Highlights 13+ We derive conditions for evolutionary branching in directionally evolving populations. 14 + The derived conditions extend those for univariate trait spaces to bivariate trait spaces. univariate traits is the existence of a convergence stable trait value at which selection is 6 locally disruptive. Real populations, however, undergo simultaneous evolution in multiple 7 traits. Here we extend conditions for evolutionary branching to bivariate trait spaces in which 8 the response to disruptive selection on one trait can be suppressed by directional selection on 9 another trait. To obtain analytical results, we study trait-substitution sequences formed by 10 invasions that possess maximum likelihood. By deriving a sufficient condition for 11 evolutionary branching of bivariate traits along such maximum-likelihood-invasion paths 12 (MLIPs), we demonstrate the existence of a threshold ratio specifying how much disruptive 13 selection in one trait direction is needed to overcome the obstruction of evolutionary 14 branching caused by directional selection in the other trait direction. Generalizing this finding, 15we show that evolutionary branching of bivariate traits can occur along evolutionary-16 branching lines on which residual directional selection is sufficiently weak. We then present 17 numerical analyses showing that our generalized condition for evolutionary branching is a 18 good indicator of branching likelihood even when trait-substitution sequences do not follow 19MLIPs and when mutations are not rare. Finally, we extend the derived conditions for 20 evolutionary branching to multivariate trait spaces. 21Keywords 22 frequency-dependent selection, speciation, adaptive dynamics, two-dimensional traits, multi-23 dimensional traits 24 25

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We constructed a sex allocation model for local mate competition considering the asymmetry of competitive abilities among sons. This model assumes two females of a parasitoid wasp oviposit on the same host in sequential order. The evolutionarily stable strategy will be in either Stackelberg or Nash equilibrium, depending on whether the females can recognize their opponent's sex ratio or not, respectively. The Nash equilibrium predicts the second female produce more males than the first. If the second female is able to know and respond to the strategy of the first (a Stackelberg equilibrium), the first will decide an optimal sex ratio assuming that the second reply to it. Under such an assumption, our model predicts that not producing sons is adaptive for the second female when the sons she produces have low competitive ability. Males of parasitoid wasps Melittobia spp. are engaged in lethal male-male combat, indicating large asymmetry in mating success among sons. If females have the ability to recognize their opponent's sex ratio, our model suggests that the severe lethal male-male combat may be one factor explaining their extremely female-biased sex ratio that is unexplainable by pre-existent models.