Species abundance correlations carry limited information about microbial network interactions

Unraveling the network of interactions in ecological communities is a daunting task. Common methods to infer interspecific interactions from cross-sectional data are based on co-occurrence measures. For instance, interactions in the human microbiome are often inferred from correlations between the abundances of bacterial phylogenetic groups across subjects. We tested whether such correlation-based methods are indeed reliable for inferring interaction networks. For this purpose, we simulated bacterial communities by means of the generalized Lotka-Volterra model, with variation in model parameters representing variability among hosts. Our results show that correlations can be indicative for presence of bacterial interactions, but only when measurement noise is low relative to the variation in interaction strengths between hosts. Indication of interaction was affected by type of interaction network, process noise and sampling under non-equilibrium conditions. The sign of a correlation mostly coincided with the nature of the strongest pairwise interaction, but this is not necessarily the case. For instance, under rare conditions of identical interaction strength, we found that competitive and exploitative interactions can result in positive as well as negative correlations. Thus, cross-sectional abundance data carry limited information on specific interaction types. Correlations in abundance may hint at interactions but require independent validation.

The joint equilibrium abundance of both species ( " ! , " " ) is determined by ! ( " ) = " ( " ). Equation (Eq. 1) shows that species 1 grows to its carrying capacity ! in the absence of interspecific interactions, i.e. if α12 = 0. Likewise, α12 > 0 allows species 1 to grow to higher abundance in the presence of species 2 than determined by its own carrying capacity, whereas α12 < 0 leads to a reduced abundance of species 1 in the presence of species 2.
Similar relations hold for the abundance of species 2 in presence of species 1, depending on α21. From Equation (Eq. 2), it can also be derived that " ! > 0 is only compatible with " " being above its carrying capacity " if at the same time α21 > 0, whereas " " being below " requires α21 < 0. Joint inspection of equations (Eq. 1) and (Eq. 2) also establishes the following, more subtle, conditions for co-existence: 1. If α12 > 0 and α21 > 0, i.e., in case of mutualism, " has a negative intercept in the Cartesian ( " , ! ) coordinate system (Figure A (panel A) in S1 Text). As both functions have a positive slope in this situation, and ! always has a positive intercept, " must have a stronger slope than ! for both to intersect in the positive Eq. 1 Eq. 2 quadrant. This boils down to ! # !" $ ! > !" ! , or equivalently α21α12 < α11α22, as % = − ! # ## by definition. This means that the product of interspecific mutualism needs to be smaller than the product of intraspecific competition for both species to co-exist, otherwise there is no control of population growth.
2. If α12 < 0 and α21 < 0, i.e., in case of competition, both functions have positive intercept and negative slope ( Figure A (panel B) in S1 Text). Intersection in the positive quadrant requires the function with the larger intercept to intersect the abscissa, i.e., the " axis where ! = 0, at a smaller value than the function with the smaller intercept. Thus, this requires | "! | > ! $ " , with " having the larger intercept. In the first instance, interspecific competition is stronger than intraspecific competition, whereas in the second instance, interspecific competition is less strong than intraspecific competition. It turns out that only the last of these conditions yields a stable equilibrium, meaning that the abundances of both species return to equilibrium after small displacements.
3. If α12 < 0 and α21 > 0, i.e., in case of exploitation of species 1 by species 2, ! has a positive intercept and negative slope, whereas " still has a negative intercept and positive slope (Figure A (panel C) in S1 Text). Intersection in the positive quadrant requires ! to intersect the abscissa at a larger value than " , the point where " intersects the abscissa. The condition for co-existence thus becomes | !" | < ! $ ! , or equivalently α12 < α22, meaning that the parasite should exert stronger inhibitory effect on growth of oneself than on that of the exploited species.
4. Conversely, in case of exploitation of species 2 by species 1, i.e., if α12 > 0 and α21 < 0, both ! and " have a positive intercept, but ! now has a positive slope whereas " has a negative slope ( Figure A (panel D) in S1 Text). Intersection in the positive quadrant then requires ! to have a smaller intercept than " . The condition for coexistence thus becomes | "! | < ! $ " , or equivalently α21 < α11, again meaning that the parasite should exert stronger inhibitory effect on growth of oneself than on that of the exploited species.
The additional requirement for stable co-existence is that the two-species system should be locally stable around the equilibria ( " ! , " " ), which can be formalized in terms of the Jacobian matrix of the Lotka-Volterra model evaluated at ( " ! , " " ). This amounts to determining trace and determinant of the matrix of the partial derivatives of the growth equations regarding either species, i.e., It can be verified that the conditions for co-existence stated under mutualism and exploitative interactions yield equilibria that are locally stable, just as the last of the conditions under competition. We will not derive these conditions here, as these are covered by textbooks on theoretical ecology [1]. In summary, the two-species Lotka-Volterra model with self-limitation has the following possibilities for stable co-existence ( Table A in S1 Text).
The condition for stable co-existence of competitors requires both species to have less effect on the growth of other species than on oneself. In case of an unstable equilibrium, either species will eventually outcompete the other; the species with initial advantage will drive the other species to extinction, a condition referred to as competitive exclusion [2,3]. This will occur, for instance, when each species produces a substance which is toxic to the other species but relatively harmless to itself. Table A. Conditions for stable co-existence in the two-species Lotka-Volterra model.

Type of interaction Condition Outcome
Mutualism Exploitative interaction type 2