Evidence of scale-free clusters of vegetation in tropical rainforests

Tropical rainforests exhibit a rich repertoire of spatial patterns emerging from the intricate relationship between the microscopic interaction between species. In particular, the distribution of vegetation clusters can shed much light on the underlying process that regulates the ecosystem. Analyzing the distribution of vegetation clusters at different resolution scales, we show the first robust evidence of scale-invariant clusters of vegetation, suggesting the coexistence of multiple intertwined scales in the collective dynamics of tropical rainforests. We use field data and computational simulations to confirm our hypothesis, proposing a predictor that could be particularly interesting to monitor the ecological resilience of the world's “green lungs.”

Figure 1

Sketch of the clustering process of a fixed number of points for different interaction distances. The upper insets illustrate the clusters at criticality for different homogeneous and inhomogeneous Poisson point processes (see [12] for further examples).

Figure 2

Barro Colorado Island. $P_{\infty }$ as a function of the distance parameter $\widehat{r}$ for (a) the community level and (c) the most abundant species: $Hybanthusprunifolius$. The solid green line shows the average phase transition over the eight censuses from 1985 to 2005, while the dashed gray lines represent individual censuses. The colored dots represent the position where the cluster size distributions are shown. Lower inset: Averaged system susceptibility. Upper inset: Spatial cluster distribution at ${\chi }_{\max }$. $H.prunifolius$ exhibits a nontrivial region with high susceptibility, ranging into $\widehat{r}\in (2,4)$. (b)–(d) Cluster size distribution averaged over all censuses for different values of $\widehat{r}$ (see legend). The community level shows a usual percolation transition with $\tau \simeq 2.0\pm 0.1$ (dashed lines are guides to the eye indicating this exponent). A broad (critical) region with variable exponents exists for the most abundant species. The vertical lines represent the total system size (i.e., the number of individuals) for each specific case.

Figure 3

SENM. $P_{\infty }$ for the most abundant species as a function of the distance parameter, $\widehat{r}$, for (a) the random regime ($\nu =5·{10}^{-6},K=25,L=2^9$) and (c) the scale-invariant regime ($\nu =5·{10}^{-6},K=5,L=2^9$). The solid green line shows the average phase transition over the ${10}^3$ independent realizations, while the dashed gray lines stand for the individual ones. The colored dots represent the position where the cluster size distributions are shown. Lower inset: Averaged system susceptibility over ${10}^3$ realizations for $L=2^9,2^{10}$, and $2^{11}$ (black, red, and blue lines, respectively). Note that $\chi$ diverges as the system size increases. For each lattice side we change $\nu$ to maintain fixed $\nu L^2$, selecting it for $L=2^9$. Upper inset: Spatial cluster distribution at ${\chi }_{\max }$. The scale-invariant regime exhibits a nontrivial extended region with divergent susceptibility. (b) and (d) $P\left(S\right)$ averaged over ${10}^3$ realizations for different values of $\widehat{r}$ (see legend). The random regime exhibits a usual percolation phase transition, with $\tau \sim 2.0\pm 0.1$ (black dashed line), whereas the scale-invariant regime shows an intermediate region with variable exponents. Now, the dashed lines are guides for the eye for $\tau \sim 2.0$ and $\tau \sim 2.5$, respectively.

Figure 4

Fractal nature of BCI. (a) Mass variance, ${\sigma }^2\left(\ell \right)$, with different radius $\ell$ for $H.prunifolius$ (blue line). Note the almost constant value, a sign of incipient criticality in the system. For the sake of comparison, the red line shows the Poisson case, with the same number of points and whose theoretical value scales as ${\sigma }^2\left(\ell \right)\propto {\ell }^{-2}$ (black dashed line). (b) Local slope for $C\left(\ell \right)$ (see inset) at the community level (orange line) and for $H.prunifolius$ (blue line). We obtain $D=1.98\pm 0.01$ for the community level and $D=1.86\pm 0.04$ for $H.prunifolius$. Gray dashed lines show the MNN distance for $H.prunifolius$ in both figures. All curves have been averaged for the eighth available censuses (the shaded region shows a sigma confidence level).

Figure 5

Species fluctuations. (a) Fluctuability index versus census number for different selected species in BCI (see legend). (b) Rate of change versus census number for the same selected species.