![The per capita population growth rate and the spatial invasion criterion
a, The scaled per capita population growth rate λ̃f(Nf)/rf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{\lambda }}_{f}({N}_{f})/{r}_{f}$$\end{document} for five example species of different forests plotted over abundance Nf. We scaled λ̃f(Nf)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{\lambda }}_{f}({N}_{f})$$\end{document} 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 λ̃f(Nf)/rf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{\lambda }}_{f}({N}_{f})/{r}_{f}$$\end{document} 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).
Source data](https://www.researchgate.net/publication/389358234/figure/fig4/AS:11431281312159848@1740628937407/The-per-capita-population-growth-rate-and-the-spatial-invasion-criterion-a-The-scaled_Q320.jpg)
February 2025
·
524 Reads
Nature
The search for simple principles that underlie the spatial structure and dynamics of plant communities is a long-standing challenge in ecology1, 2, 3, 4, 5–6. In particular, the relationship between species coexistence and the spatial distribution of plants is challenging to resolve in species-rich communities7, 8–9. Here we present a comprehensive analysis of the spatial patterns of 720 tree species in 21 large forest plots and their consequences for species coexistence. We show that species with low abundance tend to be more spatially aggregated than more abundant species. Moreover, there is a latitudinal gradient in the strength of this negative aggregation–abundance relationship that increases from tropical to temperate forests. We suggest, in line with recent work¹⁰, that latitudinal gradients in animal seed dispersal¹¹ and mycorrhizal associations12, 13–14 may jointly generate this pattern. By integrating the observed spatial patterns into population models⁸, we derive the conditions under which species can invade from low abundance in terms of spatial patterns, demography, niche overlap and immigration. Evaluation of the spatial-invasion condition for the 720 tree species analysed suggests that temperate and tropical forests both meet the invasion criterion to a similar extent but through contrasting strategies conditioned by their spatial patterns. Our approach opens up new avenues for the integration of observed spatial patterns into ecological theory and underscores the need to understand the interaction among spatial patterns at the neighbourhood scale and multiple ecological processes in greater detail.
