The dynamics of two- and three-way sexual conflicts over mating

Abstract

We consider mathematical models describing the evolutionary consequences of antagonistic interactions between male offence, male defence and female reproductive tract and physiology in controlling female mating rate. Overall, the models support previous verbal arguments about the possibility of continuous coevolutionary chase between the sexes driven by two-way (e.g. between male offence and female traits) and three-way (e.g. between male offence, male defence and female traits) inter-sexual antagonistic interactions. At the same time, the models clarify these arguments by identifying various additional potential evolutionary dynamics and important parameters (e.g. genetic variances, female optimum mating rates, strength of selection in females and the relative contributions of first and second males into offspring) and emphasizing the importance of initial conditions. Models also show that sexual conflict can result in the evolution of monandry in an initially polyandrous species and in the evolution of random mating in a population initially exhibiting non-random mating.

Figure 1

A three-way tug-of-war (after Rice 1998).

Figure 2

A two-way tug-of-war between male offence and female reproductive tract and physiology.

Figure 3

The probabilities of mating and remating. (a) The probability of mating, ψ(u). (b) The probability of remating, ϕ(v).

Figure 4

The dynamics in Example 1. (a–c) Evolution towards the maximum female mating rate (‘males win’ scenario; condition (3.4) is satisfied). (d–f) Evolution towards an intermediate female mating rate (‘dynamic compromise between the sexes’ scenario; condition (3.4) is not satisfied). (a, d) The dynamics on the $(\overline{x},\overline{y})$ phase-plane. (b, e) The dynamics of $u=\overline{y}-\overline{x}$ in time. (c, f) The dynamics of the average mating rate ψ(u) in time. The trajectories for different initial conditions are shown.

Figure 5

A two-way tug-of-war between male defence and female reproductive tract and physiology.

Figure 6

The dynamics in Example 2. (a–c) Evolution towards zero female remating rate (‘males win’ scenario; condition (4.5a) is satisfied). (d–f) Evolution towards an intermediate female remating rate (‘dynamic compromise between the sexes’ scenario; condition (4.5b) is satisfied). (g–i) The outcome depends on initial conditions (condition (4.5c) is satisfied). (a, d, g) The dynamics on the $(\overline{x},\overline{y})$ phase-plane. (b, e, h) The dynamics of $v=\overline{z}-\overline{x}$ in time. (c, f, i) The dynamics of the average remating rate ϕ(v) in time. The trajectories for different initial conditions are shown.

Figure 7

The dynamics in the second model of the three-way conflict on a (ψ(u), ψ(v))-plane. (a, c) Evolution towards a state with random mating and complete suppression of remating (condition (5.13) is not satisfied). (b, d) Evolution towards a state with random mating and no remating suppression (condition (5.13) is satisfied). The equilibrium ψ(u)=ψ^*, ψ(v)=1 does not exist in (a) and (b) (condition (5.13) is not satisfied). The equilibrium ψ(u)=ψ^*, ψ(v)=1 exists in (c) and (d) (condition (5.13) is not satisfied). The trajectories for different initial conditions are shown.

Figure 8

The blow-up of the upper right corner of figure 7d. The trajectories eventually approach the state with ψ(u)=ψ(v)=1 (random mating, no remating suppression).