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Incorporating ecological principles into statistical models for the prediction of species’ distribution and abundance

Ben Stewart-Koster, Edward L. Boone, Mark J. Kennard, Fran Sheldon, Stuart E. Bunn and Julian D. Olden

B. Stewart-Koster (bensk@uw.edu), M. J. Kennard, F. Sheldon and S. E. Bunn, Australian Rivers Inst. and eWater CRC, Griffith Univ., Nathan, QLD 4111, Australia. Present address of BSK: Univ. of Washington, School of Aquatic and Fishery Sciences, Seattle, WA 98105, USA. – E. L. Boone, Dept of Statistical Sciences and Operations Research, Virginia Commonwealth Univ., Richmond, VA 23284, USA. – J. D. Olden, Univ. of Washington, School of Aquatic and Fishery Sciences, Seattle, WA 98105, USA.

Understanding the determinants of species’ distributions and abundances is a central theme in ecology. The development of statistical models to achieve this has a long history and the notion that the model should closely reflect underlying scientific understanding has encouraged ecologists to adopt complex statistical methods as they arise. In this paper we describe a Bayesian hierarchical model that reflects a conceptual ecological model of multi-scaled environmental deter-minants of riverine fish species’ distributions and abundances. We illustrate this with distribution and abundance data of a small-bodied fish species, the Empire gudgeon Hypseleotris galii, in the Mary and Albert Rivers, Queensland, Australia. Specifically, the model sought to address; 1) the extent that landscape-scale abiotic variables can explain the species’ distribution compared to local-scale variables, 2) how local-scale abiotic variables can explain species’ abundances, and 3) how are these local-scale relationships mediated by landscape-scale variables. Overall, the model accounted for around 60% of variation in the distribution and abundance of H. galii. The findings show that the landscape-scale variables explain much of the distribution of the species; however, there was considerable improvement in estimating the species’ distribution with the addition of local-scale variables. There were many strong relationships between abundance and local-scale abiotic variables; however, several of these relationships were mediated by some of the landscape-scale variables. The extent of spatial autocorrelation in the data was relatively low compared to the distances among sampling reaches. Our findings exemplify that Bayesian statistical modelling provides a robust framework for statistical modelling that reflects our ecological understanding. This allows ecologists to address a range of ecological questions with a single unified probability model rather than a series of disconnected analyses.

Understanding patterns and drivers of species’ distributions and abundances has been a long-standing pursuit in ecology (Andrewartha and Birch 1954, Orians 1980). Considerable ecological theory has been developed to explain how envi-ronmental factors across multiple spatiotemporal scales influence the assembly of species into available habitat (Southwood 1977, Turner 1989). The inherent hierarchical structure of many ecological datasets comprising variables that vary markedly across spatial and temporal scales has long complicated the development of statistical models of species’ distribution and abundance (Latimer et al. 2006). Historically, efforts to quantify multi-scaled ecological relationships have relied on a range of statistical meth-ods including regression-based methods (Guisan and Zimmerman 2000), variation partitioning with ordination methods (Cushman and McGarigal 2002) and machine learning algorithms (Olden et al. 2008). Many of these approaches remain limited in their capacity to capture multi-scaled species–environment relationships, in particular the interactive effects of environmental variables across

scales, because of their single level mathematical structure (Diez and Pulliam 2007). Consequently, there is an ongoing and growing need for statistical models that better reflect the structure of the conceptual, or scientific model, of the system under study (Austin 2002, Joseph et al. 2009), particularly for hierarchical systems (McMahon and Diez 2007).

Although prevalent across ecosystem types, riverine landscapes exemplify the challenge of understanding the effect of multi-scaled ecological drivers operating over an environmental hierarchy. This is particularly the case for stream fishes which can be highly mobile, long-lived taxa and therefore may encounter a range of environmen-tal conditions in space and time throughout their life- span (Schlosser 1991, Fausch et al. 2002). The potential range of environmental variation encountered by stream fishes suggests that concepts from landscape ecology, emphasising the importance of scale and environmen-tal heterogeneity, should transfer readily to the aquatic realm (Fausch et al. 2002). Wiens (2002) described six cen-tral themes of landscape ecology as they apply to riverine

Ecography 36: 342–353, 2013 doi: 10.1111/j.1600-0587.2012.07764.x

© 2012 The Authors. Ecography © 2012 Nordic Society Oikos Subject Editor: Jens-Christian Svenning. Accepted 29 June 2012

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ecosystems, and three of these closely relate to the study of multi-scaled species–environment relationships: habi-tat patches differ in quality (theme 1), patch context mat-ters (theme 3), and spatial and temporal scale is important (theme 6). Patch quality, defined by local-scale biotic and abiotic variables is known to regulate species distributions and abundances and this may be affected by the nature of the surrounding landscape – the patch context (Palmer et al. 2000, Wiens 2002). The concept of scale also influences these themes because the scales at which data are collected have the potential to influence outcomes of analyses quantifying species–environment relationships (Crook et al. 2001, Wiens 2002). These three themes rep-resent core elements of a conceptual model that depicts the multi-scaled and hierarchical nature of the environmental processes that influence the distribution and abundance of fish in riverine landscapes (Fig. 1). Biotic and abiotic processes across broad (landscape) scales influence local-scale biotic and abiotic characteristics (Allen and Starr 1982, Poff 1997), supporting the proposition that stream fish assemblages are hierarchically structured by habitat factors at different scales (Fausch et al. 2002). Incorporating the structure of this conceptual model into quantitative models should greatly improve our understanding of the processes driving patterns of distribution and abundance across scales.

In this study we use the conceptual model in Fig. 1 as a framework to develop a hierarchical Bayesian zero-inflated Poisson (ZIP) model to address three common research questions related to the distributions and abundances of freshwater fish species: 1) to what extent do landscape-scale environmental variables influence species distribution (i.e. presence/absence) compared to local-scale variables? 2) How do local-scale abiotic conditions reflecting patch quality influence local-scale species’ abundances? 3) How do relationships between species’ abundances and patch qual-ity change according to landscape-scale variables, or patch context? While these questions are specifically related to

freshwater fishes, they are readily applied to organisms in terrestrial or marine systems.

Methods

Data collection and organisation

The dataset used to illustrate the model consists of distri-bution and abundance data for Hypseleotris galii, a semi-benthic species which favours deeper slow flowing water, typically in low to mid elevations across its range in coastal rivers of sub-tropical to temperate eastern Australia (Pusey et al. 2004). Twenty-eight least disturbed, wadeable stream reaches, typically between 70 and 80 m in length, were sampled in the Mary and Albert Rivers in southeastern Queensland, Australia (Fig. 2a, Kennard et al. 2006). Stream reaches were sampled three times per year (winter, spring and summer) on 7–10 occasions over a 4-yr period between winter 1994 and winter 1997. Within the sam-pling reaches, individual hydraulic units (i.e. riffles, runs or pools) were sampled separately and from here on are referred to as habitat units. Each habitat unit was blocked upstream and downstream with weighted seine nets (11 mm stretched-mesh) to prevent fish movement into or out of the study area and fish assemblages were sampled by mul-tiple pass electrofishing (Smith-Root model 12B Backpack Electroshocker). Further details of sample methods can be found in Kennard et al. (2006).

In this study, abundance data were analysed at the habi-tat unit level and distribution data (i.e. presence/absence) were aggregated at the reach level. Thus, habitat unit level data comprise observations from up to three hydraulic units within each sampling reach resulting in 420 observa-tions and reach level data comprise 247 observations for the species distribution component of the model (Fig. 2b).

A series of local-scale ecologically relevant environmen-tal characteristics were used to describe ‘patch quality’ at

Figure 1. A conceptual model illustrating the relationship between three themes of landscape ecology (Wiens 2002) and how they relate to multi-scaled environmental determinants of fish distribution and abundance in riverine landscapes.

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each reach on each sampling occasion. These were quanti-fied according to a standard protocol described in Kennard et al. (2006, 2007) and described antecedent hydrology, instream structure and local hydraulics (Table 1). Mean daily runoff (MDR), and CV of daily discharge (CVD) in the 4 months prior to each sampling occasion were used in the model to describe recent antecedent hydrologic conditions. The proportion of total area covered by sub-merged, emergent and overhanging vegetation (PLANT) and the proportion of total area covered by instream woody debris (WOOD) were selected to describe instream structures at the habitat unit-scale. Average depth (AVD) and stream gradient (GRAD) were used as descriptors of hydraulics at the habitat unit-scale. Three landscape-scale predictor variables were included in the model to describe different elements of ‘patch context’ and large-scale envi-ronmental characteristics of the sampling reaches; the dis-tance to the river mouth (DISTM), reach elevation (ELEV) and the long-term mean daily runoff (LMDRU). The square of the predictor variables DISTM and ELEV were

also included to account for possible non-linear relation-ships. All local and landscape-scale environmental predic-tor variables were minimally redundant, with the absolute Pearson’s moment correlation coefficients among predictor variables all 0.5.

Hierarchical Bayesian zero-inflated Poisson regression model

We developed a Bayesian hierarchical zero-inflated Poisson (ZIP) mixture model similar to that developed by Boone et al. (2012). The model estimates the probability of species absence as a mixture of two separate processes; a Bernoulli process that determines the species’ presence or absence and a Poisson process that determines the species abun-dance (Martin et al. 2005, Boone et al. 2012). Parameters describing ecological relationships within the model are esti-mated at several levels of the hierarchy, described as either Habitat unit, Reach, River or Region (Fig. 3). Parameters

Figure 2. (a) Location of fish sampling reaches in the Mary and Albert Rivers, southeastern Queensland (b) Schematic of the sampling design at a sampling reach. Twenty eight reaches across the Mary River (17 reaches) and the Albert River (11 reaches) were sampled up to ten times each resulting in 247 observations aggregated at the reach scale across the two rivers. Nested within the reaches were up to three contiguous habitat units that were sampled separately on each occasion, resulting in 420 observations at the habitat unit level across the two rivers.

Table 1. Predictor variables included in the models organised by scale. The classes of variables included in the different components of the most probable model are marked with an asterisk.

Scale Class Variable Description Model component

Landscape Distance to river mouth

DISTM distance of sampling reach to river mouth (km) Bernoulli and Poisson*DISTM2 quadratic of DISTM

Elevation ELEV elevation of sampling reach (m) Bernoulli*Elevation ELEV2 quadratic of ELEVHydrology LMDRU long term mean daily runoff Poisson

Local Hydraulics GRAD average gradient of habitat unit Poisson*Hydraulics AVD average depth of habitat unitHydrology MDR mean daily runoff 4 months prior to sampling Poisson*Hydrology CVD CV of daily discharge 4 months prior to samplingInstream

structurePLANT proportion of habitat unit area covered by plants Poisson*WOOD proportion of habitat unit area covered by woody debris

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Poisson process dependent on local and landscape-scale variables. The mixture of the Bernoulli and Poisson processes produces a Zero-inflated Poisson distribution given by:

P( )

( )

( )Y y

p p e y

pe

yijk ijk

ij ij ijk

ij

y

ijk

ijk

ijk

ijk5 5

2 5

2

2

2

1 0

1

µ

µ µ

iijkijky!

0

(1)

Here Yijk is the observed abundance of H. galii at the ith reach in the jth river on the kth sampling occasion; pij is the probability of the absence of H. galii at the ith reach in the jth river; and mijk is the expected abundance at the ith reach in the jth river on the kth sampling occasion. Note that the probability that the species is absent from the ith reach in the ZIP formulation is pij (12pij)e2mijk which accounts for the absences due to the Bernoulli process as well as the absences due to the Poisson. The Poisson process governs

the species abundance with probability ( )!

122

pe

yij

y

ijk

ijk

ijk

ijkµ µ.

The Bernoulli process is modelled via a logistic regression with probability of absence at the ith reach in the jth river, pij, (Eq. 2):

logit(pij) 5 Tmijbm (2)

Here Tmij is the mth predictor variable at the jth reach in the ith river from a set of landscape-scale variables that vary in

at the Habitat unit level vary in space and time and depend on parameters estimated at the Reach level; parameters at the Reach level vary in space only, both within and between the two rivers and depend on parameters estimated at the River level; parameters estimated at the River level only vary between the two rivers and parameters at the Region level are fixed effects that do not vary in space or time.

Commonly the ZIP model structure is used where there are too many species absences in the dataset which cannot be accommodated by a standard Poisson distribu-tion. However, the main purpose in this case is to examine the importance of the scale of explanatory variables with respect to the numerical resolution of the response variables (sensu Rahel 1990) thereby addressing the first research question. A species may be truly absent from a stream reach due to abiotic variables at several scales. For example, a sampling reach may be outside the geographic range of a given species, in which case it will never be observed there (Lancaster and Downes 2010). Conversely, a sampling reach may be within the geographic range of the species, but the species may be absent due to local-scale abiotic variables such as the lack of suitable microhabitat, intolerable water quality or an inability to colonise the location due to poor connectivity (Jackson et al. 2001). Therefore we used predictor variables at different scales for each of the Bernoulli and Poisson processes. The probabil-ity of species’ presence/absence is modelled as a Bernoulli process dependent on landscape-scale variables while the probability of the observed abundance is modelled as a

Figure 3. Schematic of the hierarchical structure of the model showing the two components of the model, Bernoulli and Poisson as left and right hand branches respectively. The Bernoulli component estimates the probability of presence given landscape-scale variables that vary in space alone. The Poisson component estimates the expected abundance given a hierarchy of predictor variables including local-scale habitat variables that vary through time at each reach and landscape-scale variables that vary in space alone. The level of the model at which each parameter is estimated is described as either, Habitat unit, Reach, River or Region. Parameter definitions are provided in the model description.

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of the regression coefficients ϕli, which is estimated by the exponential covariance model in Eq. 7 below. The para-meters ωci resolve question 3: if patch context did not affect local-scale species–environment relationships, we would expect the posterior distributions of ωci coefficients to have a median of zero with relatively high uncertainty. These parameters also account for any non-stationarity in the local-scale species–environment relationships and mean abundance with respect to the landscape-scale variables (e.g. distance to the river mouth) which vary in space only.

We also quantified the spatial autocorrelation in mean species abundances as well as the local-scale species– environment relationships, glij estimated in the hierarchi-cal component of the model (Boone et al. 2012). Just as these relationships may vary according to patch context, variation in these relationships may also display some level of spatial autocorrelation. An exponential spatial autocorre-lation model to quantify the spatial dependencies (Eq. 7);

Cov s s eij i j li li ij li

s sli

ij i j

( , ) ( , )′ ′ρ

ρ σ σ′ ′

5 52

2

2 2 3 (7)

where, ρli is the effective range of the spatial correlation for the mean of the Poisson regression coefficient ϕli, and ||sij 2 si ′j′|| is absolute value of the instream distance along the river between sites, ij and i ′j′ (Peterson et al. 2007). The effective range of spatial autocorrelation indicates the extent to which species abundance and local-scale species– environment relationships are similar due to spatial proximity. The distance metric used is symmetric hydro-logic distance between pairs of sites which equates to the distance along the stream network as the fish swims and allows correlation both upstream and downstream along the river network (Peterson et al. 2007). The choice of symmetric hydrologic distance is reasonable given the capacity of fishes in general and H. galii in particular to move upstream against the river flow. Note that using the instream hydrologic distance to quantify spatial autocorre-lation ensures the correlation structure is isotropic as spa-tial autocorrelation is constrained to the stream channel (Ver Hoef and Peterson 2010).

Generally, sites are arrayed quite wide apart through-out each catchment with greater distances among sites in the Mary River due to its longer length (median intersite distances 46 and 111 km in the Albert and Mary Rivers respectively). However, there are four sites in each catch-ment that are within 10 km of each other. The posterior distributions of ρli which is the effective range of auto-correlation for the local-scale regression coefficients, ϕli, identifies the instream distance between two sampling reaches where the spatial autocorrelation is negligible.

To complete the Bayesian specification of this likelihood structure, we used uninformative proper prior distributions with low precision parameters where required.

β ∼∼

σ ∼ χρ ∼

ξ ∼

mi

li

li

n

N Ia N I

Inv

( , . )( , . )

( , )exp( )

0 0 0010 0 001

1 11

2 2

2

2

IInv2χ2 1 1( , )

space but are constant in time and bm is the logistic regres-sion coefficient for the mth predictor variable and b0 is the intercept term. The coefficients that describe the relation-ship between landscape-scale variables and the probability of species’ presence/absence are regional level parameters and therefore do not vary between the two rivers (Fig. 3).

As stated above, spatiotemporal distribution and abundance at the local-scale is modelled via a hierarchical Poisson regression (Eq. 3, 4) with rate parameter mijk at the habitat unit level (Fig. 3) adjusted for the probability of the species’ absence at the reach (Eq. 2). The local level of the Poisson regression therefore estimates the species’ relationship to patch quality and is modelled by Habitat unit and Reach level predictor variables with coefficients estimated at the Reach level (Fig. 3). Species abundance is scaled by the reach area, Areaijk so that the density of indi-viduals 1022 m, aijk, (Eq. 3) can account for spatial and temporal variation in sampling area (sensu Wyatt 2002);

mijk 5 aijk 3 Areaijk/10 (3)

log(aijk) 5 Xlijkglij (4)

where mijk is the rate parameter of the Poisson distribution, in this case the expected raw count at the kth sampling occasion at the jth reach in the ith river; aijk is the cor-responding fish density 10 m22; Xlijk is the lth local-scale predictor variable at the jth reach in the ith river on the kth sampling occasion; and glij is the Poisson regression coef-ficient for the lth local-scale predictor variable describing patch quality at the jth reach in the ith river.

This model structure specifically reflects, and therefore tests, the hypothesis in question 1, that a separate ecologi-cal process generates the species’ spatial distribution within each river from that which generates the species’ spatiotem-poral distribution and abundance at the local-scale (Martin et al. 2005). Under the Bayesian framework, if the species’ spatial distribution were not driven by the landscape-scale variables in the Bernoulli component of the model, this component would have a poor classification rate compared to that of the Poisson component.

The hierarchical nature of the Poisson component addresses question 3 regarding the importance of the land-scape context of a given patch to species–environment relationships. The local-scale Poisson regression for species abundance estimates separate regression coefficients for species–environment relationships at each sampling reach glij, however, these may vary systematically with landscape level predictor variables. These relationships are described by the River level regression coefficients (Fig. 3; Eq. 5, 6);

glij ~ N (ϕli, S(ρli, s2li)) (5)

ϕli 5 Zcij ωci (6)

where ϕli is the mean of the posterior distribution of the Poisson regression parameters glij, that describe the local-scale species–environment relationships; Zcij is the cth landscape-scale variable at the jth site in the ith river; ωci is the hierarchical regression coefficient for the cth landscape-scale variable in the ith river; ω0i is the intercept term for the ith river; and S(ρli, s

2li) is the variance-covariance matrix

347

approach uses the full posterior distribution of model prob-ability thereby incorporating the uncertainty surrounding each one (Boone et al. 2005).

Model interpretation and assessment

The structure of the model quantifies how the species’ spatial distribution is related to landscape-scale abiotic vari-ables (Bernoulli) as well as how spatiotemporal variation in distribution and abundance is related to landscape and local-scale abiotic variables (Poisson). Both of these com-ponents of the model were considered when assessing the model accuracy with respect to species’ presence/absence to address question 1. The Bernoulli component quantifies the species’ spatial distribution at the Reach level (n 5 247) rather than the habitat unit of the Poisson component (n 5 420; Fig. 2, 3). To compare the classification rates of each component of the model, the observed species dis-tribution was pooled to the Reach scale to compare the extent to which observations are correctly classified as pres-ent/absent due to either the Bernoulli or Poisson process. A threshold of 0.5 was chosen to delineate the probability of presence or absence under the Bernoulli component of the model as this maximised predictive accuracy. Under the Poisson component, the species was expected to be pres-ent when the expected abundance was at least 1. To help address questions 2 and 3, we calculated the multilevel R2 of Gelman and Pardoe (2006), R 2M, to quantify the impor-tance of patch context as identified by the hierarchical regres-sion coefficients in the Poisson component of the model:

RMk

k

kk

2E

E5 21

V

V

τ

ϕ

where ϕk is the hierarchical regression coefficient of interest;

Vk is the variance estimator given by

k k kk

n

xn

x xV 52

25

11

2

1

( )∑and E V

k kτ

indicates the posterior mean, or expected value,

of the variance of the landscape-scale model errors, τk; and

ijkkVϕ is the variance of the posterior distribution of

the local-scale regression parameter. As noted by Gelman and Pardoe (2006), R 2M may yield negative values indicating an extremely poor fit where the variance in the data is smaller than model error variance.

Results

The most probable model for the distribution and abundance of H. galii had a posterior probability of 0.99 (all other models 0.01) and included all potential vari-ables except the distance to river mouth variables in the Bernoulli component and long term mean daily runoff in the landscape level of the Poisson component (Table 1). The overall fit of the model was good with a pseudo-R 2 of the most probable model (the square of Pearson’s cor-relation coefficient of observed and expected values under the model) of 0.58.

Using these relatively uninformative prior distributions means that the data provide the information for parameter estimation in the model (Gelman et al. 2004).

In summary, modelling species distribution and abun-dance as a mixture of two different processes via this ZIP structure addresses the extent that species distribution is related to landscape-scale environmental conditions while local-scale abundance is related to local-scale abiotic vari-ables (question 1). The local-scale Poisson regression quanti-fies the importance of patch quality to species abundance (question 2), while the hierarchical component of the Poisson regression can quantify the extent to which patch context influences local-scale abundance and species– environment relationships (question 3).

Parameter estimation and model selection

Parameter estimation was done by exploring the posterior distribution of parameters via Metropolis–Hastings Markov Chain Monte Carlo sampling embedded in a Gibbs sampler (Boone et al. 2012). The MCMC sampling was implemented in Octave 3.03 on a Beowulf cluster machine. Sampling times varied depending on the model complexity and the processing power of each node on the cluster with the most complex model taking about six to eight seconds per sample. To speed up the process of con-vergence and ensure high MCMC acceptance rates, we tuned the parameter estimates with a series of short pilot MCMC chains to identify sensible initial values and step values (Boone et al. 2012). After tuning these values, we ran five separate chains for 12 000 iterations to estimate each model. We discarded the first 2000 iterations as burn-in as it appeared the chains had achieved convergence after 1500 iterations. The remaining 10 000 samples from each of the five chains were combined and all inferences are based on these 50 000 posterior samples. To assess con-vergence we visually inspected the combined posterior samples and calculated the potential scale reduction factor, R, (Gelman et al. 2004), which was 1.05 for all para-meters indicating convergence. All convergence analysis and inference was done in the R statistical environment (R Development Core Team).

Given the complexity of the model we took a strategic approach to variable selection. Rather than running every possible combination of models (a total of 362 880), vari-ables were included and excluded from the models in pairs according to ‘Scale’ and ‘Class’ (Table 1). For example in a model where local-scale variables of Class ‘Instream structure’ were included and Class ‘Hydraulics’ were excluded, both PLANTS and WOOD were included as a pair and both GRAD and AVD were excluded as a pair (Table 1). Similarly, in a model where landscape-scale variables of Class ‘Elevation’ were included, both ELEV and ELEV2 were included as a pair (Table 1). With potential predictor variables being included or excluded in pairs in this way, there were a possible 63 different vari-able combinations from which to select the most probable model. We calculated the posterior model probability of each of the 63 fitted models using a uniform prior, thereby giving no model prior preference over any other. This

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Figure 4. (a) Plot of observed abundance of H. galii against the expected values under the model. Circles reflect habitat units in the Mary River and triangles reflect habitat units in the Albert River. Note the expected values are the result of the mixture of processes governing the probability of species presence at the sampling reach given the landscape-scale abiotic variables in the Bernoulli component and the expected abundance at the habitat unit given the landscape and local-scale abiotic variables in the Poisson component of the model, n 5 420. (b) Probability of presence of Hypseleotris galii across all elevations sampled in the Mary and Albert Rivers. The solid curve indicates the median of the posterior probability of presence while the dashed curves indicate the 95% credible band surrounding this probability.

Table 2. The number of local-scale regression coefficients, glij, with a 95% credible interval that is either above () or below (2) zero for each local-scale abiotic variable in each river.

GRAD AVD CVD MDR PLANT WOOD

– – – – – –

Mary (17 reaches)

1 9 5 5 1 3 7 4 1 5 1 7

Albert (11 reaches)

1 2 1 0 3 1 1 2 0 5 1 3

Multi-scaled correlates of species’ distribution

In answering the first ecological question, the results sug-gested that both local and landscape-scale variables influ-enced the species’ distribution. The correct classification rate of the Bernoulli component was 0.66 with Cohen’s Kappa 5 0.15. Of particular note was the false positive rate (FPR) of 0.32 or 80 observations. This suggests that about a third of the reach-scale absences are within the expected spatial distribution of the species, which shows a peak in probability of presence in the mid elevations and begins to decline toward the upper limit of sampling in this study with model uncertainty increasing with elevation (Fig. 4b). The relatively high FPR and the high uncertainty around the expected spatial distribution may be due to local-scale spatiotemporal habitat variables rendering the reach unsuitable at certain times. Converting expected non-zero abundances from the Poisson component to presences and aggregating at the Reach scale for direct comparison with the Bernoulli component showed an improvement in classification accuracy due to the Poisson component from 0.66 to 0.85 with Cohen’s Kappa increasing to 0.67. The FPR of the Poisson component was also lower than the Bernoulli at 0.09, or 25 observations. This indicates that there are 55 observations that were within the expected spatial distribution of H. galii, where it was absent due to unsuitable local-scale abiotic variables. These results address the first question and suggest that the landscape-scale variables do not exclusively influence the distribu-tion of the species. Rather, there is considerable temporal variation in the species’ distribution which could be explained by local-scale environmental variation.

Species’ abundance and patch quality and context

The model quantified several relationships between species abundance and patch quality, thereby addressing the sec-ond ecological question; however, these were not nec-essarily uniform across the sampling reaches. For some local-scale habitat variables, there was a fairly consistent directional relationship to abundance. For example, in the Mary River stream gradient (GRAD) was found to have a generally negative effect on abundance with the 95% credible interval of the local-scale regression coefficient being less than zero at nine out of the 17 sampling reaches and non-significant at 7 others (Table 2). Similarly, the presence of higher plant cover (PLANT) in both the Mary and Albert Rivers showed a negative relationship to abun-dance with five coefficients in each river having a 95% credible interval below zero (Table 2). In contrast, the rela-tionships between abundance and other habitat variables were quite different in direction across reaches. For example, the relationships to mean daily runoff (MDR) and average depth (AVD) varied in direction across the rivers (Table 2).

349

These results indicate that patch quality does influence species abundance, however, the variable findings both within and between rivers suggest that other factors, such as patch context, may also be influencing abundance.

Several hierarchical relationships showed very high probabilities indicating abundance was related to distance to the river mouth, both directly and indirectly via an effect of patch context (Table 3). Credible intervals for the reach level intercepts indicate a probable direct relationship between mean abundance at the Reach scale and distance to the river mouth (Table 3). The values for R 2M for the reach level intercepts, which indicate how much variation in mean abundance was explained by distance to the river mouth, were 0.21 for the Mary and 0.38 for the Albert River. This direct relationship with abundance suggests that distance to the river mouth plays a role in defining patch quality as well as patch context.

Hierarchical coefficients for the local-scale habitat vari-ables indicate that the effect of patch context depends on patch quality in some cases (Table 3). For example, in the Mary River, relationships to the two hydraulic variables follow a similar pattern with increasing average depth and stream gradient having a negative association with spe-cies abundance at low and high levels of distance to the river mouth. Towards the centre of the distance to mouth gradient, there was a positive association between aver-age depth and abundance and little or no association with gradient (Fig. 5a, b). The local level hydrologic relation-ships also show probable hierarchical associations with distance to the river mouth (Table 3; Fig. 5c, d); however, there is considerable uncertainty surrounding these relation-ships evidenced by the relatively wide intervals surround-ing regression coefficients. For example, close to the mouth of the Albert River, there was a positive association between abundance and increased mean daily runoff, which tended toward a negative association with relatively high uncertainty further upstream.

Spatial autocorrelation

The effective range of autocorrelation, ρli, was relatively low for both species’ abundances and relationships between abundance and patch quality. The medians of the posterior distributions of ρli have a range of between 3.5 and 10.5 km across both rivers (Table 3). Considering the range of pairwise distances among the sampling reaches in both rivers, this is relatively low and suggests that the spatial variation in local-scale abundance and relationships to patch quality is not due to spatial proximity of sites. With the exception of woody debris, the values are generally similar for each local-scale variable across both rivers.

Discussion

Analyses of species distributions and abundances are often limited to description of pattern without a clear under-standing of process. However, modern statistical methods provide an opportunity to increase ecological realism in post- hoc analyses of species’ distributions and abundances. While it is difficult to identify underlying mechanisms with such Tabl

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Wilson et al. 2010). Frequently these models involve sepa-rate modelling steps for each ecological process (Pearson et al. 2004). Our model achieves this within a single inte-grated framework and could easily be applied to other sys-tems or modified for other hypotheses.

Scale-related determinants of species’ distribution and abundance

Previous work has supported the hypothesis that riverine fish distributions and abundances are influenced by envi-ronmental processes variables at different scales (Fausch et al. 2002). Species’ distributions are thought to be driven primarily by landscape-scale environmental factors, while fluctuations in abundance may be related to local-scale habitat conditions (Pusey et al. 2000, Kennard et al. 2007). However, the results of the present study did not support these hypotheses in the case of H. galii. There were a con-siderable number of absences through time that were related to local-scale habitat variables, or patch quality, rather than landscape-scale variables. Consistent with the concept of hierarchical filters (Poff 1997), Kramer et al. (1997) described the selection of habitat as a hierarchical process whereby a species ‘selects’ a region, then a major habitat type. Subsequent ‘selections’ relate to successively fine- scale ‘decisions’ according to the local habitat that indi-viduals encounter. The results of the present study are not

analyses, the aim of this study was to develop a statistical model that closely mirrored a conceptual model for multi-scaled determinants of species distributions and abundances. Each link of the conceptual model (Fig. 1) was represented by a parameter in the statistical model. This provided a test of the entire conceptual ecological model and its component parts. It is well established that Bayesian hierarchical models provide an opportunity to develop more complex statistical models (Cressie et al. 2009, Webb and King 2009), and we have shown that they can be tailored specifically to fit a conceptual ecological model by modifying the ZIP model structure (Boone et al. 2012) to incorporate variables at different scales in each model component.

Our hierarchical modelling focused on a single fresh-water fish species in eastern Australia; however, multi-scale ecological questions span many disciplines of ecology (Pearson et al. 2004, Latimer et al. 2006). The concepts of landscape ecology and metacommunity ecology specifi-cally integrate the consideration of scale into conceptual models of species dynamics (Wiens 2002, Leibold et al. 2004). In addition, there may be separate processes influ-encing an ecological phenomenon such as the distribu-tion of a species across the landscape (Nielsen et al. 2005). Quantitative models that incorporate separate processes that may regulate the distribution and abundance of species have been developed for terrestrial organisms such as plants (Guisan et al. 1998) and mammals (Nielsen et al. 2005,

Figure 5. Plots of reach level coefficients along the landscape level gradient of distance to the river mouth (km) for selected local-scale variables in each river. Each set of points indicates the variation in local-scale regression coefficients for each variable through the landscape in the Albert River (triangles) and Mary River (circles). The corresponding regression line shows the relationship between the reach level coefficients and distance to the river mouth, defined by the hierarchical parameters in Table 3.

351

variation in the relationship between stream gradient and the abundance of chinook salmon in Oregon streams, with the variation likely related to competitive interactions with different species present in different parts of the catchment. A similar interaction could also reflect the relationship between water velocity and stream size, where larger streams provide more velocity refugia, in the form of large woody debris, deeper pools and smoother flow near the substrate than small streams (Deschênes and Rodriguez 2007).

Spatial autocorrelation

There was little spatial autocorrelation in either the mean abundance of H. galii or its local-scale abiotic rela-tionships in both study rivers. This finding may be a con-sequence of the sampling resolution which is relatively coarse (Dormann 2007), the method used to quantify spa-tial autocorrelation (Dormann et al. 2007) or the metric used to define pairwise distances. Spatial proximity can be quantified by several distance measures such as Euclidean distance or instream distance measures that follow the river channel (symmetric or asymmetric) as used in this study. Hydrologic distance measures can be weighted according to recent river flows to account for the effect of this important variable in aquatic systems (Peterson et al. 2007). This would be suitable when modelling water quality variables or organisms that rely on downstream drift for dispersal, however, for mobile species such as H. galii, which make upstream migrations such weighting may not be preferred. Further, in spite of the conceptual appeal of hydrologic distance in riverine landscapes, there may also be a case for the use of Euclidean distance on some occa-sions. For example, where processes related to the regional climate or geology drive autocorrelation in organism abun-dances, instream hydrologic distance metrics could overes-timate the ecological distance between two sampling sites. Determining which distance metric to use in any spatial modelling exercise should involve careful consideration of these issues.

Incorporating ecological principles into statistical models

There has been considerable debate over the merits of applying Bayesian statistical formulations to complex pro-cess models of ecological data (Lavine 2010). Traditionally, complex process models have been outside the domain of statistical modelling; however, more recently Bayesian approaches have been used to quantify parameters in such models (Heisey et al. 2010, Harrison et al. 2011, Pagel and Schurr 2012). Criticisms of these models have focused on the potential for spurious results due to a lack of under-standing of highly complex models themselves (Hodges 2010), the underestimated influence of prior distribu-tions (Lele 2010), the required assumptions for parameter estimation (Hodges 2010) and the potential for the misrepresentation of biological and ecological processes (LaDeau 2010). There exists a continuum of statistical model complexity, where at one end are simple models with relatively few assumptions and data requirements, but

inconsistent with this hypothesis. They suggest that in the Mary and Albert rivers, this species inhabits mid to upper elevations and is likely to be absent at low elevations. Within this landscape-scale spatial distribution, the pres-ence or absence of H. galii at a particular stream reach was temporally dynamic suggesting that local-scale abiotic con-ditions are not only related to local-scale abundance but also presence/absence. Hypseleotris galii occurs widely through-out eastern Australia and is thought to be a mobile spe-cies capable of relatively long distance movements within a catchment (Pusey et al. 2004). Its mobility would account for the spatial and temporal variation in presence/absence within the two rivers in response to patch quality, as indi-viduals may well move out of previously suitable habitat as patch quality changes with seasonal environmental variation.

Species’ abundance and the importance of patch quality and context

There was a relatively strong relationship between the abundance of H. galii and patch quality, as defined by the local-scale environmental variables. The credible intervals of many of the reach level regression coefficients describ-ing relationships to local-scale hydraulics, antecedent flows and instream structure were greater or less than zero, indi-cating that local-scale abundance was strongly related to these aspects of local environmental conditions. Variation in the abundance of stream fish has been shown to be strongly related to variation in local-scale abiotic conditions (Schlosser 1991, Jackson et al. 2001). While the results of this study are consistent with this expectation, they should not be viewed as conclusive evidence that these local-scale abiotic variables are the most important to H. galii distribution and abundance. The most probable model, which identified relationships to in-stream hydrau-lics, structural habitat and antecedent flows at the local-scale, is only the most probable model of the 63 tested. Other models which were not tested, for example with different combinations of predictor variables, includ-ing biotic information (e.g. presence of competitors and/or predators) or different antecedent hydrologic metrics, may identify a more probable model than the final model identified in this study. Finally, the considerable variation in the reach level relationships that can be explained by the abiotic variables at the landscape-scale suggests that patch context is also an important factor in species’ spatiotempo-ral distribution and abundance.

The context of any given habitat patch within the broader landscape may influence species distribution and abundance, as well as affecting the relationships between species abundance and patch quality (Wiens 2002). We found several relationships between the abundance of H. galii and local-scale variables defining patch quality such as stream gradient that were dependent on patch context. Such cross-scale interactions between local-scale variables and patch context have been found in other sys-tems and may result from varied ecological mechanisms (Gresswell et al. 2006, Deschênes and Rodriguez 2007). For example, Torgersen et al. (2006) identified spatial

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Acknowledgements – We thank Erin Peterson for providing the data for and assistance with the hydrologic distance metric. We thank Keith Gido, Carmel Pollino and Angela Arthington for helpful comments on earlier drafts of this manuscript. We also thank Antoine Guisan and Ingolf Kühn for excellent reviews of the manuscript. This research was financially supported by eWater Cooperative Research Centre and the former Land and Water Resources Research and Development Corporation (LWRRDC). BSK was supported by a postgraduate scholarship from eWater CRC.

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fully estimable parameters describing ecological patterns, and at the other are highly complex process models requir-ing many assumptions and large amounts of data (Hodges 2010, Harrison et al. 2011). Bayesian approaches can be used to quantify models all along this continuum, includ-ing process models of ecological dynamics. Our approach aimed to find an intermediate position between these two extremes by developing the systematic component of the model according to predominant themes in landscape ecology (Wiens 2002). In this way we developed a correla-tive model, using predictor variables that may be surrogates for mechanistic relationships driving patterns in species distribution and abundance, with a likelihood structure that should more closely reflect the system under study. This model could then be used to resolve several research questions regarding environmental determinants of the distribution and abundance of the species across multiple scales. While it is true that each of the three questions could have been answered using other analytical methods, the Bayesian hierarchical ZIP model provided a unified frame-work to resolve them within a single model rather than a series of potentially disconnected analyses.

While such models are necessarily very complex they have a number of distinct advantages for inference about ecological relationships and predicting ecological responses to future environmental change. Using a single model such as this ensures that the regression coefficients are esti-mated conditional upon both of the ecological processes and each of the environmental variables in the model. For example, the estimate of the effect of average depth on the abundance of H. galii is made while controlling for the probability that the species is present as well the effect of the other variables in the model that influence abundance. The implication of this for prediction of changes in the distribution and abundance is that we can be very specific about the likely ecological process (the Bernoulli or the Poisson) that will drive any expected changes given future environmental change. For example, habitat alteration may affect the expected abundance of the species at a local-scale (the Poisson) while large-scale environmental change may alter the expected spatial distribution of the species (the Bernoulli). Fitting the model in the Bayesian frame-work provides estimates of the full posterior distribution of each parameter which ensures uncertainty is propagated throughout the model (Gelman et al. 2004).

We have shown how researchers can use a conceptual model to guide the development of the statistical model to quantify ecological relationships. The complexity of the statistical model will depend on the characteristics of the data and the conceptual understanding of the system. This may range from relatively complex models such as ours to relatively simple hierarchical models that adequately reflect the conceptual understanding of a system (Latimer et al. 2006, Webb and King 2009, Stewart-Koster et al. 2011). This can serve to quantify ecological relationships in a more realistic manner than traditional single level regression models. While these methods may not be acces-sible to all ecologists at present, we believe that develop-ing more ecologically realistic statistical models can better address multiscale ecological processes and subsequently predict ecological responses to environmental changes.

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