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Study site and experimental setup
Our experiment was conducted at the La Hampa field station of the Spanish National Research Council (CSIC) in Seville, Spain (37°16′58.8″ N, 6°03′58.4″ W), 72 m above sea level. The climate is Mediterranean, with mild, wet winters and hot, dry summers. Soils are loamy with pH = 7.74, C/N = 8.70 and organic matter = 1.16% (0–10-cm depth). Precipitation totaled 532 mm during the experiment (September 2014–August 2015), similar to the 50-y average. We used ten common annual plants, which naturally co-occur at the study site, for the experiment. These species cover a wide phylogenetic and functional range and include members of six of the most abundant families in the Mediterranean grasslands of southern Spain (Table 1). Seeds were provided by a local supplier (Semillas silvestres S.L.) from populations located near to our study site. Our experiments were located within an 800 m2 area, which had been previously cleared of all vegetation and which was fenced to prevent mammal herbivory. Landscape fabric was placed between plots to prevent growth of weeds.
Theoretical background for quantifying niche and fitness differences
Here we summarize the approach developed in ref. 38 to quantify the stabilizing effect of niche differences and average fitness differences between any pair of species. Both these measures are derived from mathematical models that capture the dynamics of competing annual plant populations with a seed bank19,39. This approach has been used in the past to accurately predict competitive outcomes between annual plant species38. Population growth is described as:

$$frac{{N_{i,t ,+, 1}}}{{N_{i,t}}},=,left( {1,-,g_i} right)s_i,+,frac{{lambda _ig_i}}{{1,+,alpha _{ii}g_iN_{i,t},+,{mathrm{{Sigma}}}_{j = 1}^{mathrm{S}}alpha _{ij}g_jN_{j,t}}},$$
(1)

Where ({textstyle{{N_{i,t + 1}} over {N_{i,t}}}}) is the per capita population growth rate, and Ni,t is the number of individuals (seeds) of species i before germination in the fall of year t. Changes in per capita growth rates depend on the sum of two terms. The first describes the proportion of seeds that do not germinate (1 − gi) but survive in the seed soil bank (si). The second term describes how much the per germinant fecundity, in the absence of competition (λi), is reduced by the germinated density of conspecific (giNi,t) and various heterospecific (left( {{mathrm{{Sigma}}}_{j = 1}^{mathrm{S}}g_jN_{j,t}} right)) neighbors. These neighbor densities are modified by the interaction coefficients describing the per capita effect of species j on species i (αij) and species i on itself (αii).
Following earlier studies14,38, we define niche differences (1 − ρ) for this model of population dynamics between competing species as:

$$1,-,rho,=,1,-,sqrt {frac{{alpha _{ij}}}{{alpha _{jj}}}frac{{alpha _{ji}}}{{alpha _{ii}}}} .$$
(2)

The stabilizing niche differences reflect the degree to which intraspecific competition exceeds interspecific competition. 1 − ρ is 1 when individuals only compete with conspecifics (i.e., there is no interspecific competition) and it is 0 when individuals compete equally with conspecifics and heterospecifics (i.e., intra and interspecific competition are equal). Niche differences between plant species can arise for instance from differences in light harvesting strategies29,37,38,39, or in soil resource use and shared mutualisms40.
The average fitness differences between a pair of competitors is ({textstyle{{kappa _j} over {kappa _i}}})38, and its expression is the following:

$$frac{{kappa _j}}{{kappa _i}},=,frac{{eta _j,-,1}}{{eta _i,-,1}}sqrt {frac{{alpha _{ij}}}{{alpha _{ji}}}frac{{alpha _{ii}}}{{alpha _{jj}}}} .$$
(3)

The species with the higher value of ({textstyle{{kappa _j} over {kappa _i}}}) (either species i or species j) is the competitive dominant, and in the absence of niche differences excludes the inferior competitor. This expression shows that ({textstyle{{kappa _j} over {kappa _i}}}) combines two fitness components, the “demographic ratio” (left( {{textstyle{{eta _j – 1} over {eta _i – 1}}}} right)) and the “competitive response ratio” (left( {sqrt {{textstyle{{alpha _{ij}} over {alpha _{ji}}}}{textstyle{{alpha _{ii}} over {alpha _{jj}}}}} } right)). The demographic ratio is a density independent term and describes the degree to which species j has higher annual seed production, per seed lost from the seed bank due to death or germination, than species i

$$eta _j,=,frac{{lambda _jg_j}}{{1,-,left( {1,-,g_j} right)s_j}}.$$

The competitive response ratio is a density-dependent term, which describes the degree to which species i is more sensitive to both intra and interspecific competition than species j. Note that the same interaction coefficients defining niche differences are also involved in describing the competitive response ratio, although their arrangement is different. Because of this interdependence, a change in interaction coefficients (( {alpha _{ji}^prime s} )) simultaneously changes both stabilizing niche differences and average fitness differences21.
With niche differences stabilizing coexistence and average fitness differences promoting competitive exclusion, the condition for coexistence (mutual invasibility) is expressed as14,38:

$$rho, More

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