Competition hierarchy, transitivity and additivity: investigating the effect of fertilisation on plant–plant interactions using three common bryophytes

Abstract

The way competition structures plant communities has been investigated intensely over many decades. Dominance structures due to competitive hierarchies, with consequences for species richness, have not received as much experimental attention, since their manipulation is a large logistic undertaking. Here the data from a model system based on bryophytes are presented to investigate competition structure in a three-species system. Grown in monocultures, pairwise and three-species mixtures under no and high nitrogen supply, the three moss species responded strongly to treatment conditions. One of them suffered from nitrogen fertilisation and hence performed better in mixtures, where the dominant species provided physical shelter from apparently toxic nitrogen spray. Accordingly, no linear competitive hierarchy emerged and qualitative transitivity remains restricted to the unfertilised treatments. Faciliation also affected other properties of the competition structure. The reciprocity of competition effects could not be observed. Moreover, the performances in three-species mixtures were not well predictable from the knowledge of monocultures and pairwise mixtures because competitive effects were not additive. This had implications for community stability at equilibrium: all two-species systems were stable, both fertilised and unfertilised, while the three-species system was only stable when fertilised. This stability under fertilisation has probably to do with the facilitative effect of the two dominant species on the third. In this experiment, little support for commonly held ideas was found about the way competition in plant communities is structured. On the other hand, this study shows that moss communities are ideal model systems to test predictions of theoretical models concerning properties and consequences of competition in plant communities.

Appendix

Deriving the equilibrium solution for Lotka–Volterra-type competition between two species

Starting with a two species LV-model of species A and B, with growth rate r, carrying capacity K, competition coefficient α and performance measure w (e.g. biomass, RGR or alike):

$$ {\hbox{d}}w_{{\hbox{A}}{\hbox{B}}} {\hbox{/d}}t{\hbox{ = }}r_{\hbox{A}} w_{{\hbox{A}}{\hbox{B}}} {\hbox{/}}K_{\hbox{A}} {\hbox{(}}K_{\hbox{A}} -w_{{\hbox{AB}}} - \alpha_{{\hbox{AB}}} w_{{\hbox{B}}{\hbox{A}}} {\hbox{)}} $$

$$ {\hbox{d}}w_{{\hbox{B}}{\hbox{A}}} {\hbox{/d}}t{\hbox{ = }}r_{\hbox{B}} w_{{\hbox{B}}{\hbox{A}}} {\hbox{/}}K_{\hbox{B}} {\hbox{(}}K_{\hbox{B}} - w_{{\hbox{BA}}} -\alpha_{{\hbox{BA}}} w_{{\hbox{A}}{\hbox{B}}} {\hbox{)}}. $$

At equilibrium dw/dt = 0. As r, w and K > 0, it follows that

$$ (K_{\hbox{A}} - w_{{\hbox{A}}{\hbox{B}}} - {\hbox{ $ \alpha $ }}_{{\hbox{AB}}} w_{{\hbox{B}}{\hbox{A}}} ) = 0\quad {\hbox{and}}\quad (K_{\hbox{B}} - w_{{\hbox{B}}{\hbox{A}}} - {\hbox{ $ \alpha $ }}_{{\hbox{BA}}} w_{{\hbox{A}}{\hbox{B}}} ) = 0, $$

hence α_AB = (K _A − w _AB)/w _BA and α_BA = (K _B − w _BA)/w _AB

Relating α_ij from the Lotka–Volterra equation to ɛ_ij from Freckleton and Watkinson (2001b)

Freckleton and Watkinson (2001b) start with the hyperbolic growth model, which is based on the number of individuals as performance measure. It cannot be directly applied to biomass, as it assumes a trade-off between individual biomass and density. In this study I have no information about the number of individuals in the trays and hence again use the Lotka–Volterra formulation:

$$ {\hbox{d}}w_{{\hbox{A}}{\hbox{B}}} {\hbox{/d}}t{\hbox{ = }}r_{\hbox{A}} w_{{\hbox{A}}{\hbox{B}}} {\hbox{/}}K_{\hbox{A}} {\hbox{(}}K_{\hbox{A}} - w_{{\hbox{AB}}} - {\hbox{ $ \alpha $ }}_{{\hbox{AB}}} w_{{\hbox{B}}{\hbox{A}}} {\hbox{)}}.$$

However, now I introduce α_ii to indicate intraspecific competition:

$$ {\hbox{d}}w_{{\hbox{A}}{\hbox{B}}} {\hbox{/d}}t{\hbox{ = }}r_{\hbox{A}} w_{{\hbox{A}}{\hbox{B}}} {\hbox{/}}K_{\hbox{A}} {\hbox{(}}K_{\hbox{A}} -{\hbox{ $ \alpha $ }}_{{\hbox{AA}}} w_{{\hbox{AB}}}- {\hbox{ $ \alpha $ }}_{{\hbox{AB}}} w_{{\hbox{B}}{\hbox{A}}} {\hbox{)}}.$$

In analogy to Freckleton and Watkinson (2001b), I transform:

$$ {\hbox{d}}w_{{\hbox{A}}{\hbox{B}}} {\hbox{/d}}t{\hbox{ = }}r_{\hbox{A}} w_{{\hbox{A}}{\hbox{B}}} {\hbox{/}}K_{\hbox{A}} {\hbox{(}}K_{\hbox{A}} -{\hbox{ $ \alpha $ }}_{{\hbox{AA}}} (w_{{\hbox{AB}}}- {\hbox{ $ \alpha $ }}_{{\hbox{AB}}} /{\hbox{ $ \alpha $ }}_{{\hbox{AA}}} w_{{\hbox{B}}{\hbox{A}}} {\hbox{))}}$$

The equivance coefficient ε_ij is now defined as the ratio of α_ij and α_ii. Since I have no information on the strength of intraspecific competition, I set α_ii = 1 and hence ε_ij = α_ij. Both in the analysis of Freckleton and Watkinson (2001b) and in this study, ε_AB represents the competitive effect of species B on species A in equivalents of A. If, for example, adding 1 g of species A would lead to a reduction in final biomass of A by 0.5 g, while adding 1 g of species B would lead to a 2 g reduction, ε_AB = 4, as 1 g of B has the same effect as 4 g of A.

For perfect transitivity of a competitive hierarchy, Freckleton and Watkinson (2001b) show that ε_ik = ε_ij · ε_jk. In this study, I use competitive coefficients α_ij instead.

Estimating competition coefficients under the assumption of perfect transitivity

The competition coefficient α_AB is defined as α_AB = (K _A − w _AB)/w _BA (see above).

In analogy: α_BC = (K _B − w _BC)/w _CB.

Perfect transitivity assumes that α_AC = α_AB · α_BC, hence:

$$ {\hbox{ $ \alpha $ }}_{{\hbox{AC}}} = (K_{\hbox{A}} - w_{{\hbox{A}}{\hbox{B}}} )/w_{{\hbox{B}}{\hbox{A}}} (K_{\hbox{B}} - w_{{\hbox{B}}{\hbox{C}}} )/w_{{\hbox{C}}{\hbox{B}}}.$$

When calculating the estimated α_AC along this formula, errors in all six variables propagate into the estimate of α_AC. There are 5⁶ = 15625 possible combinations of values from this study (5 replicates for the 6 variables). I used the mean and standard error (standard deviation divided by the square root of 5) of these 5⁶ values in Fig. 4.

In contrast, the observed α_AC depends only on three variables (K _A, w _AC and w _CA) and can be determined with greater accuracy (5³ = 125 values).