Multi-scale analysis of species introductions: combining landscape and demographic models to improve management decisions about non-native species

Authors

  • Kerry A. Brown,

    Corresponding author
    1. Department of Ecology, Evolution, and Environmental Biology, Columbia University, 1200 Amsterdam Avenue, New York, NY 10027, USA;
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    • Current address: Division of Science and Mathematics, University of the Virgin Islands, St Thomas, USVI 00802.

  • Sacha Spector,

    1. Center for Biodiversity and Conservation, American Museum of Natural History, Central Park West, New York, NY 10024, USA; and
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  • Wei Wu

    1. Department of Environmental Studies, College of Environmental Science & Forestry, State University of New York, 1 Forestry Drive, Syracuse, NY, 13210 USA
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    • Current address: Department of Coastal Sciences, The University of Southern Mississippi, Ocean Springs, MS 39564, USA.


*Correspondence author. E-mail: kbrown2@uvi.edu

Summary

  • 1Non-native, invasive species can affect biological patterns and processes at multiple ecological scales. The multi-scalar effects of invasions can influence community structure, ecosystem processes and function, and the nature and intensity of ecological interactions. Consequently, efforts to assess the spread of invasive species may benefit from a multi-scale analytic approach.
  • 2We analysed results from landscape- and population-scale models for Syzygium jambos, a non-native tree in the Luquillo Mountains of Puerto Rico, to demonstrate a multi-scale approach that can be used to inform management decisions about invasive plants. At the landscape-level, we used an Ecological Niche Modelling approach to predict environmentally suitable habitats for the target plant. At the population-level, we constructed matrix projection models to determine the finite rate of population increase (λ) for S. jambos. We then extrapolated λ values to the landscape-scale to obtain a distribution map of λ values for the Luquillo forest.
  • 3The landscape analyses suggested that the most environmentally suitable habitats were those most similar to where S. jambos had already been observed. The population-level analyses showed that four of the seven populations had λ values less than 1, indicating that they were projected to be below replacement. The λ distribution map showed that S. jambos growth was highest in areas where it was most common and lowest in areas where it was most rare.
  • 4Our analyses further suggested that the importance of different drivers of invasion and the environmental variables that mediate them appear to be strongly scale-dependent. Past disturbances seemed most important for controlling invasions at fine-spatial scales; while abiotic environmental variables modulated coarse-scale invasion dynamics.
  • 5Synthesis and applications. We have shown that a multi-scale analytic approach can be used to manage invasive species by simultaneously targeting susceptible life stages and rapidly growing populations in a landscape. The utility of this approach stems from an ability to: (i) map the distribution of habitats that can potentially sustain λ values above replacement; (ii) identify populations to manage or monitor during selected stages of an invasion; (iii) forecast the probability for a target species to increase above a critical threshold abundance; and (iv) set priorities for control and monitoring actions.

Introduction

Successful invasions are facilitated or restricted by patterns and processes operating at several ecological scales (Mack et al. 2000; Pauchard & Shea 2006). Each stage of an invasion – colonization, establishment, naturalization, and spread (sensu Richardson et al. 2000; Henderson, Dawson, & Whittaker 2006) – is defined by ecological processes at different spatial scales (Shea & Chesson 2002). The effects of invasions are also multi-scalar and can influence community structure, ecosystem processes and function, and the nature and intensity of ecological interactions. Such scale-dependent invasion dynamics are an important, but often overlooked element of the invasion process (for exceptions, see Allen & Shea 2006; Davies et al. 2007). However, it has become increasingly apparent that multi-scale analyses of invasions, particularly at local and regional scales, may improve development of management strategies to control populations of non-native species.

Landscape- and population-level studies of species invasions have generally focused on processes operating at different spatial scales. Landscape ecologists have commonly focused on integrating information over large, heterogeneous landscapes to understand coarse-grain distribution patterns and habitat specializations for target invasive species. One approach for determining species’ potential distribution across a landscape has been Ecological Niche Modelling (ENM), a technique that develops predictive models of a species’ distribution based on a set of relevant environmental variables (Pearson et al. 2006). A primary assumption with ENMs is that the species’ fundamental niche is determined by interactions between multiple environmental variables, whereas the realized niche is circumscribed by environmental conditions and biological and historical realities, such as competitive and predatory interactions, which narrow the distribution of the invading species (Hutchinson 1957). Individuals found within areas of appropriate environmental, biological, and historical conditions are assumed to be able to sustain viable populations, reproduce, and spread to other similar habitats, whereas individuals inhabiting ranges in which the conditions are inappropriate are not (Peterson 2003). Population-level studies tend to focus, although not exclusively, on ecological scales that regulate colonization and establishment during the invasion process. These types of studies integrate information about the effects of biological processes at finer spatial scales. Commonly, age- or stage-structured matrix projection models are used to assess a target species’ population dynamics. The underlying assumption here is that an accurate assessment of a population's finite rate of increase, λ, is a reliable indicator of the intensity and rate of spread of an invader.

Regardless of the mechanisms involved in promoting an invasion, it has become increasingly clear that efforts to assess the spread and expansion of invasive species may benefit from a multi-scale analytic approach. Niche opportunities for invaders will be determined by the physical environment (e.g. temperature or moisture) as well as biological interactions (e.g. herbivory or competition), both of which vary along different spatio-temporal dimensions. The long-term predictability of invasions will be enhanced when information about the invader's population and landscape dynamics are integrated, because colonization, fertility, and growth do not occur independent of habitat; nor are invasive species spread and expansion independent of biotic interactions.

Our study links results from species distribution models with stage-structured population projection models to assess the expansion potential of a locally dominant non-native tree Syzygium jambos, (L. Alston, Myrtaceae), in a tropical forest in north-eastern Puerto Rico. We developed a landscape-scale distribution map for S. jambos based on six climate and biological variables. At the population-level, we developed stage-structured transition models that assessed the plant's future abundance and response to simulated perturbation experiments. We then integrated results from the population and landscape models to explore how each approach may, individually and synergistically, inform understanding of the invasion process at different spatial scales. We used these two modelling approaches to explore: (i) the distribution of favourable environments for the target non-native plant; (ii) the distribution of populations with positive finite rates of increase (λ); (iii) whether environmental drivers of the plant's expansion are congruent at the landscape and population levels; and (iv) whether multi-scale analyses can be utilized to develop more effective or efficient management protocols for non-native, invasive species.

Material and methods

focal species

Syzygium jambos is a pan-tropical invasive tree that occurs throughout Puerto Rico. The plant is native to South-east Asia and was introduced to tropical botanical gardens throughout the world and into the West Indies during the 18th century (Descourtilz 1829, as seen in Wadsworth 1943). Birds are responsible for localized dispersal of S. jambos fruits, while frugivorous bats and river currents may be responsible for long-distance dispersal. For a more detailed description of S. jambos’ ecology see Brown, Scatena & Gurevitch (2006).

study area

Our study area was the Luquillo Experimental Forest (henceforth referred to as Luquillo), within the Mountains of north-eastern Puerto Rico (18°18′N, 65°50′ W) – one of the Long-Term Ecological Research sites in the USA (Supporting Information Fig. S1). Mean annual rainfall increases from 1000 mm year−1 in the lowlands to nearly 5000 mm year−1 at the highest elevations (García-Martino et al. 1996). Corresponding to this climatic gradient are changes in life-zone designation, forest composition, and structure (Ewel & Whitmore 1973; Lugo & Lowe 1995). Hurricanes are the dominant natural disturbance affecting the area (Scatena & Larsen 1991). Additional information on the ecology and management of the site can be found elsewhere (Odum & Pigeon 1970).

Landscape-modelling algorithm

modelling invasive plant distributions

Modelling the predicted distribution of an invasive species usually involves parameterizing the ecological niche model using environmental variables that limit the distributional area of the species in its native range, then re-projecting that distribution onto the species’ invaded range (Peterson 2003, 2005). Because the ecological conditions of the sampled or current invasive populations may not reflect their entire potential distribution in their novel habitat, this approach minimizes underestimation of the potential distribution of invasive species, and assumes that the resultant distribution is based on equilibrium occurrence data (but see Peterson 2005 for a discussion of the issue of non-equilibrium vs. equilibrium niche modelling).

Our study focuses on determining the current distribution of S. jambos within Luquillo as well as the plant's demographic parameters in that habitat. Therefore, we constructed the niche model using occurrence records from Luquillo. In this way, we developed an integrated landscape model that reflects the invasive population dynamics as a function of S. jambos’ current distribution and includes land-use history in Luquillo, which strongly influences S. jambos establishment (Brown et al. 2006).

environmental variables

We used six climatic and biological variables to parameterize the niche model for S. jambos: measurements of rainfall, temperature, solar insolation above the forest canopy, evapotranspiration rate, Leaf Area Index (LAI), and an estimate of 1936 canopy cover percentage in Luquillo (see Supporting Information Table S1). Estimates of rainfall, temperature, LAI, and solar insolation above the forest canopy were derived from a topography-based climate model, TOPOCLIM by Wang et al. (2003). Estimates of evapotranspiration rate was modelled for Luquillo by Wang et al. (2003) and Wu et al. (2006). The 1936 canopy cover percentages were based on Foster, Fluet & Boose (1999), who estimated forest cover and forest type in 1936 from aerial photographs. This data layer gave an indication of canopy cover in Luquillo in 1936, which was associated with land uses ranging from pasture to shade coffee plantations to private farms. Data layers for all environmental variables were compiled in a GIS and re-sampled to the same datum and coordinate system (NAD 1927 State Plane, Puerto Rico) at 30-m resolution.

geographical distribution of s. jambos

We used the maximum entropy method (Maxent version 3·0·6, for free download see http://www.cs.princeton.edu/~schapire/maxent/) to model the geographical distribution of suitable habitats for S. jambos within Luquillo. Maxent is robust for data sets with relatively few presence points (Hernandez et al. 2006), which was the case in our study; further, when using Receiver Operator Characteristic curves (ROC) with Maxent, absence data is not required (see below). Phillips, Anderson & Schapire (2006) present a detailed mathematical description of Maxent and its application to species distribution modelling. Maxent is a general-purpose method for characterizing probability distributions from incomplete information (Pearson et al. 2006; Phillips et al. 2006) and estimates the realized niche of a target species based on presence-only occurrence records and environmental variables known to describe important aspects of the habitat that influence the species’ presence. Maxent estimates the unknown probability distribution that defines a species’ distribution across a landscape assuming that the estimated distribution must agree with everything inferred from the environmental conditions at the presence localities (using background pixels), but avoids placing any unfounded constraints on the resultant distribution (Phillips et al. 2006). The resulting probability distribution is the one closest to uniform, while still subject to constraints imposed by the observed occurrence of the species and environmental conditions across the study area.

The Maxent algorithm for determining habitat suitability assigns a probability to each pixel in the study area. Maxent computes a species’ distribution based on the probability assigned to each pixel and the value of pixels are constrained by environmental conditions at presence localities. The use of Maxent is an effective method for generating unique probability distributions based on an incomplete state of knowledge, which recently has been applied to species distribution modelling (Elith, Graham & Species Modelling Group 2006; Pearson et al. 2006).

We used 10 000 background pixels to represent the environmental conditions present in the study region. We used Maxent's default settings for the convergence threshold and maximum number of iterations (500). The program automatically selected necessary parameters to minimize over-fitting (i.e. regularization values). The Maxent model predictions were presented as cumulative probabilities, wherein the value of a pixel was the sum of the probabilities of that pixel and all other pixels with equal or lower probability multiplied by 100 to yield a percentage. However, the Maxent output ranged from 0–100 and indicated relative suitability (not probability of occurrence).

decision threshold

Decision thresholds assist with model interpretation by establishing objective criteria that delineates unsuitable and suitable areas. We used the method described by Phillips et al. 2006 for applying analysis of ROC curves to presence/random data. The ROC curve is a plot of the probability of true positive (sensitivity) as a function of the probability of obtaining a false positive (1-specificity). The Area Under the ROC curve (AUC) gave a measure of predictive power based on an estimate of the probability that the predictions and the outcomes are in agreement. The AUC was interpreted as the probability that a random false positive and a random true positive were correctly predicted by the Maxent model, with an AUC value of 0·5 indicating that model predictions were no better than random guessing. (This holds for ROC curves derived from presence-random data, as was the case in our analysis.) While there is no consensus in the literature as to desirable values of AUC, values > 85% are considered a baseline for model accuracy (Pearce & Ferrier 2000). The AUC provided a single measure of model performance, independent of any particular choice of threshold.

We examined the following thresholds to distinguish between suitable and unsuitable areas: (i) no threshold value, wherein the continuous suitability distribution from the Maxent model output was analysed without setting a cut-off threshold; and (ii) fixed threshold – the fixed threshold was set at 20 (e.g. FT20). This value was chosen following the initial analysis of the ROC curves and omission and commission estimates. It is the point at which all pixels predicted as being at least as suitable as those in which species are known to occur already or the minimum area designated as suitable while preserving zero omission error in the occurrence records used to construct the distribution (Pearson et al. 2006). To assess the importance of each environmental variable in the model, we used Maxent's jackknife test function on each variable to measure the ‘training gain’ or the improved predictability of the model based on the incorporation of a particular variable (or decrease based on its omission).

sampling points

We used occurrence records from 21 sites to parameterize the landscape model. Twelve of the sites were first sampled in January 2001. Presence data for seven of the sites were obtained from the Luquillo Experimental Forest, Long-Term Ecological Research Project (http://luq.lternet.edu/), in particular, the Forest Dynamics Plot (Thompson et al. 2002). The remaining two sites were surveyed in January 2001 as part of a pilot study (Brown 2004). Each quadrat that included at least one S. jambos individual was counted as a presence locality.

Population model

abundance, spatial distribution and population structure

From January 2001–January 2003, one of us (K.A.B.) conducted yearly population censuses of seven of the 21 S. jambos sites mentioned above (see Supporting Information Fig. S1). The closest populations were 360 m apart, while the farthest was 11 km apart. At each of the seven sites, one permanent plot 200 m2 in area was established. In January 2001, individuals of S. jambos were tagged, measured for height, diameter at breast height (DBH), and their locations were mapped. Tagged individuals were re-censused in January 2002 and 2003. Trees with multiple stems connected near the base were counted as single individuals and assigned the DBH of the largest stem. Seedlings were surveyed by randomly placing twenty 1-m2 quadrats in each plot. All S. jambos individuals 1–25 cm in height were tagged, mapped, and censused during the same time period as the adult plots. The diameter at the base of each seedling was measured at 20 mm from the forest floor and recorded to nearest 0·1 mm (accuracy to ±0·0015 mm). Seedlings that were ≤ 30 mm were measured between 5–10 mm at the base.

stage classification

We used the criteria developed by Moloney (1986) to divide S. jambos individuals into six life-history stages based on plant height and life-cycle stage. This technique ensured that plant stages reflect meaningful life-history changes and individuals in each group behave in a relatively cohesive manner. Moreover, fluctuations in herbivory and breakage, which may be important ecological processes that affect growth and survival in this system, were best estimated with height rather than DBH. Sampled individuals were assigned to stages based on height: (i) seedlings (1–25 cm) classified as new germinants (those with cotyledons still present) or established seedlings; (ii) juveniles were non-reproductive individuals measuring 26–50 cm; (iii) adults (reproductive individuals ≥ 50 cm) were subdivided into three classes: small adults (51–100 cm); medium adults (101–400 cm); and large adults (401–600 cm).

flowering and fruiting phenology

Plant phenology was recorded and germination rates were measured to estimate fecundity. The production of flowers and fruits was evaluated on a bi-monthly basis from January to August 2003. We estimated the number of seeds by following the phenology of five individuals from each reproductive stage class within each of the seven populations. All flowers and fruits were counted on these individuals for the duration of the reproductive season. Additionally, on a chosen subset of branches, we followed specific fruits that we permanently marked and placed mesh bags around to ensure capture. Once the fruits matured and fell into the bag, seeds were extracted and stored in a laboratory at ambient temperature. Fecundity was estimated by multiplying the average number of seeds produced per plant in a given stage by the total number of fruits produced by each population.

germination

We estimated percentages of germination from trials carried out with seeds collected in mesh bags from individuals belonging to the three reproductive size classes. A total of 272, 923, and 1444 seeds from the small, medium, and large adult size classes, respectively, were germinated under field conditions in August 2003. The number of seeds that germinated was recorded 14 days after seeds had been sown. To obtain a conservative estimate of germination rates, we considered any seed with a visible radicle as having germinated successfully.

population dynamics

Transition matrix models for stage-structured populations were used to estimate population growth rates (λ) (Lefkovitch 1965). Yearly transitions were calculated according to the proportion of individuals within each stage subject to different fates (i.e. stasis, growth and transition or dying). Data from the life-cycle diagram for each population were arranged in m × m matrices, where m was the number of life-history stages (m = 7; Fig. 1). We assessed population dynamics for each of the seven populations individually. A mean population matrix was calculated for the integrated population–landscape analysis.

Figure 1.

Mean life-cycle diagram for the population of Syzygium jambos (seven populations, n = 7 matrices) and the corresponding transition matrix A (aij), fecundity (F, first row), stasis (S, main diagonal), and growth (G, subdiagonals). The nodes (circles) indicate stage categories. Arrows indicate possible transitions (permanence, transition, or regression) and numbers beside each arrow indicate the transition probability of an individual that follows that path. The average number of seeds produced per plant per size class is given in the upper arrows connecting nodes.

Environmental stochasticity was approximated by randomly sampling population vital rates (i.e. survival and fecundity) from a log-normal distribution, in which the mean and standard deviation were defined by the entries of the stage matrix and a standard deviation matrix, respectively (Akçakaya et al. 2004).

elasticity analysis

We performed elasticity analyses (i.e. prospective analyses) to determine the effects of changes in survival and fecundity of particular life-history stages on lambda (λ) and S. jambos population abundance over time (Caswell 2001). The predictability of structured population models decreases as one project further into the future; thus, for this model we ran the simulations for 20 time steps, with each time step equivalent to 1 year.

population simulations

The population simulations were carried out with RAMAS Metapop (Akçakaya & Root 2002). Incorporating stochasticity allowed us to estimate population trajectories and build an explosion model that simulated the probability that S. jambos populations would increase above a maximum threshold abundance beyond which populations are likely to grow exponentially (until they asymptote). This upper-limit threshold was set at a value of 3 × 105 individuals within 20 years based on model simulations of the highest average cumulative abundance for all seven populations.

Spatial scales

population to landscape

We used a Geographical Information System (GIS) to develop a predictive map of population growth rates (i.e. λ) based on the landscape-level environmental variables. First, we extracted the value for a specific environmental variable from each GIS map layer for each of the seven populations. Next, we used these values as independent variables to derive a multiple regression model with λ as the response variable. Environmental variables that were strongly co-linear were excluded from the multiple regression. The resultant model was projected back on to the original map to produce a surface modelling the distribution of predicted λ values across Luquillo. We assessed the accuracy of the λ distribution map by comparing predicted λ values with those observed from each site.

integrating landscape and population models

We determined the extent to which perturbations in target transition rates influenced S. jambos population dynamics in plots with different habitat suitability values. S. jambos populations were split into two categories based on the predicted suitability value of the habitat (from the Maxent analysis) in which they were located. The suitability categories for each population ranged from 20–50 (henceforth referred to as low suitability) and 50–100 (henceforth referred to as high suitability). Three sites were chosen from each suitability category. We randomly excluded one of the sites in the high suitability category to control for area. The limit for the lowest suitability category (e.g. 20) was based on the fixed decision threshold described above; while the limit of the high suitability category (e.g. 50) was presumed to be a reasonable cut-off point for suitable S. jambos habitats.

Different management scenarios were considered by simulating 10% mortality (i.e. 10% decrease in medium adult survival, λd) for all populations in a specific suitability category (see Supporting Information Table S2). We ran simulations with the original parameters and then ran another round of simulations in which we changed the target transition value for all populations in a specific suitability category. The parameters not in the focal suitability class were held at their original values.

Results

landscape model and environmental variables

The Maxent geographical distribution model output predicted that the most environmentally suitable habitats were those most similar to where S. jambos had already been observed. The lower elevations occurring at the edges of Luquillo and the northern region exhibited the most suitable areas (Fig. 2a). The discrimination capacity of the model (i.e. the AUC) was calculated as 0·947, which indicated that the model can discriminate between sites with and without S. jambos 94·7% of the time. As expected, the 20·1% fixed threshold value (FT20) gave a more conservative prediction for the study area (Fig. 2b). With the FT20 threshold, the patches with the highest suitability index values were located at northern edges of Luquillo.

Figure 2.

(a) Simulated distribution of environmentally suitable habitats for S. jambos in Luquillo. A continuous predicted distribution for the species is shown. (b) Simulated predicted distribution for S. jambos in Luquillo for the fixed decision threshold at FT20 (habitats are interpreted as being habitable for S. jambos if suitability values are above 20). Darker areas are an indication of higher predicted suitability values.

The environmental variable with the highest training gain was evapotranspiration rate. Rainfall and 1936 canopy cover were the next two most influential variables. Model predictions were only slightly improved when temperature and solar insolation were included. Leaf Area Index, when used as the sole predictor variable, had practically no training gain and was not a good predictor of the plant's distribution.

population model and elasticity analysis

The annual population growth rate estimated from the transition matrices for the seven populations ranged from 0·971 to 1·06 (Supporting Information Table S2). The λ values for four of the seven populations were less than 1, indicating that they were projected to be at or below replacement over the next 20 years. Increases or decreases in abundance of S. jambos populations were consistently most responsive to changes in the transition probabilities for the three reproductive stages. Elasticity analysis gave varying outcomes for each of the seven populations. The analyses showed that λ was most sensitive to changes in medium adult survival for four of the populations and fluctuations in seeds and seedlings had the least impact (Espiritu-1, Espiritu-2, Mameyes-2, and Sabana-4). For Sabana-5 and Mameyes-3, λ was most sensitive to changes in large adult survival rates, and for Sabana-3, λ was most affected by changes in small adult survival.

integrating model results

The distribution map of finite rate of population increase (λ) indicated the habitats throughout Luquillo where S. jambos populations could maintain positive or stable growth rates. The λ distribution map, which was based on the landscape-level environmental variables, produced good predictions. Rainfall, evapotranspiration rate, 1936 canopy cover percentage, and LAI explained 95% of the variation in the model (adjusted R2 = 0·951). Canopy cover percentage in 1936 was a more important predictive variable than the other three variables, suggesting that invasion patterns associated with population-level, fine-scale interactions was modulated by past land use (e.g. ecological disturbance). Temperature and solar insolation were co-linear and were removed from the regression. The map accurately predicted positive finite rates of increase for three (Espritu-1, Sabana-3, and Sabana-4) and negative rates for two of the populations (Sabana-5 and Mameyes-2). The direction of change, as measured by λ, for the remaining two populations was incorrectly predicted. The λ distribution map showed that most of the positive values were concentrated at the lower elevations, around the perimeter of Luquillo, and there were scattered patches of positive λ values in the interior of the forest (Fig. 3).

Figure 3.

Distribution of finite rate of population growth rate (λ) in the Luquillo Mountains of Puerto Rico. The grey areas exhibit λ values at or below replacement and darker areas show values at or above replacement (positive population growth). Regression equation, y = 0·7565 + 0·0011 rain – 0·000004 EvapTrans + 0·0077 LAI + 0·0888 cover 1936.

Simulating 10% mortality in medium adult survival led to a decrease in abundance for populations in both suitability classes over the 20-year simulation (see Supporting Information Fig. S2). The decrease in population size was substantially sharper from the original abundance when medium adult survival for populations in the high suitability categories were decreased, compared with when that target transition probability was lowered for populations in the low suitability category (Supporting Information Fig. S2). Moreover, the explosion model showed that the probability of increasing above the critical threshold abundance of 3 × 105 individuals was 0·79 when there was a decrease in medium adult survival for populations in the low suitability category (Fig. 4). However, simulating that same 10% mortality for populations in the high suitability category led to a 0·03 probability of exploding above the critical threshold abundance (Fig. 4).

Figure 4.

Three explosion curves showing the probability of increasing above critical threshold abundance of 3 × 105 individuals over a 20- year period and the resulting shift in probability after simulating 10% mortality for medium adult survival. The dashed curve shows the explosion probability using the initial parameters. The grey curve shows the probability of increasing above the threshold abundance after simulating 10% mortality for medium adult survival for populations in the low suitability category. The black curve shows the probability of increasing above the threshold abundance after simulating 10% mortality for medium adult survival for populations in the high suitability category. The critical threshold abundance is shown in dashed lines. Dotted lines show 95% confidence intervals.

Discussion

landscape and populations models for s. jambos

Combining results from landscape- and population-level models to analyse the dynamics of S. jambos invasion in Luquillo permitted a more comprehensive interpretation of the plant's ecology and allowed for a clearer understanding of controls on invasive species in general. The ecological niche model-based suitability map suggested that favourable environments for S. jambos were restricted to mostly lowland areas, with some scattered habitats in the interior of the forest. Although lower elevation environments were close to the boundary of Luquillo where historically forest access has been easiest (e.g. high disturbance), the geographical distribution of habitats most suitable for the plant was mediated by rainfall and evapotranspiration rate. These variables may have played an important role in controlling the spread of S. jambos across Luquillo. It should be noted, however, that the intensity of propagule pressure from source locations may be a better predictor of a system's invasibility than environmental factors (Von Holle & Simberloff 2005). The intensity of propagule pressure was not incorporated in the ENM, because quantifying the function of arriving propagules on forest invasibility is often difficult, since there is little or no documentation of failed colonization events.

Results from the population-level study indicated that the abundances for four of the seven populations surveyed were projected to decrease over the next 20 years. The remaining three populations exhibited λ values above one. The extent and degree to which this scenario establishes a source-sink dynamics between populations with high and low λ values is not known. At the very least, the results indicated that dominance of S. jambos populations within Luquillo may be transient. The high regression probabilities for juvenile to seedling and small adult to juvenile stages are most probably a reflection of the impacts of herbivory and perturbations that cause breakage, and are unlikely to be an artefact of how the plant's life-history stages were divided. The role of herbivory on survival and growth for S. jambos populations is an area of continuing enquiry.

combining results of landscape and population models

The distribution map of λ values represented an integrated, multi-scale analysis that extrapolated population-level dynamics to the landscape scale. By examining both models, we were able to define habitats where S. jambos were not predicted to be present, but can establish and maintain viable populations. There was remarkable agreement in the overlap between the continuous distribution map of suitable environments and positive (or static) λ values. Despite this outcome, there was no immediately obvious a priori reason to expect both approaches to have converged on similar results, particularly because the models were parameterized using variables collected and sampled from different spatial and temporal scales. The fact that this convergence occurred indicated that our approach captured much of the variation required to adequately model the landscape and population dynamics of the plant. The explosion model indicated that S. jambos population abundance was unlikely to increase above a critical explosion threshold when the high suitability populations were strategically controlled. Small decreases in the plant's vital rate (e.g. 10%) produced substantial reduction in the probability of increasing above the critical abundance.

The most challenging obstacle for the practical use of this approach to manage S. jambos populations in Luquillo (and introduced species in general) will be properly incorporating the inherently complex interactions that develop between non-native and native species into the landscape and population models. Moreover, populations of invasive species tend to be spatially aggregated, depending upon the modes of dispersal and initial pathways of colonization. This may lead to some degree of spatial autocorrelation among collection localities, which would bias model results (Lichstein et al. 2002). Nonetheless, this multi-scale analytic approach produced a more nuanced assessment of the invasion process, one that would provide flexibility in the development of potential management plans (see below).

insights into the invasion process

Our results suggested that key predictors of an introduced species’ success changes across spatial scales, indicating that the drivers of invasions can be strongly scale-dependent. At regional scales, the variability or average differences of climate variables often play an important role in regulating an invader's distribution; however, at localized scales, past disturbances can drive the observed spatial distribution. Disturbance is often thought to be one of the most important variables for facilitating plant invasions (Von Holle & Motzkin 2007). However, this study indicated that disturbance was possibly most important at controlling invasions at fine-spatial scales, but lost its significance at coarser-spatial scales. Accordingly, we suggest that a more nuanced perspective on the role of disturbance in facilitating species invasions be adopted. In general, the importance of different drivers of invasion (i.e. predator escape, disturbance) and the environmental variables that mediate them are probably defined by the spatio-temporal scale at which they are sampled.

Similar scale-dependent findings of invasion dynamics have been highlighted in other studies, particularly those that have described the diversity–invasibility paradox (Shea & Chesson 2002; Davies et al. 2007). Our results are consistent with the hypothesis that processes that drive invasions operate at different spatial scales and their consequential effect on community structure (i.e. diversity patterns) also exhibits scale-dependence. In Luquillo, past disturbances possibly played an important role in driving diversity patterns at fine spatial scales, because perturbations (i.e. land use change) will presumably modulate localized, spatially explicit interactions that lead to niche partitioning and competitive exclusion (Davies et al. 2005). Heterogeneity in landscape-level climate variables, such as solar irradiance and rainfall, on the other hand, may arguably be most important at structuring native and non-native plant diversity patterns at larger scales. Invasive species range distributions at the landscape level will most probably be regulated by large-scale environmental gradients, while at the population-level, limits to species distributions will be controlled by biotic interactions (Kinlan & Hastings 2005) and fine-grain patterning of abiotic factors (Kulmatiski, Beard & Stark 2006). It should be noted that biotic interactions that occur at fine spatial scales can play an important role in determining species distributions, but since landscape models are applied at macro-scales where climatic influences on species distributions are dominant, they can minimize the impact of localized interactions (Pearson & Dawson 2003).

relevance for conservation planning

This study represents the first multi-scale analysis for managing non-native, invasives, which we believe has profound conservation implications for controlling the spread of introduced species. Integrating the results from the landscape and population modelling schemes provides more robust assessments of the spread and expansion potential of target non-natives and more efficient means of directing efforts to control the spread of introduced species. These practical improvements over more limited approaches hold particular relevance for implementing invasive species management and monitoring strategies in highly invaded regions (sensu Buckley 2008).

A multi-scale strategy, for instance, can more precisely guide decisions about management of introduced species by concentrating control efforts on specific phases of an invasion. Landscape-scale analyses allow for detection during the nascent phases of an invasion (e.g. establishment); while control efforts for well-established populations are best resolved by population-level analyses. In tandem, a multi-scale strategy can precisely guide management of introduced species by simultaneously targeting susceptible life stages and vulnerable populations in the landscape. This approach may not ensure eradication of target species, especially for widespread plant populations, but it may assure ‘maintenance control’ at acceptable population sizes (Mack et al. 2000). Another practical application emerges from the potential to triage control efforts. Accordingly, in real-world circumstances of limited monetary and labour resources, conservation managers can formulate plans that prioritize their efforts across scales by targeting stage-specific vital rates of populations in predetermined habitats. Moreover, management schemes that use a multi-scale analytic approach may more readily influence policy decisions, because such analyses offer multi-pronged solutions spanning from local to regional scales (Eisinger & Thulke 2008). From a practical perspective, effective management of species introductions require a regional-based strategy that is able to predict susceptible habitats for future establishment, and also focus on control rather than eradication of vulnerable populations of already widespread invaders.

In practice, managers should be able to use this approach to: (i) map the distribution of habitats that can potentially sustain λ values above replacement (population growth potential); (ii) identify populations to manage or monitor based on overlap between suitable habitats and λ distribution maps as well as a specific life-history stage within a particular habitat type (habitat expansion potential); (iii) forecast the probability for a specific population within a particular habitat type to increase above a critical threshold abundance or ‘breakout’ abundance; and (iv) set priorities for control and monitoring actions based on a comprehensive view of the likely trajectories of single populations or geographical areas. In summary, we believe that multi-scale approaches to invasive species management will provide conservation benefits while creating immediate management efficiencies and higher success rates.

Acknowledgements

We thank F.N. Scatena and J. Gurevitch for guidance and support during this project; M. Miriti for valuable criticism of this manuscript; S. Moya, G. Cox, and L. Phillips for help with data collection; and R. Pearson and S. Phillips provided guidance on the use of Maxent. Thanks to the research assistants at Sabana Field Station and the International Institute for Tropical Forestry (IITF) for logistical support. This manuscript benefited from the comments of two anonymous reviewers. This project was supported by funds from the IITF, US Department of Agriculture, Forest Service, award 012005 and the Columbia Frontiers of Science Fellowship.

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