BACKGROUND According to the classical model of Macevicz and Oster, annual eusocial insects should show a clear dichotomous "bang-bang" strategy of resource allocation; colony fitness is maximised when a period of pure colony growth (exclusive production of workers) is followed by a single reproductive period characterised by the exclusive production of sexuals. However, in several species graded investment strategies with a simultaneous production of workers and sexuals have been observed. Such deviations from the "bang-bang" strategy are usually interpreted as an adaptive (bet-hedging) response to environmental fluctuations such as variation in season length or food availability. To generate predictions about the optimal investment pattern of insect colonies in fluctuating environments, we slightly modified Macevicz and Oster's classical model of annual colony dynamics and used a dynamic programming approach nested into a recurrence procedure for the solution of the stochastic optimal control problem. RESULTS 1) The optimal switching time between pure colony growth and the exclusive production of sexuals decreases with increasing environmental variance. 2) Yet, for reasonable levels of environmental fluctuations no deviation from the typical bang-bang strategy is predicted. 3) Model calculations for the halictid bee Lasioglossum malachurum reveal that bet-hedging is not likely to be the reason for the graded allocation into sexuals versus workers observed in this species. 4) When environmental variance reaches a critical level our model predicts an abrupt change from dichotomous behaviour to graded allocation strategies, but the transition between colony growth and production of sexuals is not necessarily monotonic. Both, the critical level of environmental variance as well as the characteristic pattern of resource allocation strongly depend on the type of function used to describe environmental fluctuations. CONCLUSION Up to now bet-hedging as an evolutionary response to variation in season length has been the main argument to explain field observations of graded resource allocation in annual eusocial insect species. However, our model shows that the effect of moderate fluctuations of environmental conditions does not select for deviation from the classical bang-bang strategy and that the evolution of graded allocation strategies can be triggered only by extreme fluctuations. Detailed quantitative observations on resource allocation in eusocial insects are needed to analyse the relevance of alternative explanations, e.g. logistic colony growth or reproductive conflict between queen and workers, for the evolution of graded allocation strategies.

中文翻译：

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年度真社会昆虫的适应性动态资源分配：环境变化不一定促进分级控制。

背景 根据 Macevicz 和 Oster 的经典模型，一年生真社会性昆虫在资源配置上应该表现出明显的二分性“bang-bang”策略；当纯菌落生长时期（工人的排他性生产）之后是一个以排他性繁殖为特征的生殖期时，群体适应度会最大化。然而，在几个物种中，已经观察到同时生产工人和性行为的分级投资策略。这种与“bang-bang”策略的偏差通常被解释为对环境波动（例如季节长度或食物供应的变化）的适应性（对冲）响应。为了预测波动环境中昆虫群体的最佳投资模式，我们稍微修改了 Macevicz 和 Oster 的年度群体动力学经典模型，并使用嵌套在递归过程中的动态规划方法来解决随机最优控制问题。结果 1) 纯菌落生长和有性繁殖之间的最佳转换时间随着环境变异的增加而减少。2) 然而，对于合理水平的环境波动，预计不会偏离典型的 bang-bang 策略。3) halictid 蜜蜂 Lasioglossum malachurum 的模型计算表明，在该物种中观察到的性别与工人的分级分配不太可能是赌注对冲的原因。4）当环境方差达到临界水平时，我们的模型预测从二分行为到分级分配策略的突然变化，但群体生长和有性繁殖之间的过渡不一定是单调的。环境方差的临界水平以及资源分配的特征模式都在很大程度上取决于用于描述环境波动的函数类型。结论 到目前为止，对冲作为对季节长度变化的进化反应一直是解释年度真社会性昆虫物种分级资源分配的实地观察的主要论据。然而，我们的模型表明，环境条件适度波动的影响并没有选择偏离经典的 bang-bang 策略，并且分级配置策略的演变只能由极端波动触发。