Interactions between nurse plants and parasitic beneficiaries: A theoretical approach to indirect facilitation.

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.