Fig. 1: Negotiation over parental care in repeated bouts.

A Two parents invest in a common set of offspring over a discrete number of time periods. At the start of each time period, each parent adjusts its provisioning rate in the new period to their partner’s provisioning rate in the previous period. B This adjustment is based on heritable behavioural reaction norms (or negotiation rules) that evolve over the generations. In our model, the reaction norms are described by logistic functions, which are determined by two heritable parameters: a parameter α determining the location of the inflexion point and a parameter β determining the slope at the inflexion point. For each sex, two example reaction norms are shown. The solid lines are increasing functions (\({{\rm{\beta }}} > 0\)), implying that the provisioning rate elicited by the partner increases with the partner’s provisioning rate in the previous time step. The dashed lines illustrate decreasing reaction norms (\({{\rm{\beta }}} < 0\)) where a higher provisioning rate by the partner elicits a lower provisioning rate in the next time period. C The negotiation process is illustrated by overlaying the solid reaction norms of a male and a female in a single graph (thereby swapping the axes of the female reaction norm in graph B). For any combination of provisioning rates in the previous time period, the light blue arrow indicates the response of the male, while the yellow arrow indicates the response of the female. D In the example shown (\({{\rm{\beta }}} > 0\) in both parents), the repeated mutual adjustment of parental provisioning rates converges monotonically to stable sex-specific provisioning rates. The figure was created in Adobe Illustrator and the vector images were taken from Adobe Stock.