Fig 1.
Conceptual models of biparental inbreeding.
Females (F1 and F2) produce n offspring. The focal male (M1) is related by rM1,F1 to the focal female (F1). The fitness of M1 and F1’s n offspring is decreased by inbreeding depression δM1,F1. Whether or not M1 inbreeds or avoids inbreeding with F1 does not affect his opportunity to outbreed with an unrelated female (F2). In Parker’s [24, 25] model (A), if M1 avoids inbreeding, then F1 is assumed to outbreed with unrelated male (M2). In our extended model (B), if M1 avoids inbreeding, F1’s alternative mate M2 may be related to M1 (rM1,M2) or F1 (rM2,F1). If rM2,F1 > 0, then the fitness of M2 and F1’s offspring will be decreased by inbreeding depression (δM2,F1).
Fig 2.
Zones of parameter space in which inbreeding versus avoiding inbreeding is predicted to increase male and female inclusive fitness when all other potential mates are unrelated.
The x-axis shows the relatedness (rM1,F1) between two focal potential mates, male M1 and female F1, where rM1,F1 = 0 equates to outbreeding, and rM1,F1 = 1 equates to self-fertilisation. The y-axis shows the magnitude of inbreeding depression (δM1,F1) below which inbreeding is beneficial for each sex. Areas where neither sex, both sexes, and males only benefit from inbreeding are shown in white, grey, and black, respectively. The intersections between black and white areas and black and grey areas respectively demarcate the thresholds below which M1 and F1 benefit by inbreeding [24, 25].
Fig 3.
Three illustrative scenarios of biparental inbreeding in which the inclusive fitness benefits to a focal male M1 and female F1 depend on their relatedness to the female’s alternative mate M2.
F1 produces n offspring, whose fitness is reduced by inbreeding depression δM1,F1 or δM2,F1 if she mates with M1 or M2, respectively. In scenario 1 (A and B), M1, F1, and M2 are all equally related such that rM1,F1 = rM1,M2 = rM2,F1 = r. In scenario 2 (C and D), M1 and F1 are half-siblings, M1 and M2 are first cousins, and M2 and F1 are unrelated. In scenario 3 (E and F), M1 and F1 are half-siblings, M1 and F1 are first cousins, and M1 and M2 are unrelated.
Fig 4.
Zones of parameter space in which inbreeding versus inbreeding avoidance is predicted to increase male and female inclusive fitness given varying kinship between M1, F1, and M2.
Inbreeding depression thresholds (y-axis shown on a natural log scale) illustrate the values below which M1 and F1 have higher inclusive fitness by inbreeding instead of avoiding inbreeding. If M1 and F1 do not inbreed, F1 is assumed to breed with M2, who may or not be related to M1 or F1. The kinship between M1 and F1 (fM1,F1) increases along the x-axis of all plots. fM2,F1 and fM1,M2 increase through 0, 0.0625, and 0.125 across left to right columns and top to bottom rows, respectively. Areas where neither sex, both sexes, and males only benefit from inbreeding are shown in white, grey, and black, respectively. Negative threshold values are mathematically possible for some parameter combinations, but are biologically unrealistic because they require that offspring fitness increases monotonically with inbreeding. Where negative thresholds would be required, inbreeding is therefore assumed to never be beneficial. For simplicity, these examples assume focal individuals are outbred.
Table 1.
A general payoff matrix for male relatives (M1, M2) for either inbreeding or avoiding inbreeding with a female (F1) of equally close relatedness (r).