Fig. 4: The per capita population growth rate and the spatial invasion criterion.
From: Latitudinal scaling of aggregation with abundance and coexistence in forests
a, The scaled per capita population growth rate \({\widetilde{\lambda }}_{f}({N}_{f})/{r}_{f}\) for five example species of different forests plotted over abundance Nf. We scaled \({\widetilde{\lambda }}_{f}({N}_{f})\) with the reproduction rate rf to obtain a quantity that is comparable among forest plots. The species include Castanopsis acuminatissima of the MST plot (bf = −0.3), Ficus langkokensis of the Xishuangbanna plot (bf = −0.52), Carya tomentosa of the Tyson Research Center plot (bf = −0.75), Ostrya virginiana of the Wabikon plot (bf = −0.92) and Maackia amurensis of the Changbaishan plot (bf = −1.08). We also show the theoretical values of \({\widetilde{\lambda }}_{f}({N}_{f})/{r}_{f}\) for bf = 0 (black) and −1 (grey). b, Same as a, but for the averages over all focal species in tropical, subtropical and temperate forests. c, Trade-off between the exponent bf of the aggregation–abundance relationship (x axis) and the maximal risk factor ρf,max (equation (19)) that just satisfies the spatial invasion criterion (equations (2) and (18)): temperate forests (smaller bf) show smaller risk factors ρf than tropical forests (larger value of bf). The lines show ρf,max for a small scaled growth rate δ = 0.0035 (see equation (19)) and example ratios of Ns/Nf* = 5/50 (red) and Ns/Nf* = 5/5,000 (black), where Ns (=5) is the small invasion abundance and Nf* (=50 and 5,000). The circles show for each species the risk factor ρf and the cyan discs mark the 33 out of 720 species that do not satisfy the criterion (19 species with ρf > 350 are not visible). The data are from scenario 1 (that is, no niche differences, no immigration and observed equilibrium abundances) (Extended Data Table 1).
