When a nurse species facilitates the density of more than one species, strong indirect interactions can occur between the facilitated, or beneficiary, species, and these could lead to cascading interactive effects on community dynamics. In this context, negative effects of beneficiaries on the growth or reproduction of nurses are much more common than positive effects. This suggests beneficiaries frequently act as parasites of their nurses, and the consequences of this are largely unexplored. Our general aim is to analyze whether competition between parasitic beneficiaries can lead to indirect facilitation to nurse species and how this influences nurse-beneficiary systems. We explored potential outcomes of such reciprocal interactions in the general case of one facilitator and two facultative parasitic beneficiary species with different strategies for competing for space, one having a high carrying capacity but low maximum intrinsic growth rate (K-species), and the other having low carrying capacity but a higher intrinsic growth rate (r-species). These are defined in terms of the logistic equation, and reflect the abundances they can reach when growing alone. By considering a set of ordinary differential equations for the abundances of the nurse and the two parasitic beneficiaries in the mean-field approximation (where spatial correlations do not play a role), we first show analytically that coexistence of the three species is only possible when the r-species beneficiary is, at the same time, more harmful than the K-species and receives more benefit from the nurse. We then show that only the K-species can indirectly facilitate the nurse in such system. These are general, analytic results, independent of particular values of the parameters. We then explore these results using a 2-D lattice model informed by cushion plants in alpine ecosystems, and their interactions with beneficiaries with r and K strategies. Interesting spatial effects emerge in this case, such as a seeding effect: facilitation by the nurse increases beneficiary abundances also outside nurse patches. These in turn generate a negative feedback to the nurse, due to local competition for space near its edge. Spatial distribution effects are also crucial for relaxing the conditions for the survival of the r-species, allowing an r-strategist with weaker parasitic effects to indirectly facilitate the nurse through suppression of a more harmful K-species. Unexpectedly, this also has an indirect positive effect on the K species because of increased abundance of nurses. In the case of the r-species representing a ruderal invader, our lattice results would suggest that invaders have the potential to benefit both nurse and native beneficiary species via indirect facilitation. More generally, our results indicate that facilitation of more than one other species varying in competitive ability and which act as parasites on a nurse, can in turn promote indirect facilitation effects. This form of indirect facilitation has not been explicitly studied before, although it may create substantial conditionality in the outcomes of interactions among multiple species and the dynamics of nurse-beneficiary systems.

中文翻译：

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护理植物和寄生受益者之间的相互作用：间接促进的理论方法。

当一个保育物种促进一个以上物种的密度时，被促进或受益的物种之间可能会发生强烈的间接相互作用，这可能导致对群落动态的级联交互影响。在这种情况下，受益人对护士成长或繁殖的负面影响比正面影响要普遍得多。这表明受益人经常充当护士的寄生虫，其后果在很大程度上尚未得到探索。我们的总体目标是分析寄生受益人之间的竞争是否会导致对养育物种的间接促进，以及这如何影响保育受益系统。我们在一个促进者和两个兼性寄生受益物种具有不同的空间竞争策略的一般情况下探索了这种相互相互作用的潜在结果，一个具有高承载能力但低最大内在增长率（K 物种），另一个承载能力低但内在增长率较高（r-species）。这些是根据逻辑方程定义的，反映了它们单独生长时可以达到的丰度。通过考虑平均场近似中护士和两个寄生受益人丰度的一组常微分方程（其中空间相关性不起作用），我们首先分析表明，这三个物种的共存只有当同时，r 种受益人是 比 K 种更有害，并从护士那里获得更多好处。然后我们表明，只有 K 物种可以间接促进这种系统中的护士。这些是一般的分析结果，与参数的特定值无关。然后，我们使用由高山生态系统中垫层植物提供的二维晶格模型探索这些结果，以及它们与受益者与 r 和 K 策略的相互作用。在这种情况下出现了有趣的空间效应，例如播种效应：护士的促进也增加了护士补丁之外的受益人丰度。由于局部竞争边缘附近的空间，这些反过来会对护士产生负面反馈。空间分布效应对于放宽 r 物种的生存条件也至关重要，允许具有较弱寄生效应的 R 策略师通过抑制更有害的 K 物种来间接促进护士。出乎意料的是，由于护士数量增加，这也对 K 物种产生了间接的积极影响。在代表粗暴入侵者的 r 物种的情况下，我们的格子结果表明入侵者有可能通过间接促进使护士和本地受益物种受益。更一般地说，我们的结果表明，促进竞争能力不同并充当护士寄生虫的不止一种其他物种，反过来可以促进间接促进效应。这种形式的间接促进以前没有被明确研究过，