Statistical mechanics of ecological systems: Neutral theory and beyond

The simplest theories often have much merit and many limitations, and, in this vein, the value of neutral theory (NT) of biodiversity has been the subject of much debate over the past 15 years. NT was proposed at the turn of the century by Stephen Hubbell to explain several patterns observed in the organization of ecosystems. Among ecologists, it had a polarizing effect: There were a few ecologists who were enthusiastic, and there were a larger number who firmly opposed it. Physicists and mathematicians, instead, welcomed the theory with excitement. Indeed, NT spawned several theoretical studies that attempted to explain empirical data and predicted trends of quantities that had not yet been studied. While there are a few reviews of NT oriented toward ecologists, the goal here is to review the quantitative aspects of NT and its extensions for physicists who are interested in learning what NT is, what its successes are, and what important problems remain unresolved. Furthermore, this review could also be of interest to theoretical ecologists because many potentially interesting results are buried in the vast NT literature. It is proposed to make these more accessible by extracting them and presenting them in a logical fashion. The focus of this review is broader than NT: new, more recent approaches for studying ecological systems and how one might introduce realistic non-neutral models are also discussed.

Figure 1

Visual scheme of two important macroecological patterns (see Table 1): RSA and SAR. The functional shape of the RSA depends on the spatial scale considered, while the SAR generally displays a triphasic behavior (see Sec. 4). There is a growing appreciation that the various descriptors of biodiversity are intrinsically interrelated, and substantial efforts have been devoted to understand the links between them. From [10].

Figure 2

Tree relative species-abundance data from the Barro Colorado Island (BCI), Yasuni, Pasoh, Lambir, Korup, and Sinharaja plots for trees that are 10 cm in stem diameter at breast height. The frequency distributions are plotted using Preston’s binning method as described by [186] and the bars are the observed number of species binned into $\log (2)$ abundance categories. The first histogram bar represents $⟨{\varphi }_1⟩/2$, the second bar $⟨{\varphi }_1⟩/2+⟨{\varphi }_2⟩/2$, the third bar $⟨{\varphi }_2⟩/2+⟨{\varphi }_3⟩+⟨{\varphi }_4⟩/2$, the fourth bar $⟨{\varphi }_4⟩/2+⟨{\varphi }_5⟩+⟨{\varphi }_6⟩+⟨{\varphi }_7⟩+⟨{\varphi }_8⟩/2$, and so on. Here $⟨{\varphi }_n⟩$ is the number of species with an abundance $n$. As examples, we show the fits of a density-dependent symmetric model (black line), which will be studied in greater detail in Sec. 2. From [187].

Figure 3

Different neutral models of community ecology. In all these models, $n_i$ represents the abundance of the species $i$ in the community, and $J_L$ and $J_M$ the total abundance in the local and metacommunities, respectively. (a) Hubbell’s zero-sum neutral model ([85]). In the local community, each death is immediately followed by a birth or an immigration event. Speciation (or, equivalently, immigration) enables diversity to be maintained in the metacommunity. (b) Local community with immigrants from a metacommunity ([178]). The local community now interacts with the metacommunity through a migration process ($m$). (c) Coalescent-type approach ([57]), where community members are traced back to the ancestors that once immigrated into the community. (d) Joint RSA of many local communities ([187]). The whole metacommunity RSA distribution is built by considering the joint RSA distributions of multiple local communities.

Figure 4

STD for the interval 1990–1995 in the BCI forest. The main panel shows the results for individuals of more than 10 cm dbh, and the inset the results for individuals of more than 1 cm dbh (Center for Tropical Forest Science website). The black line represents the analytical solution given by Eq. (46). From [13].

Figure 5

Schematic representation of persistence time (or lifetime) of a species $\tau$ and survival times ${\tau }_s$, defined as the time to local extinction of a species randomly sampled among the observed assemblages at a certain time $T$. From [169].

Figure 6

Comparison between persistence empirical distributions for North American breeding birds, Kansas grasslands, New Jersey BSS forest, and an estuarine fish community and the corresponding theoretical species persistence-times pdfs. The circles and solid lines show the observational distributions and fits, respectively. The finiteness of the time window $\mathrm{\Delta }{T}_w$ imposes a cutoff in the maximum observable persistence time and thus, only lifetimes where $\tau <\mathrm{\Delta }{T}_w$ have been considered and the theoretical predictions have been adjusted appropriately (see Appendix pp2). Adapted from [167].

Figure 7

Triphasic shape of the species-area relationship. The left panel shows the three behaviors on different scales. At a local scale the relationship is linear, becoming a power-law relationship at the regional scale and returning to linear at very large intercontinental scales. The right panel shows empirical data of species diversity at the regional scale (from the BCI forest). From [10].

Figure 8

Microscopic moves of the multispecies voter model in a 2D lattice and with a nonregular network ([29]). In the 2D structure, each site is occupied by one and only one individual, whose color represents the species it belongs to. At each time step, one random individual is replaced by a daughter of one of its neighbors with probability $1-\nu$ (black lines). The probability that a speciation event occurs is $\nu$, wherein the individual is replaced by an individual of a new species (red lines). In the case of a nonregular network (right panel), the number of neighbors depends on the site considered.