their function and that Fig. 1. Survival probability of gametes and juvenile zygotes as a function of 243 any excess mass, mi − 1, represents resources provisioned for their excess mass allocated to survival, m−1. A survival of 50% is reached 244 survival. The survival probability of a gamete sg is given by fol- when the excess mass equals the half-saturation constant K marked out by the dashed line. 245 lowing saturating function sg(mi) = (mi − 1)/(Kg + mi − 1),

Siljestam and Martinossi-Allibert December 18, 2020 | vol. XXX | no. XX | 3 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

The second term gives the increase in apparent size due to 329 a. b. ) i

) 330 6 i 6 investment in competition, and can be divided into two prod- 10 10 Male gametesn y 

Female gametes(n x ) ucts. First mi(ri/(1 − ri), giving energy invested into the 331 α Juvenile zygotes(n z) m increased apparent size and second αrmi , giving how many 332 5×105 5×105 times cheaper the artificial mass is compared to increasing the 333 actual gamete size mi. For this trait we also refer to αr and αm 334 as competition investment efficiency and size-dependent in- 335

0 Gamete and zygotecount(n 0 vestment efficiency, respectively. 336 Gameteand zygote count(n 0 tend 0 tend Matingperiod(t) Matingperiod(t) Analysis 337

Fig. 2. Number of gametes and newly formed juvenile zygotes per We investigate the coevolution of two traits, gamete size mi 338 adult zygote over the mating period under a. gamete limitation (low ga- and relative competition investment ri, in the two mating 339 mete density) and b. gamete competition (high gamete density). These types x and y. We assume these traits to be coded by au- 340 graphs are given by equation A6 with parameters: initial female gamete 5 6 tosomal chromosomes, such that the traits of a genotype is 341 count nx,0 = 5 × 10 , initial male gamete count ny,0 = 10 and the product of pdσxyvxy (which is proportional to the gamete fertilisation rate) characterised by the vector T = (mx, rx, my, ry) giving the 342 −5 −7 equals 10 in a. and 10 in b. two traits of the two mating types, to which we refer as the 343 strategy of the genotype. Note that each individual will only 344 express the two traits corresponding to its own mating type. 345 289 flagella-like mobility structure that allows gametes to move Initially, we assume evolution to be under isogamic constraint, 346 290 and encounter a fusion partner. meaning that the two mating types x and y have equal gamete 347 291 The amount of energy spent on the competition trait per size mx = my = mc and equal relative competition investment 348 292 gamete is given by miri/(1 − ri) (see Eq. 1). To determine the rx = ry = rc. Under this isogamic constraint, the strategy of a 349 293 gamete speed vi we add three parameters to this expression: genotype is thereby given by only these two traits Tc = (mc, rc), 350

351 αr where c denotes the traits being under the constraint. αm  ri  294 vi(mi, ri) = v0mi , [2] In our analysis we consider a large resident population 352 1 − ri starting with a single strategy T , to which we iteratively in- 353 0 0 0 0 0 295 where αm and αr determine how gamete speed is affected by troduce an initially rare mutant strategy T = (mx, rx, my, ry) 354 296 gamete size mi and the relative investment ri, respectively that has a slight deviation in one or more of the traits of the 355 297 (with αr > 0). v0 is a velocity scale factor. resident strategy T (for evolution under isogamic constraint, 356 0 0 298 We refer to αr as competition investment efficiency. It replace T with Tc and T with Tc). The expected long term 357 0 299 describes how profitable it is to invest more energy into growth of a rare mutant strategy T is given by the fertilisation 358 0 300 the competition trait rather than gamete mass. When- success of the rare mutant strategy fi (Eq. A14) divided by 359 301 ever αr < 1, there is diminishing rate of return in speed vi the fertilisation success of the resident strategy fi (Eq. A9), 360 302 with increasing ri, while αr = 1 results in a linear return in averaged for the two mating types x and y (Shaw and Mohler, 361 303 speed and αr > 1 results in increasing rate of return with 1953), 362

304 increasing investment ri. We refer to αm as the size-dependent 0 0 0 0 0 0 0 1  fx(mx, rx,T ) fy(my, ry,T )  305 (competition) investment efficiency, which determines whether w(T ,T ) = + . [4] 363 2 fx(T ) fy(T )) 306 there is a positive or negative relationship between gamete 307 size and return from investing into the competition trait. Note that the resident fertilisation success fi is not affected by 364 308 0 When αm > 0, larger gametes are faster than smaller gametes the mutant strategy T as the mutant is assumed to be rare. 365 309 for a given proportional investment ri, while αm < 0 results 0 For the same reason, the mutant fertilisation success fi is not 366 310 in smaller gametes being faster for the same ri. These affected by the mutant traits of the other mating type. We 367 311 0 two investment efficiency parameters give flexibility in the refer to w(T ,T ) as the invasion fitness of the mutant strategy 368 312 description of the competition trait, allowing the model to and the mutant is able to increase in frequency and thereby 369 313 0 cover a variety of possible scenarios. invade whenever w(T ,T ) > 1. 370 314 PREPRINT 315 (ii) Competition through fusion partner capture: In Adaptive dynamics. By assuming rare mutations of small effect 371 316 this scenario, we assume that all gametes are in suspension in we can use the adaptive dynamics framework to predict the 372 317 water currents, and have similar motility due to the movement evolutionary dynamics (Metz et al., 1992; Dieckmann and Law, 373 318 of their environment with gamete speed given by just the basal 1996; Geritz et al., 1998). Small mutational effect ensures that 374 319 velocity v0. The competition trait is here represented by an a mutant strategy that invades will always replace and become 375 320 extension of apparent gamete size (for example by producing the new resident strategy as long as the population is under 376 321 a jelly coat around the gamete, or filaments extending from directional selection (Dercole and Rinaldi, 2008, Appendix B) 377 322 the gamete) which allows to capture potential fusion partners and rare mutations ensures that a mutant strategy will either 378 323 (see e.g. Tilney and Jaffe, 1980). This trait is modelled as fixate or go extinct before a new mutant is introduced. If a 379 324 increasing the apparent gamete size mˆ i. This means that mutant strategy fixates it will become the new resident strat- 380 325 gamete size is effectively increased in terms of collision target egy T of the population. This results in a trait substitution 381 326 size (Eq. A1), but not in terms of survival. Apparent size is sequence where the strategy of the population T evolves in a 382 327 defined as step-wise manner for each invading mutant strategy. 383 The selection gradient β gives the direction in the trait 384 αm+1 ri 328 mˆ i(mi, ri) = mi + αrmi . [3] space of T in which mutants have highest invasion fitness, 385 1 − ri

4 | Siljestam and Martinossi-Allibert bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Fig. 3. Four examples of simulation runs depicting qualitatively all possible evolutionary outcomes of the model: a. no evolution of sexual dimorphism in gamete size or competition, b. evolution of anisogamy with female-biased competition investment, c. evolution of anisogamy with male-biased competition investment and d. oscillating evolutionary dynamics (limit cycle) alternating between female-biased and male-biased competition investment. In each graph, the horizontal axis represents evolutionary time in equation 6, and the vertical axis gives trait value for gamete mass (top panels), competition investment (middle panel) and how fitness changes (Eq. A9) relative to the isogamic attractor marked out by a black dot (bottom panels). Values are given for the two mating types x (blue, full line) and y (orange, dashed line). If stable anisogamy evolves (cases b. and c.), the mating type with the larger gamete size is called female and denoted x (blue) while the one with smaller gametes is called male and denoted y (red). Parameters for figure 4 a.-c.: Kg = 10 , Kz = 0 and δ = 0.01, ar = 0.2 and am = (0, −0.4, 0.4) respectively for a.,b. and c (parameter combinations indicated by stars in 4 figures 4 and S1). Parameters for figure d.: Kg = 30, Kz = 10 and δ = 10, ar = 0.6 and am = −0.4 (parameter combination indicated by a star in figure S1).

∗ ∗ ∗ 386 thus giving the expected direction of evolution. The selection constrained evolution Tc = (mc , rc ) happens to also always 416 387 gradient β is given by the gradient of the invasion fitness be a singular strategy at its corresponding point in the uncon- 417 0 ∗ ∗ ∗ ∗ ∗ 388 (Eq. 4) when the mutant trait T is similar to the resident strained evolution T= = (mc , rc , mc , rc ) (Van Dooren et al., 418 ∗ ∗ 389 traits T 2004). This gives each isogamic singular strategy pair (Tc ,T=) 419 0 390 β(T ) = w(T ,T ) . [5] a total of four stability properties: the attractiveness and 420 O 0 T =T invadability of both the constrained and the corresponding 421 391 Within the limit of small mutational effect, iterating this unconstrained evolution. 422

392 mutation-invasion process results in an gradual evolutionary We first allow evolution to occur under the isogamic con- 423 ∗ 393 path given by straint until it reaches an attractor (singular point) Tc . Then, 424 dT ∗ we release the constraint starting evolution from T and ob- 425 394 = Cβ(T ), [6] = dt serve how gamete size and the competition trait in both mating 426

395 (Dieckmann and Law, 1996; Champagnat et al., 2006; Durinx types coevolve. Under this setting we can, as described below, 427 396 et al., 2008), where C is the product of the mutational variance- classify the evolutionary dynamics at each singular strategy 428 ∗ ∗ 397 covariance matrix, the mutation rate and population size. pair (Tc ,T=) into four different scenarios: it can either be a 429 point that repels isogamic constrained evolution, or it can be 430 398 Predicting the evolutionary outcomes. We can predict the out- an attractor of the constrained evolution in which case three 431 399 comes of the evolutionary path (as given by Eq. 6) using the outcomes are possible: evolution transitions from isogamy to 432 400 adaptive dynamics framework. StrategiesPREPRINTT where directional anisogamy, evolution comes to a halt, or genetic polymorphism 433 401 selection ceases, i.e β(T ) = 0, are of special importance. These evolves. 434 ∗ 402 strategies are called singular strategies and we denote them T ∗ 403 (or Tc for evolution under the isogamic constraint). A singular Classification of the evolutionary dynamics at isogamic singular 435 ∗ ∗ 404 strategy T has two important stability properties. First, T point. We assume evolution to start under the isogamic con- 436 405 can be either an attractor or a repeller of the evolutionary straint and in order to classify the evolutionary dynamics 437 ∗ ∗ 406 dynamics. Determining this tells us whether evolution will around an isogamic singular strategy pair (Tc ,T=) we first 438 ∗ 407 approach it or not, and we refer to this property as the at- look at the attractiveness of Tc , which tells whether the con- 439 ∗ ∗ 408 tractiveness. Second, T can be invadable or uninvadable by strained evolution will approach Tc in the first place or not. 440 ∗ 409 nearby mutant strategies and we refer to this property as the If Tc is a repeller it will not be approached by evolution and 441 ∗ 410 invadability (more details are given in Appendix 5.2)(Leimar, the other three stability properties (the invadability of Tc and 442 ∗ 411 2009). the attractiveness and invadability of T=) have no relevance. 443 ∗ 412 Each strategy of the isogamic constrained evolution On the other hand, if Tc is an attractor, the isogamic 444 ∗ 413 Tc = (mc, rc) has a corresponding isogamic strategy for the constrained evolution will approach it. After Tc is reached 445 414 unconstrained evolution, namely T = (mc, rc, mc, rc) and we by evolution, two or potentially three different evolutionary 446 415 denote this strategy T=. Also, each singular strategy of the outcomes can follow depending on the other three stability 447

Siljestam and Martinossi-Allibert December 18, 2020 | vol. XXX | no. XX | 5 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

448 properties: In addition, isogamic polymorphism is a sub-optimal strategy 509 ∗ compared to anisogamy in terms of fertilisation success: the 510 449 1) An attractor of the isogamic constrained evolution Tc ∗ most likely type of gamete collision, between a large and a 511 450 is an evolutionary end-point if both Tc and its correspond- ∗ small gamete, has a 50% probability of occurring between 512 451 ing T= are attracting and uninvadable. Then when evolution ∗ gametes of the same mating type and can thus not result 513 452 reaches Tc no further mutants can invade in either the con- 514 453 strained or unconstrained trait space and there is no selection in fertilisation. Anisogamy is thereby up to two times more 515 454 for removing the isogamic constraint. Hence, evolution stops efficient in terms of fertilisation rate. It is therefore possible 516 455 and the population stays isogamic. that if a pseudo-isogamic genetic polymorphism evolves first, ∗ it will eventually be replaced by the more efficient anisogamy. 517 456 2) An isogamic attractor Tc is a point where anisogamy ∗ ∗ For the scope of this manuscript we will report whenever the 518 457 evolves if Tc is uninvadable but T= is an invadable repeller sub-optimal pseudo-isogamic genetic polymorphism can evolve, 519 458 for the unconstrained evolution. Then it is a so called saddle but we will focus on the evolution of anisogamy assuming it 520 459 point being a fitness maximum in the isogamic constrained to be the final evolutionary outcome. 521 460 trait manifold but a fitness minimum in orthogonal directions By classifying the evolutionary dynamics in this way for all 522 461 (where the gamete traits of the mating types diverge) in the isogamic singular strategy pairs we can then predict the evo- 523 462 unconstrained trait manifold. This means that as long as the ∗ lutionary dynamics more generally. To do this we first choose 524 463 evolution is under the isogamic constraint Tc is an attractor the values of the model parameters, and we then numerically 525 464 that can not be invaded by any further mutants when reached. ∗ solve for the singular strategies (Appendix 5.1) and then get 526 465 However, as evolution gets close to Tc there in an increasing the attractiveness and invadability (Appendix 5.2) for both the 527 466 disruptive selection for gamete size mi and relative competition isogamic constrained and unconstrained evolution. We report 528 467 investment ri to diverge between the two mating types x and y if these attractors are points where evolution either (1) stops 529 468 for the unconstrained evolution. This disruptive selection resulting in stable isogamy, (2) transitions into anisogamy, 530 469 acts as a selection pressure for the removal of the isogamic (3) transitions into either anisogamy or pseudo-isogamic ge- 531 470 constraint. As soon the the isogamic constraint is removed netic polymorphism. We iterate this procedure for a wide 532 471 gamete size mi and relative competition investment ri of the range scenarios by systematically varying the parameters of 533 472 two mating types x and y diverge and the population evolves ∗ the model. 534 473 anisogamy as it repels away from the isogamic strategy T=. A special case occurs if there is a single attractor of the 535 474 The mating type with larger gametes (arbitrarily) gets the ∗ isogamic constrained evolution Tc,a and either all repellers 536 475 index x, and is defined as female while the other mating type y ∗ ∗ (Tc,r1,Tc,r2,...) are on the edge of the trait space (occurs 537 476 is defined as male (mx > my). ∗ 538 ∗ for minimum gamete size mc,ri = 1, no competition invest- 477 3) If an attractor of the isogamic constrained evolution Tc is ∗ ∗ ment rc,ri = 0, or full competition investment rc,ri = 1) or 539 478 invadable by nearby mutants it follows that its corresponding ∗ 540 ∗ alternatively if there are no repellers. Then Tc,a is an unequiv- 479 unconstrained T is an invadable repeller (Van Dooren et al., = ocal destination of the isogamic evolution. In all other cases, 541 480 2004). Here, evolution can lead to two different outcomes. 542 ∗ there might be multiple evolutionary outcomes, and which one 481 First, because T is a repeller of the unconstrained evolution, = is reached might depend on the initial trait value from where 543 482 anisogamy can evolve removing the isogamic constraint, just as 544 ∗ evolution starts and the mutational covariance matrix C. 483 described above in case (2). Second, because T is an invadable c If anisogamy evolves (cases (2) and (3)) the isogamic con- 545 484 attractor for the constrained evolution, it is a special point 546 ∗ straint is removed and the number of evolving traits is in- 485 where isogamic genetic polymorphism can evolve: if Tc is creased from two: Tc = (mc, rc), to four: T = (mx, rx, my, ry). 547 486 reached by evolution and the isogamic constraint remains, We can numerically solve for the isogamic singular strate- 548 487 nearby isogamic mutant strategies can invade and coexist ∗ ∗ 549 ∗ gies (Tc,1,Tc,2,...), but we can no longer solve for the singular 488 with Tc resulting in two isogamic genotype strategies. The ∗ ∗ strategies of the unconstrained evolution (T1 ,T2 ,...) as the 550 489 strategies of these two isogamic genotypes then diverge as dimension of their trait space is too high. To get the evolu- 551 490 evolution proceeds (evolutionary branching, Geritz et al., 1998). tionary outcome after anisogamy evolves we have to rely on 552 491 One genotype then evolves larger gametes (with equal size simulation of the evolutionary path. 553 492 for both mating types) and the other evolves smaller gametes 493 (still equal size for both mating types). Hence, this introduces PREPRINTSimulating the evolutionary path. In parallel with analysing 554 494 genetic variation in gamete size in the population becoming the isogamic singular strategies predicting the evolutionary 555 495 pseudo-isogamic, as this gamete size polymorphism within outcomes, we also numerically simulate the evolutionary path 556 496 mating types can be seen as an anisogamous situation, but given by Eq. 6 (as described in Appendix 5.3). We start 557 497 with each genotype remaining isogamous. In summary, at the simulation of the evolutionary path under the isogamic 558 498 such an invadable isogamic attractor there are two potential constraint and let it run until it stabilises at an isogamic 559 499 evolutionary outcomes: anisogamy if the isogamic constraint is ∗ ∗ ∗ attractor Tc,a = (mc,a, rc,a). We can then verify if the 560 500 removed, or a pseudo-isogamic polymorphism if the constraint simulated result matches the predictions from the analysis 561 501 remains. (presented above) of the attractiveness and the trait value 562 502 In this case if the isogamic constraint is removed because of the singular strategies. Then, we continue the numeri- 563 503 anisogamy is favored by selection, then the possibility for cal simulation without the isogamic constraint, starting at 564 504 isogamic genetic polymorphism is ruled out. There is indeed the corresponding singular point of the unconstrained evo- 565 ∗ ∗ ∗ ∗ ∗ 505 evidence supporting that the evolution of isogamic polymor- lution T= = (mc,a, rc,a, mc,a, rc,a) allowing for anisogamy to 566 506 phism is unlikely if anisogamy is allowed to evolve, as one evolve (ignoring the possibility for isogamic polymorphism to 567 ∗ 507 might expect anisogamy to evolve first prohibiting the evo- evolve in case (3) as mentioned above). If T= is an attractor 568 508 lution of isogamic polymorphism (Van Dooren et al., 2004). no evolution follows, the population stays isogamic and we 569

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∗ 570 conclude that Tc,a was an isogamic end-point of evolution. Figures S1-S2 representing a a wide range of scenarios (with 629 ∗ 571 On the other hand, if T= is a repeller, trait values for the subsets presented in Figures 4-5). 630 572 different mating types diverge away and anisogamy evolves. 573 We can here verify again that the simulated result matches the Transition from isogamy to anisogamy. We find that for any 631 574 predictions gained from the stability analysis of the singular set of parameters within our investigated parameter ranges 632 575 strategies of whether anisogamy evolves or not. (Figures 4-5 and S1-S2) there is always a single isogamic strat- 633

576 Finally, we let the simulation of the unconstrained evolution egy to which the population converges. This occurs because 634 577 run until the evolutionary path stabilises at an anisogamic among the singular strategies of the isogamic constrained 635 ∗ ∗ ∗ ∗ ∗ 578 attractor Ta = (mx,a, rx,a, my,a, ry,a) giving us the anisogamic evolution there is always a single attractor Tc,a = (mc,a, rc,a) 636 579 end-point of evolution (a fixed point attractor), or until evolu- and the rest (one or more) are repellers on the edge of the 637 580 tion stabilise in an oscillating cycle (a limit cycle attractor). trait space (mc = 1, rc = 0, or rc = 1) for each parameter 638 581 Since we could not solve for the anisogamic singular strategies, combination. Consequently, evolution will always approach 639 582 simulations give additional results that could not be obtained the attractor Tc,a independently of from where it starts. Af- 640 583 with the stability analysis of singular point presented above. ter the isogamic constrained evolution reaches this attracting 641 strategy Tc,a, we find that the evolution either stops and 642

584 Parameter reduction and gamete density. We find that the the population stays isogamic (e.g. Figure 3a) or anisogamy 643 585 number of parameter can be reduced down to five when evolves as selection turns disruptive for the gamete traits of 644 586 analysing the evolutionary dynamics of the model. The model the two mating types. This results in divergence between two 645 587 has a total of nine parameters (d, e, Kg, Kz, p, tend, v0, αm mating types x and y of both their gamete sizes (mx and my) 646 588 and αr) and we found that five of the parameters always occur and their relative competition investments (rx and ry) (Fig- 647 589 as a product together in the invasion fitness function (Eq. 4) ure 3b-d). For both types of competition traits, we show in 648 590 and thereby also in the the selection gradient (Eq. 5) as well Figures. 4-5, S1-S2 the parameter spaces where evolution leads 649 591 as in the evolutionary path (Eq. 6). These parameters are to isogamy (within dashed contour lines) or anisogamy (outside 650 592 d, e, p, tend and v0. Hence, when analysing the evolution- dashed contour lines). This shows that anisogamy is generally 651 593 ary dynamics we can substitute their product into a single the more common outcome at low gamete densities δ and at 652 594 parameter δ = d × e × p × tend × v0 reducing the numbers of high survival constraint on either gametes Kg or zygotes Kz, 653 595 free parameters of the models down to a total of five: δ, Kg, regardless of the competition trait that gamete size coevolves 654 596 Kz, αm and αr. with. 655 597 Varying the parameter δ is the same as varying one or more For a considerable subset of the parameter region where 656 598 of the five parameters of its product (d, e, p, tend and v0). the evolution of anisogamy is expected, there is the possibility 657 599 Hence, varying δ can correspond to varying many different for the suboptimal pseudo-isogamic genetic polymorphism to 658 600 properties of the system. First, varying δ can correspond to evolve (outlined with grey contours in Figures 4-5 and S1-S2). 659 601 varying the gamete density, as it contains d × e: the two pa- In that case, evolution can result in two diverging isogamic 660 602 rameters being proportional to the initial gamete density ni,0. genotypes, one producing large gametes (of both mating types) 661 603 This means that, among many things, the value of δ regulates and the other producing small gametes. However, as soon as 662 604 whether the system tends towards gamete limitation at low ga- the isogamic constraint is removed this situation is no longer 663 605 mete densities or gamete competition at high gamete densities. stable and anisogamy evolves. We note that in these cases, 664 606 Furthermore, δ contains tend × v0 which is proportional to the anisogamy is probably the most likely outcome of evolution, 665 607 distance travelled per gamete, and also p the probability of a as it is up to twice as efficient (see Analysis section). For this 666 608 collision resulting in fertilisation. All five parameters encapsu- reason, we assume in the simulations that anisogamy evolves 667 609 lated in δ have identical effects on the evolutionary dynamics under these conditions. 668 610 (as they occur together in a product): they regulate whether 611 the system tends towards gamete limitation low values of δ, Simulation of the coevolution of gamete size and competition 669 612 or gamete competition high values of δ. From now on, for the in two mating types. Solving for the isogamic singular strate- 670 613 sake of simplicity we will refer to δ as gamete density (d × e), gies and analysing their evolutionary stability told us whether 671 614 assuming the other parameters (p, tend, and v0) to be constant. anisogamy will evolve or not. In addition, we also simulate the 672 evolutionary path (Eq. 6) with the double aim of confirming 673 615 Note that when we vary the gametePREPRINT density δ = d × e, it can 616 both correspond to changing from low to high population the result of the stability analysis and finding the trait values 674 617 density d, or from low to high amount of energy allocated to at the anisogamic endpoint of evolution T = (mx, rx, my, ry). 675 618 gamete production per individual e. Simulations do not include the possibility for pseudo-isogamic 676 polymorphisms, directly proceeding to anisogamy instead. 677 Overall, simulations confirm our predictions (with single pixel 678 619 Results differences in some places, likely due to numerical impreci- 679 620 Here, we present how two gametic traits, gamete size mi and sions). For the parameter regions where anisogamy evolves 680 621 relative competition investment ri, coevolve in a population we present simulated results for the degree of gamete size- 681 622 with two mating types x and y, where the strategy of a geno- dimorphism measured as the ratio of gamete sizes of the two 682 623 type is given by the four gamete traits T = (mx, rx, my, ry). mating types mx/my, and the sex-bias in competition mea- 683 624 We begin with some general results of the stability analy- sured as the difference rx − ry in Figures 4-5, S1-S2. We find 684 625 sis for the transition from isogamy to anisogamy. We then that four qualitatively different types of anisogamic outcomes 685 626 present the simulation results for the coevolution of gamete are possible. First, anisogamy can result in either female-bias 686 627 size and sex-bias in competition in two mating types. We (rx > ry, Figure 3) or male-bias (rx < ry, Figure 3c) in com- 687 628 perform the analysis for the parameter range presented in petition represented¯ as blue or orange regions, respectively, in 688

Siljestam and Martinossi-Allibert December 18, 2020 | vol. XXX | no. XX | 7 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Fig. 4. Sex-bias in motility (b.) and sexual dimorphism in gamete size (a.) at the evolutionary endpoint, as a function of size-dependent investment efficiency αm (horizontal axis of each subplot) and competition investment efficiency αr (vertical axis). Size-dependent investment efficiency αm modulates whether gamete size has a positive or negative effect on the efficiency of energy invested in the competition trait (αm > 0 or αm < 0, respectively). The higher the value of competition investment efficiency αr, the more profitable it is to invest more energy into motility. Graphs are shown for three values of gamete density δ (columns), which represent variation from a gamete limitation context (δ = 0.01) to a gamete competition context (δ = 10 000), and for three values of the gamete survival constraint Kg (rows). For all graphs, zygote survival constraint Kz = 0. Contour lines correspond to derivations of the stability analysis (identical for a. and b.). A black dashed line encapsulates the area where isogamy is the expected evolutionary end-point; in the remaining area anisogamy is expected, and a grey contour encapsulates the area where a pseudo-isogamic genetic polymorphism can occur before anisogamy evolves. Coloured shading gives the results from the numerical simulations. Colour intensity expresses the degree of sexual dimorphism in gamete size (a) or investment in competition (b) at the evolutionary endpoint. White represents no dimorphism and deep colours represent strong dimorphism. Stars represent the parameter combinations for which a simulation run is shown in Figure 3, with the upper-case letter (A,B,C) referring to the corresponding lower-case letter in the panels of Figure 3.

689 Figures 4-5b, S1-S2b). Then, for the fusion partner capture Each subplot of Figure 4 also depicts the effects of compe- 720 690 model there are some parameter region where competition tition investment efficiency αr and size-dependent investment 721 691 trait don’t evolve resulting in anisogamy without bias in com- efficiency αm, the two parameters that modulate the efficiency 722 692 petition (rx = ry = 0, Figure 3c) seen as white regions outside of investing energy into motility. First, the competition invest- 723 693 the dashed contours in Figures 5b and S2b). Finally, there ment efficiency αr either facilitates the evolution of anisogamy 724 694 are some restricted parameter regions where the anisogamic or has no effect. Furthermore, αr increases the sex-bias in 725 695 attractor is a limit cycle resulting in never ending oscillating competition trait whenever anisogamy evolves. Looking at 726 696 dynamics (Figure 3d) where male and female bias in competi- Figure 4b, we can see that the size-dependent investment effi- 727 697 tion alternate, represented as black regions in Figures 5, S1-S2). ciency αm dictates which sex invests more energy into motility: 728 698 In the next two subsections, we discuss for each competition when αm is positive, large gametes profit more from investing 729 699 trait separately the influence of the model parameters on the into motility and in that region of the parameter space we 730 700 evolution of anisogamy and sex-bias in competition. For each observe exclusively female-biased investment in competition 731 701 trait, we vary gamete density δ from gamete limitation to if any sex-bias is present. Inversely, when αm is negative, 732 702 gamete competition, the gamete and zygote survival constraint, indicating that smaller gametes benefit more from investing 733 703 Kg and Kz as well as competition investment efficiency αr into motility, we observe that only a male-bias in competition 734 704 and size-dependent investment efficiency αm. is possible. We can further note that this situation is not sym- 735 PREPRINTmetric, at least at higher gamete densities (Figure 4b columns 736 705 Competition through gamete motility. Figure 4 presents a subset two and three): a male-bias in competition appears more easily 737 706 of the results with zygote survival constraints absent (Kz = 0, (i.e. for lower values of αr) than a female-bias in competition. 738 707 all zygotes formed survive) for simplicity as it has only a At the highest gamete density, a female-bias in competition is 739 708 minor effect. We can see that anisogamy can evolve under not observed at all, and anisogamy may only evolve if smaller 740 709 a wide range of gamete densities δ, provided that there is a gametes benefit more from investing in motility. However, for 741 710 strong constraint on gamete survival (large Kg, Figure 4a, top the lowest gamete density (Figure 4b column 1), the effect 742 711 row). If the constraint on gamete survival is somewhat relaxed of αm is rather symmetric and it is not clear whether the 743 712 (Figure 4a rows two and three), anisogamy may only evolve sex-bias in competition appears more easily in males or in 744 713 under lower gamete densities. This result is reminiscent of females (it appears to vary slightly with Kg). Another trend 745 714 the classical finding that a constraint on gamete survival en- worth noting is the fact that when anisogamy evolves with a 746 715 courages the evolution of anisogamy in the gamete competition male-bias in competition, gamete size difference is of larger 747 716 context (Bulmer and Parker, 2002). A comparison between magnitude than when it evolves along with a female-bias in 748 717 Figure 4a and b shows that the evolution of anisogamy for competition. This intuitively makes sense because investment 749 718 this competition trait is always connected with the evolution in competition will divert energy away from gamete mass to 750 719 of a sex-bias in motility.

8 | Siljestam and Martinossi-Allibert bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Fig. 5. Sex-bias in fusion partner capture (b.) and sexual dimorphism in gamete size (a.) at the evolutionary endpoint, as a function of size-dependent investment efficiency αm (horizontal axis of each subplot) and competition investment efficiency αr (vertical axis). Size-dependent investment efficiency αm modulates whether gamete size has a positive or negative effect on the efficiency of energy invested in the competition trait (αm > 0 or αm < 0 respectively). The higher the value of competition investment efficiency αr, the more profitable it is to invest energy into motility. Graphs are shown for three values of gamete density δ (columns), which represent variation from a gamete limitation context (δ = 0.01) to a gamete competition 4 context (δ = 100), and for three values of the gamete survival constraint Kg (rows). For all graphs, zygote survival constraint Kz = 10 . Contour lines correspond to derivations of the stability analysis (identical for a. and b.). A black dashed line encapsulates the area where isogamy is the expected evolutionary end-point; in the remaining area anisogamy is expected, and a grey contour encapsulates the area where a pseudo-isogamic genetic polymorphism can occur before anisogamy evolves. Coloured shading gives the results from the numerical simulations. Colour intensity expresses the degree of sexual dimorphism in gamete size (a) or investment in competition (b) at the evolutionary endpoint. White represents no dimorphism and deep colours represent strong dimorphism.

751 competition: when males spend more on competition than seems required for anisogamy to evolve; for this reason, we 783 752 females, this will magnify gamete size difference by making show a subset of the parameter space with high zygote survival 784 753 the small gametes smaller, and vice versa. Both Kz, the zy- constraint in Figure 5. A final point of similarity between 785 754 gote survival constraint and Kg, the gamete survival, have a the two competition scenarios (motility and fusion partner 786 755 positive effect on the evolution of anisogamy in high gamete capture) is the fact that a male-bias in competition appears 787 756 densities (see Supplementary Figure S1). more likely at high gamete density (gamete competition) than 788 757 We want to highlight three main points from this section at low density (gamete limitation). There are also striking 789 758 of the results: (i) when anisogamy evolves it seems to almost differences between the motility and fusion partner capture 790 759 always be accompanied by a sex-bias in competition, (ii) the scenarios: when fusion partner capture is considered as a 791 760 nature of the competition trait (modulated by αr and αm) competition trait, the possibility for anisogamy to evolve at 792 761 influences the possibility for the evolution of anisogamy as well high gamete density is more restricted than in the motility 793 762 as the direction of the resulting sex-bias in competition, and case. Also, at lower gamete densities, anisogamy is in most 794 763 finally (iii) whereas only male-biased competition can evolve cases accompanied by a strong female-bias in competition and 795 764 under high gamete density (gamete competition), both male male-bias competition is almost absent. This again shows 796 765 and female bias in competition can evolve under low density the great influence of the competition trait considered for the 797 766 (gamete limitation). possibility for anisogamy to evolve and the resulting sex-bias 798 in competition. The influence of αr and αm on the evolution 799 767 Competition through fusion partner capture. Here, we investigate PREPRINTof anisogamy and the sex-bias in competition appears less 800 768 the coevolution of gamete size with a second competition trait straightforward than in the motility scenario. Increasing αr 801 769 which increases apparent gamete size in order to capture nearby sometimes facilitates and sometimes prevents the evolution 802 770 gametes to achieve fertilisation. An increase in apparent size of anisogamy. Whenever male-bias and female-bias co-exist 803 771 will improve the target size of the gamete but will not increase within the same combination of parameters Kg, Kz and d 804 772 survival probability. Similarly to the scenario with motility as (Supplementary Figure S2c and e, top rows), the transition 805 773 a competition trait, we find that the evolution of anisogamy is from one to the other isn’t defined by a single pivot value 806 774 mostly paralleled by the evolution of a sex-bias in competition of αm, as was the case for the motility scenario. Instead, 807 775 (Figures5,S2). A noticeable difference however, is that when the transition from female-bias to male-bias appears to be a 808 776 both αm and αr are small it is frequent that none of the mating function of αm, αr and survival constraints. 809 777 types invest in the competition trait, because it is not energy- 778 efficient enough (see for example lower left corner of each We have seen that contrary to the motility trait, fusion 810 779 subplot in Figure 5b). As in the motility scenario, high survival partner capture evolves overall more easily in larger gametes. 811 780 constraints on gametes and zygotes (high values of Kg and Kz) This result may appear obvious, because the cost/benefit re- 812 781 also facilitate the evolution of anisogamy, with the difference lationship for this trait is by construction in favour of larger 813 782 that with this competition trait a higher overall constraint gametes that can expand their apparent size at a limited 814

Siljestam and Martinossi-Allibert December 18, 2020 | vol. XXX | no. XX | 9 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

815 survival cost. However, we advocate that this result is still zygote (as in Bulmer and Parker, 2002). On the other hand, a 875 816 meaningful, in showcasing the variety of competition trait that gamete limitation context allowed both sexes to invest in com- 876 817 may arise by chance, some more efficient in large gametes and petition, with the final sex-bias determined by the nature of 877 818 some in small gametes. Examining this diversity is necessary the competition trait, more specifically whether it was defined 878 819 to fully understand the relationship between the evolution as more energy-efficient in smaller or larger gametes. This 879 820 of anisogamy and that of intrasexual competition. On the result makes evident that the evolution of anisogamy needs not 880 821 contrary, making the restrictive assumption that competition result in the evolution of a male-bias in competition, but may 881 822 should generally favour small gametes (by only considering lead to a variety of outcomes depending on the circumstances 882 823 motility with negative αm for example) can lead to the conclu- under which anisogamy initially evolved (gamete limitation or 883 824 sion that males should invest more into competition, but this gamete competition) and which competition trait is considered. 884 825 view leaves many possible scenarios aside. With this second 826 competition trait, we show that the evolution of anisogamy Competition traits in the model and in nature. We imple- 885 827 and sex-bias in competition is also possible when competition mented in our model two possible competition traits. Al- 886 828 happens through a trait favouring female strategies (large though those traits are gamete-level traits, we remind the 887 829 gamete). We elaborate more on this point in the discussion. reader that our model is focusing on individual-level selection 888 and not gamete-level selection. Indeed, the competition traits 889 represent competition investment at the individual level, as 890 830 Discussion they only influence individual-level reproductive success. Both 891 831 Two pathways to anisogamy and resulting sex-bias in compe- traits are flexible with the addition of two parameters that 892 832 tition for fertilisation. We have presented a mechanistic model allow to describe freely how energy investment and gamete 893 833 of the coevolution of gamete size and a competition trait at size relate to the efficiency of the trait. This allows us to avoid 894 834 the gamete level that increases individual reproductive suc- a priori assumptions regarding which sex should benefit more 895 835 cess. Our work shows that anisogamy can evolve along with from investing in a given trait. We have shown that the nature 896 836 a sex-bias in intrasexual competition. This is true for the of the efficiency relationships of the traits greatly influences 897 837 two different scenarios of competition traits investigated here, the resulting sex-bias in selection. 898

838 first a gamete motility competition trait and second a fusion With the first trait, gamete motility, we have seen that it 899 839 partner capture competition trait. Previous work has shown evolves in males in the gamete competition context and can 900 840 that anisogamy may have evolved under gamete competition evolve in either sex in the gamete limitation context, depending 901 841 (high gamete density) or gamete limitation (low gamete den- on the relationship between gamete size and trait efficiency 902 842 sity) (e.g. Bulmer and Parker, 2002; Lehtonen and Kokko, (αm). In short, if small gametes benefit more from invest- 903 843 2011; Hoekstra et al., 1984). We believe that this distinction ing in that trait then motility will be male-biased and vice 904 844 is important because the forces of selection that would have versa. The fact that few swimming eggs have been observed 905 845 acted to produce gamete size differences in these two scenarios in nature so far (Motomura and Sakai, 1988; Klochkova et al., 906 846 are not the same. This has implications for the evolution of 2019) suggests that small gametes may generally swim more 907 847 intrasexual competition, which we investigate here. This ques- efficiently than large ones. We note however that a positive 908 848 tion is relevant to the initial evolution of anisogamy from an relationship between size and speed of gamete has been re- 909 849 isogamous state, but also to the maintenance and continuous ported in a unicellular algae (Seed and Tomkins, 2018), but 910 850 evolution of gamete size in contemporary species as we will this relationship may not hold for size differences of a thou- 911 851 discuss later. sand fold or more as is often the case in anisogamous systems. 912 852 In the gamete competition context, under high gamete den- Regardless, it is easy to imagine other traits that could be 913 853 sity, the forces that are expected to drive the evolution of less costly to develop for larger gametes. With that in mind, 914 854 anisogamy are size-dependent survival selection and competi- we incorporated in our model a second trait, fusion partner 915 855 tion for fertilisation success. In that context, anisogamy only capture. This trait increases the apparent size of the gamete, 916 856 evolves under a size constraint on survival. If anisogamy does making it a larger fertilisation target without increasing its 917 857 evolve, under high gamete density, all large gametes become survival probability. In our model, this trait develops mainly 918 858 fertilised but smaller ones are in excess, leading to the evolu- in the mating type producing the larger gamete, resulting in 919 859 tion of male-biased investment inPREPRINT intrasexual competition. In female-biased competition. This trait can be compared to the 920 860 the gamete limitation context, under low gamete density, the egg jelly coat found in several species of broadcast spawning 921 861 main force expected to drive the evolution of anisogamy is a marine invertebrates (e.g. Podolsky, 2002; Farley and Levitan, 922 862 disruptive selection pressure on gamete size that maximises 2001), and seems therefore quite realistic. It it also possible 923 863 encounter rates of gametes from the two mating types. In that that the trait we model describes somewhat accurately the 924 864 context, both sexes could be expected to engage in competition release of chemoattractant by gametes, which is another way of 925 865 because they may both have unfertilised gametes at the end increasing apparent size. This strategy is found in the eggs of 926 866 of a mating event. By adjusting gamete density, our model several species, both external and internal fertilisers, including 927 867 allowed the exploration of gamete competition and gamete mammals (Eisenbach and Giojalas, 2006). 928 868 limitation scenarios. We have seen again that anisogamy may 869 evolve in both cases, but with different outcomes in terms of Prevailing forces in the evolution and maintenance of 929 870 the sex-bias in competition. A gamete competition context anisogamy in animals. In our model, the evolution of 930 871 clearly favoured the evolution of a male-bias in competition, re- anisogamy was almost always accompanied by a sex-bias 931 872 gardless of the competition trait considered. But the evolution in competition. Gamete competition and gamete limitation 932 873 of anisogamy was more restricted in that context, relying on a shaped very different sex-biases in competition. This means 933 874 high level of size-dependent survival constraint on gamete or that an understanding of the evolutionary pathways that initi- 934

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935 ated anisogamy is key in the relationship between this initial fertilisation that causes males to compete more than females, 996 936 anisogamy and any resulting sex-bias in competition. The in the second case it is a sex-bias in PRR, which could arise 997 937 current theory on the evolution of anisogamy in animals pro- from anisogamy as well as from other causes. For example, 998 938 poses that it likely originated in sessile broadcast spawners PRR is highly sensitive to ecological factors, and field studies 999 939 (Parker, 2014), a type of animals that are often subjected to have shown that sex-biases in PRR can switch within the 1000 940 gamete limitation (Levitan, 1996). This then suggests that course of single mating season (Almada et al., 1995; Forsgren 1001 941 gamete limitation may have been important in the evolution et al., 2004, reviewed in Ahnesjö et al., 2008; Kvarnemo and 1002 942 of an initial sexual dimorphism in gamete size. According Ahnesjo, 1996). 1003

943 to our results, it seems therefore unlikely that the evolution If it seems likely that gamete limitation would have played 1004 944 of anisogamy should have resulted in male-biased intrasex- an important role in the evolution of anisogamy in ancestral an- 1005 945 ual competition in all anisogamous animals. Indeed, in our imals, we can also wonder what are the evolutionary forces that 1006 946 model the gamete limitation scenario can lead to either female maintain anisogamy in contemporary species. We have sug- 1007 947 or male-biased competition, depending on the nature of the gested above that the evolution of internal fertilisation should 1008 948 competition trait considered. have created a gamete competition context favourable to male 1009 competition. This claim should, however, be taken with care. 1010 949 Whether the sex-bias in competition at the initial A recent study in mammals has shown an inverse relationship 1011 950 anisogamy has an influence on the patterns of sex-specific between body size and sperm cell size (Lüpold and Fitzpatrick, 1012 951 competition in contemporary anisogamous species is unclear. 2015), a trend suggesting that in larger mammals gamete lim- 1013 952 The complex patterns of sex-specific selection that we observe itation may happen to some extent, leading to the evolution 1014 953 today may be subjected to an array of confounding factors of smaller more numerous sperm cells, less competitive but in- 1015 954 that we do not account for, such as evolutionary history or creasing chances of fertilisation in low gamete density. Sperm 1016 955 ecological constraints. Nevertheless, we want to comment, limitation, or failure of females to get all of their eggs fertilised 1017 956 with our findings in perspective, the fact that in the majority may also be more common than expected in insects (reviewed 1018 957 of natural systems studied today more intense intrasexual in García-González, 2004). Finally, many contemporary inver- 1019 958 competition is found in males than in females (Janicke et al., tebrate marine species are broadcast spawners that are often 1020 959 2016; Janicke and Morrow, 2018). First, the claim of a general subjected to gamete limitation (Levitan and Petersen, 1995). 1021 960 male-bias in competition should be nuanced: although males In these species, variance in reproductive success may become 1022 961 experience more intense intrasexual competition in a majority higher in either sex depending on gamete density (Levitan, 1023 962 of species (Janicke et al., 2016; Janicke and Morrow, 2018), 2004), indicating that intrasexual competition may easily arise 1024 963 there is a lot of variation among taxa as clearly visible from in either sex, as suggested by our results. Furthermore, a 1025 964 Figure 1 in Janicke et al.(2016). Females do experience non- wealth of laboratory and field studies in broadcast spawners 1026 965 negligible levels of intrasexual competition in most species and have reported female traits that increase fertilisation success 1027 966 in some cases more than males (reviewed in Hare and Sim- and are therefore involved in female intrasexual competition. 1028 967 mons, 2018). Female competition is favoured if males provide In three species of sea urchins (Levitan, 1993, 1998a) egg traits 1029 968 costly nuptial gifts or parental care, but they may also com- are shown to evolve to maximise fertilisation success. In the 1030 969 pete to increase fertilisation success in sperm limited-contexts sea urchin Lytechinus variegatus experimental removal of a 1031 970 (Hare and Simmons, 2018), which are not restricted to external jelly coat around the eggs lowers fertilisation success due to 1032 971 fertilisers (see for example a study in gorillas Niemeyer and reduced target size (Farley and Levitan, 2001). In the sand 1033 972 Anderson, 1983, and one in saiga antelopes Milner-Gulland dollar Dandraster excentricus a jelly coat that increases up 1034 973 et al., 2003). Second, we propose two non-mutually exclu- to sixfold the size of the egg increases fertilisation success 1035 974 sive hypotheses that may explain the apparent inconsistency (Podolsky, 2002), and finally in the tunicate Styela plicata 1036 975 between our theoretical claim about the origin of anisogamy (Crean and Marshall, 2008) both sexes adapt their gametes 1037 976 and empirical observations of today’s natural systems. (i): a to population density, with female gametes becoming notably 1038 977 large proportion of animals studied by biologists are internal larger at low density, a strategy that increases fertilisation 1039 978 fertilisers. The evolution of internal fertilisation reduces the success but comes at a cost for zygote survival. 1040 979 chances for gamete limitation to appear, which in turn should 980 favour the evolution of male competition.PREPRINT (ii): our model Conclusion. It seems that gamete limitation may be impor- 1041 981 assumes that individuals invest the same amount of energy tant in shaping anisogamy, which means that the evolutionary 1042 982 into reproduction regardless of their sex or mating type. If forces that are responsible for gamete size differences cannot 1043 983 that assumption does not hold, and if the cost of producing be expected to always result in the evolution of male-biased 1044 984 larger gametes does not scale linearly with gamete size, it is competition. We thus question the claim that anisogamy 1045 985 possible that a difference in potential reproductive rate (PRR, necessarily leads to male-biased competition. Some of the evo- 1046 986 time required for an individual to produce offspring and return lutionary forces that can spur the evolution of anisogamy in 1047 987 to the mating pool, as in Clutton-Brock, Clutton-Brock and the gamete competition context do clearly favour male-biased 1048 988 Parker, 1992) arises between the sexes. A sex-bias in PRR is competition; but evolutionary forces that shape anisogamy 1049 989 likely to cause a sex-bias in intrasexual competition, with the in the gamete limitation context may produce male-biased or 1050 990 sex with the highest PRR competing for mating opportunities female-biased competition, with the outcome mostly decided 1051 991 with the other sex (Clutton-Brock and Parker, 1992; Kokko by the nature of the competition trait and how its efficiency 1052 992 et al., 2006). We note that these two hypotheses (i) and (ii) relates to gamete size. There are good reason to think that 1053 993 do not undermine the fact that anisogamy (gamete size dimor- gamete limitation played an important role in the animals that 1054 994 phism) is not likely to explain directly sex-biases in intrasexual first evolved the male and female sex. Even though the evolu- 1055 995 competition: in the first case it is the evolution of internal tion of internal fertilisation in a wide range of species suggests 1056

Siljestam and Martinossi-Allibert December 18, 2020 | vol. XXX | no. XX | 11 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

1057 that gamete competition is common in nature, there is evidence The number of gametes per zygote of the two mating types declines 1120 1058 claiming that even in internal fertilisers the evolutionary forces at the same rate (dnx,t/dt = dny,t/dt in Eq. A2) as each fertilisation 1121 always removes one gamete of each mating type. The difference in 1122 1059 of gamete limitation are still at play. Together, this suggests the number of gametes between the two mating types ny,t − nx,t 1123 1060 that, as far as anisogamy is concerned, females should benefit thereby stays constant over the mating period and equals the initial 1124 1061 from competing for increased fertilisation success, too, a claim difference ny,0 − nx,0. The initial number of gametes per zygote of 1125 1062 that is supported by the observation of female intrasexual mating type i equals 1126

1063 competition in many species, an in particular with egg compe- e(1 − ri) mi − 1 n = n sg = , [A3] 1127 1064 tition traits in broadcast spawning species. Evidence showing i,0 i mi Kg + mi − 1 1065 more intense competition in males in a majority of species where ni is the number of gametes produced per zygote of mating 1128 1066 (a general pattern but not a general rule) is not proof that type i and sg the survival probability of each gamete (see main 1129 1067 anisogamy is the cause for that pattern. We suggest that, for text). Hence, the number of gametes per zygote of one mating type 1130 1068 example, potential reproductive rate (PRR) may be a better at time t is given by the number of gametes of the other mating 1131 1069 predictor of intrasexual competition, as it captures important type plus their initial difference 1132

1070 ecological influences on competition. In turn, PRR may be ni,t = nj,t + ni,0 − nj,0. [A4] 1133 1071 related to gamete size in some species, if larger gametes require where i is the focal mating type (x or y) and j is the other mating 1134 1072 more time and energy to be produced but this does not need type. 1135 1073 to be the case. Our results challenge the classical view that By using the substitution in Eq. A4 we can express equation A2 1136 1074 anisogamy alone is enough to explain more intense intrasexual as a single differential equation (as the initial gamete densities ni,0 1137 and nj,0, Eq. A3, are constants) 1138 1075 competition in the male sex. dni,t = −crni,t(ni,t + nj,0 − ni,0), [A5] 1139 1076 Appendix dt where cr = pdσxyvxy is a constant proportional to the fertilisation 1140 1077 rate. Eq. A5 is a separable differential equation with the following 1141 1078 Note: In the main part we present how the variables of the model solution 1142 1079 are functions of the gamete traits (mx, rx, my and ry). We will not 1080 detail this in the appendix for the sake of readability. One exception nj,0 − ni,0 0 ni,t = ni,0 , [A6] 1143 1081 is made for the case of fertilisation rates f and f (Eqs. A9 and
i i exp cr(nj,0 − ni,0)t nj,0 − ni,0 1082 A14) that are used to obtain the expected evolutionary path of 1083 these gamete traits (Eqs. 4, 5, 6 in the main text). (step-wise calculations in Supplementary material 1.1, Eq. S3). 1144

1084 1. Collision frequency. To mechanistically model the fertilisation 2.1. Gamete fertilisation dynamics under isogamy. For the case of 1145 1085 dynamics between the gametes of the two mating types x and y evolution under isogamic constraint, the gametes of the two mating 1146 1086 in the mating pool, we make use of classical collision theory from types are constrained to equal trait values. Hence, we have that 1147 1087 chemistry and physics (McNaught et al., 2014). Hence, we assume nx,t = ny,t and the change in the number of gametes over time 1148 1088 that gametes of the two mating types x and y travel at constant Eq. A5 simplifies to 1149 1089 speeds vx and vy with trajectories approximated by straight lines dni,t 2 = −crn . [A7] 1150 1090 at a local scale. We obtain the following frequency of collisions dt i,t 1091 between gametes of the two mating types (per unit of time and per Solving for ni,t in Eq. A7 gives the number of gametes per zygote 1151 1092 unit of volume) 2 of mating type i at time t under the isogamic constraint 1152 1093 d σxyvxynx,tny,t. [A1] ni,0 1094 Here, ni,t is the number of gametes per zygote of mating type i at ni,t = . [A8] 1153 c n t + 1 1095 time t, and d is the zygote population density (number of zygotes per r i,0 1/3 1/3 2 1096 mating type per unit of volume). Also, σxy = π(mˆ x +mˆ y ) is the (step-wise calculations in Supplementary material 1.2, Eq. S6). 1154 1097 area of trajectories which would result in a collision between gametes 1098 of the two mating types, representing a disk with a radius equal to 3. Zygote fertilisation success. When the mating period is over, the 1155

1099 the combined radiuses of the two gametes (i.e, their collisional cross number of fertilised gametes per zygote equals ni,0 −ni,tend (Eqs. A3 1156 1100 section), assuming spherical gametes and excluding the motility and A6), (i.e., the number of gametes entering the mating pool 1157 p 2 2 subtracted by the number of unfertilised gametes at t = tend, the 1158 1101 machinery from the collision target. Lastly, vxy = vx + vy is the end of the mating period). Each fertilisation produces a juvenile 1159 1102 average relative velocity between gametes of the two mating types. zygote (Fig. 2) that has a probability of survival to the next genera- 1160

1103 2. Gamete fertilisation dynamics. Gametes of the two mating types tion of sz = (mx + my − 1)/(Kz + mx + my − 1) (as presented in 1161 the main text). Hence, the number of fertilised gametes per zygote 1162 1104 x and y collide at a frequency givenPREPRINT by Eq. A1 and each collision 1105 has a probability p of resulting in a fertilisation event. Fertilisation multiplied with their survival probability sz gives the per capita fer- 1163 1106 generates a juvenile zygote and removes the colliding gametes from tilisation success fi (or simply: fertilisation success) corresponding 1164 1107 the mating pool. to the absolute fitness of a zygote, 1165 1108 Multiplying Eq. A1 with p gives the fertilisation rate (per unit of
fi(T ) = ni,0(mi, ri) − ni,t (T ) sz(mi, mj ), [A9] 1166 1109 time and per unit of volume). Dividing the fertilisation rate by the end 1110 gamete density of mating type i, d n , gives the fertilisation rate i,t where i denotes the focal mating type (x or y) and j the other mating 1167 1111 per gamete of mating type i per unit of time at time t. Multiplying type, and where the trait vector T = (mx, rx, my, ry) represents all 1168 1112 this rate with the number of gametes per zygote of mating type i, four gamete traits. 1169 1113 ni,t, gives the fertilisation rate per zygote of mating type i per unit 1114 of time at time t. 4. Fertilisation dynamics of a rare mutant strategy. To introduce evo- 1170 1115 Because gametes that achieve fertilisation are removed from the lution into our model we assume that mutation events are rare, each 1171 1116 mating pool, the number of gametes present at time t per zygote introducing a single mutant zygote. The strategy of the mutant 1172 1117 ni,t decreases according to their fertilisation rate (negative term on 0 0 0 0 0 T = (mx, rx, my, ry) deviates in one or more of the four traits 1173 1118 the right hand side of the equation) from the resident strategy T = (mx, rx, my, ry) (if evolution occurs 1174 0 0 0 0 dnx,t under the isogamic constraint T is replaced with with Tc = (mc, rc) 1175 = −pdσxyvxynx,tny,t, dt and T with Tc = (mc, rc)). The resident population is assumed 1176 1119 [A2] 0 dny,t to be large, and as the mutant strategy T is introduced at the 1177 = −pdσxyvxynx,tny,t. minimum frequency (one individual) its effect on the fertilisation 1178 dt

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1179 dynamics can be neglected, as any gamete is very unlikely to collide singular strategies, we numerically solve for when the first and the 1235 1180 with these rare mutant gametes. One can thereby consider gametes second element of β(Tc) are equal to zero, separately (i.e. we solve 1236 1181 to only collide with gametes of the resident strategy, as long as for both isoclines). We then find their intersections, which gives the 1237 1182 the mutant is rare. This simplification is used when deriving the singular points. 1238 1183 invasion fitness of the a mutant strategy. To obtain the invasion 1184 fitness of the mutant (Eq. 4), we need its zygote fertilisation suc- 5.2. Stability analysis of singular strategies. To predict the evolu- 1239 ∗ ∗ 1185 cess (Eq. A9), which in turn requires a solution for the gamete tionary dynamics in the vicinity of a singular strategy T (or Tc 1240 1186 fertilisation dynamics of the mutant (Eq. A12), given below. under the isogamic constraint), we look at two stability properties 1241 of the singular point, namely the attractiveness and invadability. 1242 ∗ 1187 4.1. Gamete fertilisation dynamics of a rare mutant strategy. Here, First, a singular strategy T is uninvadable by nearby mutants if 1243 1188 we derive the gamete fertilisation dynamics of a rare mutant strategy. the four-dimensional Hessian matrix H of the selection gradient 1244 1189 The mutant gametes are removed from the mating pool at each (Eq.5) with entries 1245 1190 successful fertilisation, and the number of gametes per mutant δ2 w(T 0,T ) 1191 zygote ni0,t (of mating type i at time t) decreases according to their O hij = 0 0 [A15] 1246 1192 fertilisation rate (negative term on the right hand side) δTi δTj T 0=T =T ∗ dn 0 0 0 i ,t is negative definite (where T and T gives the ith and the jth ele- 1247 1193 = −pdσi0j vi0j nj,tni0,t, [A10] i j dt 0 0 0 0 0 ment of the mutant trait vector T = (mx, rx, my, ry)). Otherwise, 1248 ∗ 1194 where nj,t is the number of gametes per resident zygote of the other the singular strategy T is invadable by nearby mutant strategies 1249 1195 mating type j, σi0j the collisional cross section between the mutant (Leimar, 2009). We refer to this property as the invadability. 1250 1196 gamete of mating type i and the resident gamete of mating type j, Note: if isogamic constraint is considered, replace the gamete 1251 0 0 1197 and vi0j is average relative velocity (see Appendix 1) between the strategies T and T with the constraint strategies Tc and Tc, re- 1252 1198 mutant gamete of mating type i and the resident gamete of mating spectively. This gives the two-dimensional Hessian matrix of the 1253 1199 type j. constraint evolution. The same holds for the Q-matrix and Jacobian 1254 1200 To solve for ni0,t we first substitute the number of resident J presented below. 1255 ∗ 1201 gamete nj,t in Eq. A10 with Eq. A6 giving Whether a singular strategy T is an attractor of the evolutionary 1256 path or a repeller depends on the Jacobian matrix J of the selection 1257 dni0,t cmnj,0(ni,0 − nj,0) gradient (Eq. 5). The Jacobian is given by J = H + Q where Q is a 1258 1202 = −
ni0,t, [A11] dt exp cr(ni,0 − nj,0)t ni,0 − nj,0 four-dimensional square matrix with entries 1259 2 0 1203 where cm = pdσi0j vi0j is a constant proportional to the collision δ Ow(T ,T ) qij = [A16] 1260 1204 probability for the mutant gametes of mating type i. Eq. A11 is a 0 δTi δTj T 0=T =T ∗ 1205 separable differential equation with the following solution where Tj gives the jth element of the resident trait vector 1261 T = (mx, rx, my, ry)) 1262  ni,0 −nj,0 cm c ∗ 1206 ni0,t =ni0,0 exp(cmni,0t) r , [A12] If J is negative definite T is an attractor meaning that evolution 1263 ni,0 exp(crni,0t)−nj,0 exp(crnj,0t) ∗ will converge towards T . On the other hand, if J is positive 1264 ∗ 1207 giving the number of unfertilised gametes per mutant zygote definite T is a repeller of the evolutionary dynamics. Lastly, if J is 1265 1208 at time t (step-wise calculations in Supplementary material 1.3, indefinite the evolutionary dynamics will depend on the mutational 1266 ∗ 1209 Eq. S11). covariance matrix C. Then T can be an attractor for specific 1267 1210 If evolution occurs under the isogamic constraint in Eq. A10 we mutational covariances of C (such that the Jacobian of Cβ(T ) is 1268 1211 instead substitute the resident gamete number nj,t with Eq. A8 negative definite), but is otherwise a repellor of the evolutionary 1269 1212 and we get dynamics (see Leimar, 2009). 1270 dni0,t cmni,0 For our analysis, we do not consider any mutational covariance 1271 1213 = − ni0,t. [A13] dt crni,0t + 1 between our four traits (mx, rx, my, ry) (or two traits (mc, rc) for 1272 evolution under isogamic constraint) corresponding to C being a 1273 which also is a separable differential equation and has the following ∗ diagonal matrix, and consequentially T is only an attractor if J 1274 solution cm is negative definite, and is otherwise an repeller. For simplicity 1275  1  cr ni0,t = ni0,0 , we choose C to be equal the identity matrix meaning that the 1276 crni,0t + 1 evolutionary path Eq.6 equals the selection gradient Eq. 5. 1277 1214 giving the number of unfertilised gametes per mutant zygote at 1278 1215 time t in an isogamic population (step-wise calculations in Supple- 5.3. Numerical simulation of the evolutionary path. As a complement 1279 1216 mentary material 1.4, Eq. S14) to the stability analysis, we numerically simulate the evolutionary path given by Eq. 6 using the Runge-Kutta method with adaptive 1280 1217 4.2. Zygote fertilisation success of a rare mutant strategy. The fertil- step size of order 4 and 5. First, we simulate the evolutionary path 1281 0 1218 isation success of a rare mutant strategy fi (of mating type i) follows under the isogamic constraint. As we find that there is always 1282 1219 the same logic as for the resident fertilisation success (Eq. A9) and a single attracting strategy for the parameter range we investi- 1283 1220 is given by PREPRINTgate (see Results section), the starting point has no particular 1284 meaning and we start evolution from an arbitrary point chosen to 1285 0 0 0 1 X 0 0 0 0
0 1221 f (m , r ,T ) = ni,0(m , r ) − n 0 (m , r ,T ) sz(m , mj ), [A14] −5 i i i 2 i j i ,tend i i i be Tc = (mc = 100, rc = 10 ). We run the numerical simula- 1286 i∈{x,y} tion until the evolutionary equilibrium is reached at an attractor 1287 a a a Tc = (mc , rc ). Then, we continue the numerical simulation without 1288 1222 where i denotes the focal mating type (x or y) and j the other mating the isogamic constraint with a small initial divergence  in either 1289 1223 type, and where the trait vector T = (mx, rx, my, ry) represents all gamete size m or relative competition r investment traits between 1290 1224 four gamete traits. a a a a the two mating types such that T = (mc + , rc , mc − , rc ) or 1291 a a a a T = (mc , rc +, mc , rc −). If evolution in both cases leads back to- 1292 1225 5. Predicting the evolutionary dynamics. Below we describe our pro- a a a a wards T = (mc , rc , mc , rc ) we conclude that the isogamic attractor 1293 1226 cedure for predicting the evolutionary dynamics of the gamete traits a T is also an attractor of the corresponding unconstrained evolution. 1294 1227 mx, rx, my and ry (or mc and rc under the isogamic constraint). c In this case, there is stabilising selection preventing the evolution 1295 a 1228 5.1. Solving for isogamic singular strategies. We assume evolution of anisogamy and the isogamic attractor Tc is the end-point of 1296 1229 to start under the isogamic constraint (where mx = my = mc and evolution. Otherwise, if the trait values of the two mating types 1297 a 1230 rx = ry = rc and the isogamic strategy is given by Tc = (mc, rc)). diverge away from Tc , it is a repellor of the unconstrained evolution 1298 1231 To predict the evolutionary path (given by Eq. 6) we first numeri- and anisogamy evolves. We then run the numerical simulation until 1299 ∗ 1232 cally solve for the isogamic singular strategies Tc , i.e. where the anisogamic evolution reaches an attractor, most often it is a fixed 1300 ∗ a a a a a 1233 selection gradient is zero, β(Tc ) = 0. The singular strategies rep- point attractor T = (mx, rx, my, ry) giving an evolutionary end- 1301 1234 resent candidates for attracting strategies. To find these isogamic point (see Fig. 3b-c), but for a small subset of the parameter values 1302

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1499 Supplementary information 1.2. Solving for gamete fertilisation under isogamy. For the case of the isogamic constraint, the number of gametes per zygote of mating 1500 1. Step-wise calculations. type i at time t decreases according to (Eq. A7) 1.1. Solving for gamete fertilisation. In the mating pool, the number dni,t 2 = −crn , of gametes per zygote of mating type i at time t decreases as dt i,t fertilisation events occur according to (Eq. A5) which is a separable differential equation: dni,t = −crni,t(ni,t + nj,0 − ni,0), dt dni,t 2 = − crni,t ⇐⇒ which is a separable differential equation: dt dni,t dni,t 2 dt = − crni,t − cr(nj,0 − ni,0)ni,t ⇐⇒ 2 = − cr, dt ni,t dni,t dt = − 1, and can thereby be solved by integrating both sides with respect 2 to t crni,t + cr(nj,0 − ni,0)ni,t and can thereby be solved by integrating both sides with respect Z dni,t Z dt to t 2 dt = − cr dt ⇐⇒ ni,t Z dni,t Z dt Z 1 Z 2 dt = − 1 dt ⇐⇒ c n + c (n − n )n dni,t = − cr dt ⇐⇒ r i,t r j,0 i,0 i,t n2 Z i,t 1 1 dni,t = − (t + c). cr (nj,0−ni,0) 2 − = − (crt + c) ⇐⇒ (cr + )n ni,t ni,t i,t 1 The left-hand side integral can be solved with the fol- ni,t = [S4] lowing substitution u = cr + cr(nj,0 − ni,0)/ni,t and crt + c 2 2 du = −cr(nj,0 − ni,0)/ni,tdni,t ⇐⇒ dni,t = −ni,t/(cr(nj,0 − ni,0))du, Solving for c in Eq. S4 with initial condition t = 0 and ni,t = ni,0 resulting in gives 1 Z 1 − du = − (t + c) ⇐⇒ 1 ni,0 = ⇐⇒ cr(nj,0 − ni,0) u c 1506 1 1 log(u)=t + c. c = [S5] cr(nj,0 − ni,0) ni,0

Substituting back u = cr + cr(nj,0 − ni,0)/ni,t gives 1507 Substituting c (Eq. S5) into Eq. S4 gives cr(nj,0 − ni,0) log(cr + ) =cr(nj,0 − ni,0)(t + c) ⇐⇒ ni,t 1 ni,t = 1 . c (n − n ) crt + r j,0 i,0
ni,0 = exp cr(nj,0 − ni,0)(t + c) − cr ⇐⇒ ni,t Simplifying this gives our explicit expression for the number of 1508 cr(nj,0 − ni,0) gametes of mating type i at time t under isogamy 1509 ni,t =
. exp cr(nj,0 − ni,0)(t + c) − cr ni,0 ni,t = . [S6] 1510 [S1] crni,0t + 1

Solving for c in Eq. S1 with initial condition t = 0 and ni,t = ni,0 1.3. Solving for gamete fertilisation of a rare mutant strategy. In the gives mating pool, the number of gametes per zygote with a rare mutant strategy, of mating type i at time t, decreases according to (Eq. A12) cr (nj,0 − ni,0) ni,0 = ⇐⇒ dni0,t cmnj,0(ni,0 − nj,0) exp(cr (nj,0 − ni,0)c) − cr = − ni0,t, dt expc (n − n )t
n − n 1501 cr (nj,0 − ni,0) r i,0 j,0 i,0 j,0 + cr = exp(cr (nj,0 − ni,0)c) ⇐⇒ ni,0 which is a separable differential equation 1 cr (nj,0 − ni,0)
c = log + cr dn 0 c (n − n )n c (n − n ) n i ,t m i,0 j,0 j,0 r j,0 i,PREPRINT0 i,0 = − ni0,t ⇐⇒ [S2] dt exp(cr(ni,0 − nj,0)t)ni,0 − nj,0 dni0,t 1502 dt cm(ni,0 − nj,0)nj,0 Substituting c (Eq. S2) into Eq. S1 gives = − , ni0,t exp(cr(ni,0 − nj,0)t)ni,0 − nj,0 cr(nj,0 − ni,0) ni,t = ⇐⇒ and can thereby be solved by integrating both sides with respect cr (nj,0−ni,0) to t exp(cr(nj,0 − ni,0)t)( n + cr) − cr i,0 dn Z i0,t Z cm(ni,0 −nj,0)nj,0 (nj,0 − ni,0) dt dt = − dt ⇐⇒ ni,t = ⇐⇒ n 0 exp(c (n − n )t)n − n (nj,0−ni,0) i ,t r i,0 j,0 i,0 j,0 exp(cr(nj,0 − ni,0)t)( + 1) − 1 ni,0 Z 1 Z 1 ni,0(nj,0 − ni,0) dni0,t = −cm(ni,0 −nj,0)nj,0 dt ⇐⇒ ni,t = . ni0,t exp(cr (ni,0 −nj,0)t)ni,0 −nj,0 exp(cr(nj,0 − ni,0)t)((nj,0 − ni,0) + ni,0) − ni,0 Z 1 1503 Simplifying this once more gives our explicit expression for the log(n 0 ) + c = −cm(ni,0 −nj,0)nj,0 dt i ,t exp(c (n −n )t)n −n 1504 number of gametes of mating type i at time t r i,0 j,0 i,0 j,0 1511 ni,0(nj,0 − ni,0) The right hand side integral can be solved by first ap- 1505 ni,t =
. [S3] exp cr(nj,0 − ni,0)t nj,0 − ni,0 plying the following substitution u = cr(ni,0 − nj,0)t where

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du = cr(ni,0 − nj,0)dt ⇐⇒ dt = 1/cr(ni,0 − nj,0)du, resulting Solving for c in Eq. S9 with initial condition t = 0 and ni0,t = ni0,0 in gives

Z − cm c (n − n )n 1 c m i,0 j,0 j,0 ni0,0 =(ni,0 − nj,0) r exp(c) ⇐⇒ log(ni0,t) + c = − du ⇐⇒ cr(ni,0 − nj,0) exp(u)ni,0 − nj,0 cm 1520 c
Z c = log ni0,0(ni,0 − nj,0) r ⇐⇒ cmnj,0 1 log(n 0 ) + c = − du, cm i ,t c = log(n 0 ) + log(n − n ) [S10] cr exp(u)ni,0 − nj,0 i ,0 i,0 j,0 cr

1512 and then apply the following substitution v = exp(u) where 1521 1513 dv = exp(u)du ⇐⇒ du = (1/v)dv, resulting in Substituting c (Eq. S10) into Eq. S9 gives

cm
Z exp(c (n − n )t) exp log(n 0 ) + log(n − n ) cmnj,0 1 m i,0 j,0 i ,0 cr i,0 j,0 1514 log(n 0 ) + c = − dv, [S7] n 0 = ⇐⇒ i ,t c v(n v − n ) i ,t cm r i,0 j,0
cr ni,0 exp(cr (ni,0 − nj,0)t) − nj,0 1515 and the right hand side can be simplified using partial fraction n − n cm i,0 j,0
cr 1516 decomposition: ni0,t =ni0,0 exp(cm(ni,0 − nj,0)t) ni,0 exp(cr (ni,0 − nj,0)t) − nj,0 1517 Simplifying this once more gives our explicit expression for the 1522 1 A B number of gametes of the mutant strategy of mating type i at time t 1523 = + ⇐⇒ v(ni,0v − nj,0) v ni,0v − nj,0

 ni,0 − nj,0 cm 1 =A(ni,0v − nj,0) + Bv c ni0,t =ni0,0 exp(cmni,0t) r [S11] ni,0 exp(crni,0t) − nj,0 exp(crnj,0t) 1518 where v = nj,0/ni,0 gives the solution B = ni,0/nj,0, and v = 0 1524 gives the solution A = −1/nj,0. Hence, 1525 1 ni,0 1 = 2 − [S8] v(ni,0v − nj,0) ni,0nj,0v − nj,0 nj,0v 1.4. Solving for gamete fertilisation of a rare mutant strategy under isogamy. For the case of the isogamic constraint, the number of 1519 gametes per zygote of mating type i with a rare mutant strategy at Substituting Eq. S8 into Eq. S7 gives time t decreases according to (Eq. A12) Z Z cmnj,0 ni,0 cmnj,0 1 dni0,t cmni,0 log(n 0 ) = − dv + dv + c ⇐⇒ = − n 0 , i ,t c n n v − n2 c n v i ,t r i,0 j,0 j,0 r j,0 dt crni,0t + 1 Z Z cm 1 cmni,0 1 which is a separable differential equation log(ni0,t) = dv − dv + c ⇐⇒ cr v cr ni,0v − nj,0 dni0,t cmni,0 c c n Z 1 = − ni0,t ⇐⇒ m m i,0 dt crni,0t + 1 log(ni0,t) = log(v) − dv + c, cr cr ni,0v − nj,0 dni0,t dt cmni,0 where the integral on the right hand side can be solved with the = − ni0,t crni,0t + 1 following substitution w = ni,0v − nj,0 where dw = ni,0dv ⇐⇒ dv = 1/ni,0dw, resulting in and can thereby be solved by integrating both sides with respect Z of t cm cm 1 log(n 0 ) = log(v) − dw + c ⇐⇒ dn i ,t Z i0,t Z cr cr w cmn dt dt = − i,0 dt ⇐⇒ cm cm n 0 crni,0t + 1 log(n 0 ) = log(v) − log(w) + c ⇐⇒ i ,t i ,t c c r r Z 1 Z 1 cm v
dn 0 = − cmn dt ⇐⇒ log(n 0 ) = log + c. i ,t i,0 i ,t n 0 crni,0t + 1 cr w i ,t Z Substitution back w = n v − n , and then v = exp(u), and then 1 i,0 j,0 log(ni0,t) = − cmni,0 dt + c. u = cr(ni,0 − nj,0)t gives crni,0t + 1

cm v
The right hand-side of the integral can be solved with the fol- 1526 log(ni0,t) = log + c ⇐⇒ lowing substitution u = c n t + 1 where du = c n dt ⇐⇒ 1527 cr ni,0v − nj,0 PREPRINTr i,0 r i,0 dt = 1/(crni,0)du, resulting in 1528 cm exp(u)
log(ni0,t) = log + c ⇐⇒ cr ni,0 exp(u) − nj,0 Z cmni,0 1 cm exp(cr(ni,0 − nj,0)t)
log(n 0 ) = − du + c ⇐⇒ log(n 0 ) = log + c, i ,t i ,t crni,0 u cr ni,0 exp(cr(ni,0 − nj,0)t) − nj,0 cm which simplifies to log(ni0,t) = − log(u)+ c ⇐⇒ cr

log(n 0 ) =c (n − n )t exp(c) i ,t m i,0 j,0 0 ni ,t = cm ⇐⇒ cm exp(log(u)) cr − log(ni,0 exp(cr(ni,0 − nj,0)t) − nj,0) + c ⇐⇒ cr exp(c) ni0,t = c . log(ni0,t) =cm(ni,0 − nj,0)t m u cr  cm 
cr − log ni,0 exp(cr(ni,0 − nj,0)t) − nj,0 + c ⇐⇒ Substituting back u = crni,0t + 1 gives

exp(cm(ni,0 − nj,0)t) exp(c) exp(c) 0 0 ni ,t = cm . [S9] ni ,t = cm . [S12]
c c ni,0 exp(cr(ni,0 − nj,0)t) − nj,0 r (crni,0t + 1) r

Siljestam and Martinossi-Allibert December 18, 2020 | vol. XXX | no. XX | 17 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Solving for c in Eq. S12 with initial condition t = 0 and ni,t = ni,0 gives

1529 ni0,0 = exp(c) ⇐⇒

c = log(ni0,0). [S13]

1530 1531 Substituting c (Eq. S13) into Eq. S12 gives our explicit expression 1532 for the number of gametes of the mutant strategy of mating type i 1533 at time t under isogamy 1 0 0 1534 ni ,t = ni ,0 cm . [S14] (crni,0t + 1) cr

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18 | Siljestam and Martinossi-Allibert bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

PREPRINT

Fig. S1. Sex-bias in gamete motility (b,d,f) and sexual dimorphism in gamete size (a,c,e) at the evolutionary endpoint, as a function of size-dependent investment efficiency αm (horizontal axis of each subplot) and competition investment efficiency αr (vertical axis). Size-dependent investment efficiency αm modulates whether gamete size has a positive or negative effect on the efficiency of energy invested in the competition trait (αm > 0 or αm < 0 respectively). The higher the value of competition investment efficiency αr, the more profitable it is to invest energy into motility. Graphs are shown for three values of gamete density δ, which represent variation from a gamete competition context (δ = 10 000, a,b) to a gamete limitation context (δ = 0.01, e,f). Three values of the gamete survival constraint Kg (rows) and zygote survival constraint Kz (columns) are also given. Contour lines correspond to stability analysis of the isogamic singular point. A black dashed line encapsulates the area where isogamy is the expected evolutionary end-point; in the remaining area anisogamy is expected, and a grey contour encapsulates the area where a pseudo-isogamic genetic polymorphism can occur before anisogamy evolves. Coloured shading gives the results from the numerical simulations. Colour intensity expresses the degree of sexual dimorphism in gamete size (a,c,e) or investment in competition (b,d,f) at the evolutionary endpoint. White represents no dimorphism and deep colours represent strong dimorphism. Stars represent the parameter combinations for which a simulation run is shown in Figure 3, with the upper-case letter (A,B,C,D) referring to the corresponding lower-case letter in the panels of Figure 3.

Siljestam and Martinossi-Allibert December 18, 2020 | vol. XXX | no. XX | 19 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.18.423382; this version posted December 18, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

PREPRINT

Fig. S2. Sex-bias in fusion partner capture (b,d,f) and sexual dimorphism in gamete size (a,c,e) at the evolutionary endpoint, as a function of size- dependent investment efficiency αm (horizontal axis of each subplot) and competition investment efficiency αr (vertical axis). Size-dependent investment efficiency αm modulates whether gamete size has a positive or negative effect on the efficiency of energy invested in the competition trait (αm > 0 or αm < 0 respectively). The higher the value of competition investment efficiency αr, the more profitable it is to invest energy into motility. Graphs are shown for three values of gamete density δ, which represent variation from a gamete competition context (δ = 100, a,b) to a gamete limitation context (δ = 0.01, e,f). Three values of the gamete survival constraint Kg (rows) and zygote survival constraint Kz (columns) are also given. Contour lines correspond to stability analysis of the isogamic singular points. A black dashed line encapsulates the area where isogamy is the expected evolutionary end-point; in the remaining area anisogamy is expected, and a grey contour encapsulates the area where a pseudo-isogamic genetic polymorphism can occur before anisogamy evolves. Coloured shading gives the results from the numerical simulations. Colour intensity expresses the degree of sexual dimorphism in gamete size (a,c,e) or investment in competition (b,d,f) at the evolutionary endpoint. White represents no dimorphism and deep colours represent strong dimorphism.

20 | Siljestam and Martinossi-Allibert