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We thank the Bren School of Environment Science and Management of the University of California Santa Barbara for support leading to the initial publication. This work was also supported by a grant overseen by the French ‘Programme Investissement d’Avenir’ as part of the ‘Make Our Planet Great Again’ programme (reference: 17-MPGA-0004) and by a National Science Foundation grant (LTER-1831944). More

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Target species and basic assumptions
We developed and analyzed spatially explicit models that describe range expansion of an invader species’ population that consists of many sub-populations. Invasive species often expand their range by stratified dispersal using human-mediated long-distance dispersal in addition to local expansion of sub-populations20. Extending a model rigorously analyzed by Takahashi et al.31, which explicitly involves (1) short-distance dispersal that expands the area of current sub-populations, and (2) long-distance dispersal that establishes new sub-populations beyond existing sub-populations (Fig. 4), we consider spatially inhomogeneous factors that influence on the long-distance dispersal (e.g., vectors’ distribution). The parameters and functions used in these models are listed in Table 1.
Figure 4

A schematic of short- and long-distance dispersal modes. Human activities may influence the long-distance dispersal by: (1) changing the number of propagules starting the long-distance dispersal ((R cdot varphi (x,y))), and (2) introducing biases in their spatial locations ((psi (x,y))). By compositing these short- and long-distance dispersals, we predict population establishment at the next time step.

Full size image

Table 1 Symbols and their default values.
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We estimated the population size of the invader species as the area covered by any of these sub-populations in the study area. This simple measure of the population size was based on our assumption that the inside of a sub-population is homogeneous and these sub-populations vary only in their positions, sizes, and shapes. This assumption may oversimplify spatial architectures of the sub-populations, because even clonal colonies of perennial plants often have concentric structures that can influence reproductive output40; but this simplification is applicable when the reproductive rate of the species is high enough to reach a constant carrying capacity quickly.
Short- and long-distance dispersals
We considered stratified dispersals of the invader species. Note that we explicitly considered invader species’ dispersal but their vectors’ movement (e.g., human traffics) was included only implicitly. The short-distance dispersal expands the species’ range only by a constant velocity7. Therefore, we modeled the short-distance dispersal as radial expansion of these sub-populations with a constant speed g. Meanwhile, long-distance dispersal introduces new sub-populations into the population out of its parent sub-population. Empirical observations showed that long-distance dispersal mediated by human activities introduces a new sub-population to an area at a long distance from its source population, e.g., vehicles can move plant seeds for more than hundreds of kilometers41. Long-distance dispersal diminishes the influence of the source location on the destination of a dispersal event, so as a simplifying approximation we assumed that the destination of the long-distance dispersal is independent of the source location of a sub-population.
The assumption of source location independence of the dispersal destination allows us to describe a process of long-distance dispersal by two functions on (S): (1) a function (varphi (x,y)) describing spatial variation in disperser production rates, and (2) a function (psi (x,y)) describing the probability that a coordinate ((x,y)) is selected as a destination of the disperser. Following the notation of Jongejans et al.18, we call (varphi (x,y)) and (psi (x,y)) the source and destination functions, respectively. Note that we define the spatial average of the source function to be one (i.e., ({{int_{S} {varphi (x,y),dxdy} } mathord{left/ {vphantom {{int_{S} {varphi (x,y),dxdy} } {|S|}}} right. kern-nulldelimiterspace} {|S|}} = 1)). We define R as the regional average of the disperser production rate per unit area (a spatially homogeneous component of the disperser production rate). Using this formulation, we calculate an expected disperser production rate of a given area by integrating (Rvarphi (x,y)) over the area. The spatial integration of the destination function over the study area is one because we assume the population of the invader species will reach full carrying capacity.
We have defined a population of the invader species as a spatial union of all sub-populations in the study area because we assume sub-populations are homogeneous. Within the study area, (rho_{t} (x,y)) is 1 if the coordinates ((x,y)) are within at least one of the sub-populations of the invader species at time (t) (and otherwise 0). The integration of (Rvarphi (x,y)) is the expected disperser production rate for a given (rho_{t} (x,y)), and the integration (Rint_{S} {rho_{t} (x,y)varphi (x,y),dxdy}) is the expected total production of dispersers from a population in the t-th time step. Thus, assuming that the total number of dispersers that start long-distance dispersal at time (t) (denoted by (n_{{{text{d,}}t}})) follow a Poisson distribution, we can write a probability that the population produces k dispersers in the t-th time, Eq. (1).

$$ Pr [n_{{{text{d}},t}} = k] = frac{{lambda^{k} e^{ – lambda } }}{k!}{, }lambda = Rint_{S} {rho_{t} (x,y)varphi (x,y),dxdy} . $$
(1)

The destination of the disperser is determined by the destination function (psi (x,y)), which determines probabilities of ending a dispersal event at ((x,y)), which results in a new sub-population being established at that location. In total, the spatial distribution of new sub-populations introduced by long-distance dispersal follows a Poisson point process of which intensities are given as (psi (x,y)Rint_{S} {rho_{t} (x,y)varphi (x,y),dxdy}).
Three model types
We compared three different types of models by varying interactions between the species’ long-distance dispersal and human activities. These included: (1) a source-mediated-dispersal model assuming that the source function (varphi (x,y)) varies spatially by human activity while the destination function (psi (x,y)) is uniform, (2) a destination-mediated-dispersal model assuming that the destination function varies spatially while the source function is uniform, and (3) a full model assuming that both source and destination functions vary spatially.
Let (h(x,y)) be a function representing intensity of human activities at coordinates ((x,y)). Without loss of generality, we can assume that the function (h(x,y)) satisfies (int_{S} {h(x,y),dxdy} = 1), the total intensity over the study area is scaled to one. In the source-mediated-dispersal model, the source function is proportional to the human-activity intensity, i.e., (varphi (x,y) = |S| cdot h(x,y)), and the destination function (psi (x,y)) is uniformly equal to ({1 mathord{left/ {vphantom {1 {|S|}}} right. kern-nulldelimiterspace} {|S|}}). On the contrary, the source function of the destination-mediated-dispersal model is uniform and the destination function is (psi (x,y) = h(x,y)). The full model is a combination of the source- and destination-mediated-dispersal models in which both source and destination functions vary with area. In this study, we assume that a single factor determines both the source and destination functions, i.e., (varphi (x,y) = left| S right| cdot h(x,y)) and (psi (x,y) = h(x,y)) (Table 2).
Table 2 Definitions of model types.
Full size table

Asymptotic growth rate of a population
Spatial dimension introduces complexity, though rigorous mathematical analysis is still viable for a small population with few small sub-populations. This situation may arise with an accidentally transferred population. Here, we consider infinitesimally small populations to be rare for invasive species and derive an asymptotic value of the growth rate to the size of the area inhabited by the population.
Each sub-population includes age, so we incorporated age-structured population dynamics29,31 into the model, described by the differential equation,

$$ frac{partial n(a,t)}{{partial t}} + frac{partial n(a,t)}{{partial a}} = 0, $$
(2)

where (n(a,t)) represents the frequency of sub-populations with age a at time t. Note that Eq. (2) assumes no extinction of sub-populations. The equation has two boundary conditions: (1) (n(a,0)) represents an age distribution of the initial population, and (2) (n(0,t)) represents the number of new sub-populations (i.e., age 0 sub-populations) introduced by the long-distance dispersal at time t.
To determine the number of new sub-populations, we need to determine how many long-distance dispersers will emerge from a given population by including spatial heterogeneities. Recall that a sub-population expands outward by a constant speed g. Therefore, if we ignore overlaps among sub-populations each sub-population keeps a circular shape of radius proportional to age. In addition, if a sub-population is young, i.e., its size is small, we can regard the value of the source function (varphi (x,y)) inside the sub-population as uniform. Let ((x_{i} ,y_{i} )) and (a_{i}) be the position of the center and age of the (i)-th sub-population, respectively. With the above approximations, we can simply derive the expected number of long-distance dispersers that start dispersal from the (i)-th sub-population as (pi R cdot (ga_{i} )^{2} varphi (x_{i} ,y_{i} )).
On the other hand, existing sub-populations also originate from long-distance dispersal. Therefore, a position of the sub-population also follows the destination function (psi (x,y)). Building on the expected number of new sub-populations we described at the last paragraph, we calculate an average over the study area to calculate a mean-field approximation of the number of long-distance dispersers from an age a sub-population as (pi Rint_{S} {psi (x,y)(ga)^{2} varphi (x,y),dxdy}).
We assume that a population consists of a few small sub-populations in this formulation and a disperser will always establish outside existing sub-populations. Therefore, the total number of a new sub-population (i.e., (n(0,t))) is a summation of new sub-populations produced by each of the existing sub-populations,

$$ begin{gathered} n(0,t) = pi Rint_{0}^{t} {n(a,t)left[ {int_{S} {psi (x,y)(ga)^{2} varphi (x,y),dxdy} } right],da} \ = pi Rleft( {int_{S} {psi (x,y)varphi (x,y),dxdy} } right)int_{0}^{t} {(ga)^{2} n(a,t),da} . \ end{gathered} $$
(3)

With Eq. (3), the asymptotic growth rate of the Eq. (2) can be calculated as follows29,31,

$$ left( {2pi Rg^{2} int_{S} {psi (x,y)varphi (x,y),dxdy} } right)^{1/3} . $$
(4)

The asymptotic growth rate indicates that the integration (int_{S} {psi (x,y)varphi (x,y),dxdy}) describes the influence of the source and the destination functions in the early phases of population growth. Therefore, hereafter we call (int_{S} {psi (x,y)varphi (x,y),dxdy}) a spatial factor (F_{{text{h}}}) of the long-distance dispersal. Note that the spatial factor reduces to 1 for both source- and destination-mediated dispersal models. For the full models of which source and destination functions are (varphi (x,y) = left| S right|h(x,y)) and (psi (x,y) = h(x,y)), respectively, we can reduce the spatial factor (F_{{text{h}}}) to (int_{S} {left( {sqrt {left| S right|} h(x,y)} right)^{2} dxdy}), equivalent to Simpsons’ diversity index.
Numerical analysis
We evaluated the effects of the dispersal vector on distribution with an individual-based approach that describes colonies in a population as groups of individuals within a circular shape of various sizes (Fig. 1a,c,e for typical model outputs, see supplemental information SI 1 for detailed settings). To evaluate the effect of spatial heterogeneity on population dynamics, for each model type we generated 100 of (h(x,y)) randomly (see shaded area of Fig. 1a,c,e, and SI 2 for the algorithm used) and ran 100 independent realizations for each (h(x,y)). For each realization, we split the time course into three phases: establishment, expansion, and naturalization. The phases were based on the proportion of area inhabited by the population (Fig. 2a; less than 5%, 5% to 95%, and 95% to complete occupation of the total area, respectively), and measured the length of each phase. We linearly interpolated the time course based on area covered for each phase.
Using the same set of realized time courses, we estimated the asymptotic growth rate as the peak of a distribution of the logarithmic value of the instantaneous growth rate, defined as a difference of logarithmic values of covered area that are adjacent in a time course. We excluded periods that a population covers less than 1% or more than 50% of the total area to avoid strong demographic stochasticity of initial dynamics and a deceleration phase of S-shaped growth, respectively. We gathered these logarithmic values of instantaneous growth rates from 100 realizations with the same (h(x,y)) and dispersal type, then estimated the density distribution using Gaussian kernel estimation. Finally, we determined the maximum point of the estimated density distribution as the estimated asymptotic growth rate. More

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How rapidly can the major functional components of a soil ecosystem appear in volcanic ash?
The overall conclusion from comparing the functional composition of the developing ash-soil with natural forest soils from the same climates, is that the major biological aspects of the system have developed surprisingly quickly. Already by the 24-month and 36-month stage, the ash-soil had a similar diversity of bacterial functional genes to a forest soil (Fig. 8).
This contrasts with the findings of Fujimura et al.3, who found that ash deposited by the Miyakejima volcano still showed a very distinct and low diversity community compared to that expected of normal forest soils. This difference could be attributable to differences in the initial chemical composition of the ash: the pH value of the Miyakejima ash studied by Fujimura et al.3 was much lower than the ash we sampled, due to the persistence of acids from the initial eruption, or acidic rainfall from later outgassing. Our Sakurajima ash samples had only a slightly acidic pH (5.2) at the time of harvesting, which was similar to the average value (5.14) of 251 ash samples from Sakarajima collected from 1955 to 200125, and they remained in the 5–6 pH range at both 24 months and 36 months. Even the ash soil mesocosm samples that were situated on the lower slopes of the Sakurajima volcano, and the nearby natural forest soils at Sakurajima, remained in the pH 5–6 range and had a similar gene composition to other sites, suggesting that the area around the Sakurajima volcano is not subject to frequent highly acidic rain events. The difference between the Sakurajima ash and the Miyakejima ash hints at the importance of details of the chemical environment of the ash for determining its development into a functioning soil ecosystem.
Another factor which may explain the difference between the paths of biota development of the ash from the two volcanoes is the relative availability of propagules. Our ash soils were surrounded by developed ecosystems which could provide a rain of propagules of microorganisms and bryophytes in windblown dust or in rain splash. By contrast the Miyakejima ash field was an area of extensive ecosystem destruction. If bryophytes and lichens had not been able to establish due to unavailability of propagules, an important source of photosynthetically-fixed energy would have been unavailable to the Miyakejima system. This would have impeded the development of a decomposer food chain, of mineral weathering by organic acids and chelates, and of buffering of soil pH and water content and nutrient storage on organic matter and clay surfaces.
Was there a lower taxonomic diversity than is normally found in a developed soil?
At the broadest level, from the metagenomes, the taxonomic composition of the biota of the developing ash-soil resembled that of the developed forest soils—with archaea, fungi, protists and metazoans occurring at similar relative abundances in the ash-soils and forest soils. In this sense, the ash-soil had developed remarkably quickly in terms of the basic taxonomic framework of a functioning soil ecosystem. Bryophytes and lichens were able to establish10,26 with their propagules small enough to enter through the gauze covering, and presumably played an important role in providing carbon and extra niches to the developing ecosystem—even though they are unlikely to be as supportive of a diverse soil biota as the extensive root systems of vascular plants.
There was however—continuing a pattern seen in the 24-month results reported by Kerfahi et al.10—a lower OTU and Shannon diversity of bacteria based on the 16S rRNA amplicon sequencing. The continuing lower OTU-level 16S taxonomic diversity of the ash soils might have had multiple causes.
Firstly, limitations on dispersal and colonization of the ash soil systems might have kept OTU diversity lower. It is assumed that the biota present were mostly derived from the surrounding forest ecosystems—as windblown or rain splashed material. Dispersal limitation of soil biota is likely to play some role in all volcanic ash fields, although the existence of nearby areas of surviving ecosystem is also common20,27. Most volcanic explosions or ash fields result in no more than a few square kilometres of landscape being completely devastated, and isolated pockets of vegetation that survive are common27,28—in this respect the proximity of our mesocosms to natural vegetation may be fairly realistic. In natural ash deposits, upwards colonization by soil biota from buried ‘legacy soils’ is also possible20. In our experiment, upwards colonization from soil below the pots was not possible, and the plastic gauze covering may have prevented access by insects, birds or seeds that could have brought incidental soil biota. It is thus difficult to know for sure whether the overall role of dispersal limitation in our mesocosm systems is stronger or weaker than would generally be the case in natural ash fields. Since natural ash fields themselves are also very heterogenous in terms of area, depth and degree of devastation of natural vegetation, it is especially hard to generalise.
Secondly, the relatively extreme chemical and physical conditions of the volcanic ash itself are also likely play a major role in limiting the bacterial taxonomic diversity of the ash-soil biota in the mesocosms. Although the pH of the ash in our mesocosm systems was not extreme, it may be expected to be droughty due to low organic matter content, and low in available nutrients and organic matter that could sustain soil food chains. Even by 36 months, the organic carbon of the ash-soil was still orders of magnitude lower than in the surrounding forest soils (Supplementary Fig. S1)10,26. Thus, fewer niche types may be viable in the environment of the developing ash.
Nevertheless, although lower level taxonomic diversity of bacteria is clearly lower than in the established forest soils, the metagenome results from the ash soil mesocosms show that most higher level taxonomic groups of bacteria, archaea and eukaryotes are already present by the 24-month and 36-month stage, emphasizing that in some respects the establishment of soil biota has been rapid. While DNA of dead microorganisms blown as dust could have provided the impression of populations being present, it seems unlikely that dead material would be able to disperse in the same relative proportions as in a forest soil, to give a metagenome that resembled a developed soil in all its major groups, in roughly the same relative abundances (Please refer to Fig. 1 in Kerfahi et al.10).
Was there a lower diversity of categories of functional genes of soil biota than are normally found in a developed soil, indicating less functional complexity in the early ash-soil system?
Surprisingly, the richness and diversity of functional gene categories at Level 4 of the SEED Subsystem was not significantly different between the ash mesocosms and the forest soils. Around 97% of the metagenome reads in the ash mesocosms and the forest soil were bacterial, so the much lower OTU diversity of bacteria in the mesocosms would be expected to give a lower functional diversity of genes. The contrast between patterns of taxonomic and functional diversity in the ash mesocosms emphasises the redundancy of gene functions in soil organisms—such that losing a high proportion of taxonomic diversity apparently has no effect on functional diversity. If the level of functional gene diversity is high, this implies that in a general sense the ecosystem can potentially perform a wide variety of functions. Greater functional diversity is also considered to result in greater resilience of the ecosystem29.
We had also hypothesized that the distinct chemical environment of the developing ash would result in increased relative abundance of a number of specific gene categories:
Is there an increased relative abundance of genes (or of prokaryotic genera) associated with autotrophy (e.g. Rubisco gene, coxL gene) in the ash-soils?
We searched for potential taxa and genes which might indicate autotrophic carbon fixation through chemosynthetic oxidation processes. We found a higher relative abundance of Ktedonobacteraceae (Chloroflexi) in the ash soils compared to forest soils, which is a pattern that was also found in previous ash studies21,30. Ktedonobacteraceae is a novel taxonomic group that has only been recently added to the phylogenetic tree of Chloroflexi30,31. Their genome has been reported to be large and they are known to have a wide potential for different metabolic mechanisms32. Ktedonobacteraceae are mostly found in extreme environments, including volcanic ash and hydrothermal vents30.
We had hypothesized that the coxL (carbon monoxide dehydrogenase large chain) gene—which is implicated in autotrophy—would be more abundant in the ash soil samples. Likewise, the rbcL (ribulose bisphosphate carboxylase large chain) gene is involved in carbon fixation and we hypothesized its greater abundance in the ash-soil mesocosms. However, in fact both rbcL and coxL had higher relative abundances in the forest soils.
Although coxL has been mostly linked with chemotrophic organisms living in an extreme environment, due to its potential role for assisting heterotrophic growth in the environments lacking organic matter21,33, it is also one of the more abundant genes in natural forest soils34. Noting that natural forest is one of the largest global sinks of atmospheric CO34,35, the high relative abundance of coxL gene might be due to high abundance of CO in natural forests. Also, the coxL gene might have the potential to participate in other metabolic pathways. The presence of coxL genes in many different types of organisms (e.g. plant symbionts, animal pathogens, etc.) suggests other roles of CO-oxidation by the coxL gene33.
The higher abundance of rbcL in the forest soils in our study might be related to photosynthetic C fixation by cyanobacteria in both the forest soil and ash soils. rbcL gene also have been found in many natural forest sites. By contrast, Fujimura et al.3 found rbcL to be more abundant in developing volcanic ash soils, which might be attributable to differences in the chemical composition of ash or due to details of the mesoocosm system used in our study.
Is there increased relative abundance of genes that are associated with acquisition of nutrients from abiotic sources rather than decomposition of organic matter (e.g. nitrogen fixation genes)?
Contrary to expectations we found no differences in the relative abundance of nitrogen fixation gene (nifH) between the ash soils and forest soils. It is possible that this reflects nitrogen being relatively more limiting in the forest soils, where there is an abundance of organic carbon. It is also possible that in the forest soils, the abundance of labile carbon in the rhizosphere of N-limited plants encourages N fixing bacteria through root secretions36.
Is there increased relative abundance of stress response genes and dormancy related genes?
As hypothesized, we found that dormancy related genes were more abundant in the ash mesocosms than in the forest soils. This would seem to be adaptive in terms of survival of a well-drained ash-soil, low in organic matter, which is apt to dry out. This would depend, however, on the detailed microclimate. Field measurements suggested that in sunny conditions the microclimate in the static air in the trays in which our pots were held was generally around 0.5–1 °C higher than in the open air a meter away, although this difference disappeared in cloudy conditions and at night10. It is possible that our mesocosms (exposed in open sunlight, but covered by a gauze) would have either retained or lost moisture more easily than the soils, and that this might have affected the frequency of drying. We had also anticipated that stress response genes would be more common in the ash mesocosms. However, in this case the stress response category was less common than in the forest soil, surprisingly suggesting that at a cellular level stresses are in fact less common in the ash-soil.
Is there decreased relative abundance of cell–cell interaction related genes?
As anticipated, genes related to cell–cell interactions such as regulation and cell signalling genes and virulence, disease, and defense genes (Fig. 7a,b) were relatively less abundant in the ash mesocosms. This seems to emphasize that complex interactions are less common in the developing ash system than in a fully developed soil with its more abundant and diverse carbon sources. However, bacterial genes found as part of CRISPRs were relatively less abundant in the forests, implying lower intensity of defense mechanisms of bacteria against viruses in the forests compared to ash soils. The relatively low OTU diversity of the ash soils may perhaps allow greater spread and mortality of viruses through populations, necessitating greater investment in defences.
Since Odum37, it has been noted that developing ecosystems tend to be based on tighter and more effective interactions as succession continues. For example, Morriën et al.38 found that during secondary succession of old field ecosystems, co-occurrences of taxa become more predictable and carbon cycling in the decomposer chain more efficient. Empirically our findings seem to agree with the same paradigm, as we found cell–cell interaction genes found to be more abundant in the developed forest soils. It would be very interesting to directly measure decomposer carbon cycling efficiency in ash-soil mesocosm systems as they develop.
Although we focused on 16S rRNA amplicon sequence data for comparing OTU level diversity between samples as taxonomic assignment based on MG-RAST pipeline is very limited39,40, we had briefly compared the family level bacterial community composition assigned based on metagenome sequence data and based on 16S rRNA amplicon sequence data. It is still debatable if whole genome sequencing, which is free from primer bias, is superior in this respect to 16S rRNA amplicon sequencing40. Our data supports the idea that at the bacterial family level, metagenome sequencing covered a larger variety of taxonomic groups than 16S rRNA sequencing. More

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