Disentangling nature, strength and stability issues in the characterization of population interactions

Many recent reviews discuss the adequacy of definitions and metrics for the strength of population interactions. However, the discussion on the beneficial or detrimental nature of interactions is clearly absent, or at the most, inadvertently merged into the strength debate. This deficiency is emerging with the increasing interest in theoretical studies of interactions that shift in their nature; e.g. associations that present a mixture of mutualistic and antagonistic aspects, such as pollination; or species with changes in role, such as mutualistic ants that predate on aphid partners. By exploring these models, major controversies are revealed underlying some traditional perspectives: the original Levins'community matrix reformulated into interaction and jacobian matrices, that is, interaction coefficients reinterpreted as partial derivatives, fail to recognize the ecological context of interactions. The 'effect of one species on the other' is not necessarily quantified by 'the effect of varying species densities'; and shifts in the signs of jacobian elements do not correspond to shifts in types of interaction but to stability properties. Thus, the generalised use of these approaches must be revised. On the other hand, the comparison of ultimate performances of populations when growing alone or in association, here referred to as the relative performance approach, conceptually represents the original meaning of the community matrix. This conception, although measured at population levels, is a reflection of properties at the individual level. This article inspects and discusses the formalities and ecological contexts of these approaches to characterization by means of known population interaction models: linear and non-linear, variable and non-variable; aiming to disentangle crucial conceptions that are usually mingled in the literature: the strength (magnitude) and the nature (detrimental or beneficial) of the interaction, which are sometimes used interchangeably, and the stability properties of the system, which have been misleadingly associated with the latter.