Some social insects exhibit an exceptionally high degree of polyandry. Alternative hypotheses exist to explain the benefits of multiple mating through enhanced colony performance. This study critically extends theoretical analyses of the hypothesis that enhanced division of labour confers fitness benefits to the queen that are sufficient to explain the observed mating frequencies of social insects. The effects of widely varying numbers of tasks and matings were systematically investigated in two alternative computer simulation models. One model was based on tasks that have to be performed to maintain an optimal trait value, while the other model was based on tasks that only have to be sufficiently performed to exceed a minimum trait value to confer full fitness returns. Both model versions were evaluated assuming a broad and a narrow response threshold distribution. The results consistently suggest a beneficial effect of multiple mating on colony performance, albeit with quickly diminishing returns. An increasing number of tasks decreased performance of colonies with few patrilines but not of more genetically diverse colonies. Instead, a performance maximum was found for intermediate task numbers. The results from the two model versions and two response threshold distributions did not fundamentally differ, suggesting that the type of tasks and the breadth of response thresholds do not affect the benefit of multiple mating. In general, our results corroborate previous models that have evaluated simpler task/patriline scenarios. Furthermore, selection for an intermediate number of tasks is indicated that could constrain the degree of division of labour. We conclude that enhanced division of labour may have favoured the evolution of multiple mating but is insufficient to explain the extreme mating numbers observed in some social insects, even in complex task scenarios.

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We consider a quasilinear elliptic problem of the formwhere λ > 0 is a parameter, 1 < p < 2 and Ω is a strictly convex bounded domain in ℝN, N > p, with C2 boundary ∂Ω. The nonlinearity f : [0, ∞) → ℝ is a continuous function that is semipositone (f(0) < 0) and p-superlinear at infinity. Using degree theory, combined with a rescaling argument and uniform L∞a priori bound, we establish the existence of a positive solution for λ small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at λ = 0. We also extend our study to systems.

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