Neighborhood models of soil pathogen abundance We used likelihood methods and model selection for the analysis of our data (Johnson & Omland, 2004; Canham & Uriarte, 2006). Following the principles of likelihood estimation, we estimated model parameters that maximized the likelihood of observing the pathogen abundance measured in the field given a suite of alternative neighborhood models.
We fitted separate models for each combination of forest type (woodland and closed forest) and pathogen species (P. cinnamomi, Py. spiculum and Pythium spp.). Our analyses of soil-borne pathogen abundance estimated four terms: (1) average potential pathogen abundance (PPA, in cfu g−1) at each of the three sites, and three multipliers that quantified the effects on the average PPA of: (2) local abiotic conditions (expressed in terms of soil texture); (3) the characteristics of the tree neighborhood (expressed in terms of the size, spatial distribution, species and health status of the trees); and (4) the characteristics of the shrub neighborhood (expressed in terms of shrub size). Our full model had the following form:
- (Eqn 1)
We also tried a linear model framework in which the different effects were summed (for a similar approach, see Baribault & Kobe, 2011), but, in general, it showed a poorer performance than the multiplicative model framework (data not shown). PPASite is an estimated parameter that represents the expected pathogen abundance at each site in the absence of sand in the soil texture and of trees and shrubs in the neighborhood (i.e. the abiotic, tree and shrub effects = 1). The three effects in Eqn 1 were modeled using Weibull functions:
- (Eqn 2)
- (Eqn 3)
- (Eqn 4)
where b, c and d are parameters estimated by the analyses determining the sign and magnitude of the abiotic, tree and shrub effects, respectively.
The abiotic effect was modeled as a function of soil texture quantified as the proportion of sand content. Texture was chosen to represent the abiotic driver of pathogen abundance because it is a relatively stable soil property that influences key environmental variables for pathogens (e.g. water availability) and is not easily modified by plants, therefore being independent of the biotic effects in the equation.
The tree effect was modeled as a function of a tree neighborhood index (NITree). This index quantifies the net effect of j = 1,...,n neighboring trees of i = 1,...,s species on pathogen abundance, and was assumed to vary as a direct function of the size (dbh) and as an inverse function of the distance to neighbors following the form:
- (Eqn 5)
where α, β and γ are estimated parameters that determine the shape of the effect of the dbh (α) and distance to neighbors (β and γ) on pathogen abundance. Instead of setting α, β and γ arbitrarily, we tested two different versions of Eqn 5, fixing α to values of zero or unity and letting β and γ vary. A value of α = 1 implies that the effect of a neighbor is proportional to its dbh, whereas a value of α = 0 means that the tree influence on soil pathogen varies as a function of tree density, regardless of size.
We were particularly interested in exploring whether tree effects varied between individuals of different species or health status. For this purpose, we multiplied the net effect of an individual tree by a per capita coefficient (λ) that ranged from − 1 to 1, and allowed for differences between neighbors in their effects (negative or positive) on a target pathogen. We tested four different groupings of neighbor species in Eqn 5 with increasing complexity: a model in which all trees were considered to be equivalent (i.e. fixing λ = 1); a species-specific model that calculated two separate λ values, one for Q. suber and one for the coexisting tree species (either O. europaea or Q. canariensis); a model that also took into account the health status of Q. suber trees, and therefore calculated four separate λ values (healthy Q. suber, slightly defoliated Q. suber, highly defoliated Q. suber and the coexisting tree species); and a model that not only considered alive trees of different species and health status, but also the legacy effect of dead Q. suber trees, calculating five separate λ values.
The shrub effect was modeled as a function of the shrub neighborhood index (NIShrub). This index is a simplified version of the NITree, and quantifies the net effect of j = 1,...,n neighboring shrubs of i = 1,...,s species on pathogen abundance following the form:
- (Eqn 6)
NIShrub was assumed to vary as a direct function of the size (crown area) of neighbor shrubs in a neighborhood of 5 m radius. We decided not to include distance in the calculation of the index, given the already restricted area over which shrubs were mapped and to keep the number of parameters in the models manageable.
Finally, in order to test whether any of the three effects studied (i.e. texture, trees and shrubs) varied among sites of a given forest type, we tried variations of the full model in which the slopes of each effect (i.e. parameter b, c or d) were allowed to vary among sites.
Effect of soil-borne pathogens on seedling emergence and survival We fitted models that estimated seedling emergence or survival at each subplot as a direct function of the pathogen abundance in the soil. We tried both a multiplicative and a linear model framework, the latter offering a better fit to the data. Thus, for each combination of forest type, tree species and pathogen species, seedling emergence and survival were predicted as:
- (Eqn 7)
- (Eqn 8)
where PSESite and PSSSite are the potential seedling emergence and survival, respectively, at each site in the absence of pathogens, and b and c are the slopes of the regressions determining the pathogen effect. We explored the existence of site-dependent pathogen effects by fitting models that allowed the parameters b and c to vary among sites of a given forest type.
Parameter estimation and model selection Following the principle of parsimony, we followed the strategy of systematically reducing the number of parameters in the full model to the simplest model that was not a significantly worse fit than any more complicated model. We used the Akaike Information Criterion corrected for small sample sizes (AICc) to select the best model, with lower AICc values indicating stronger empirical support for a model (Burnham & Anderson, 2002). Pathogen abundance values were modeled using a Poisson error distribution, and seedling emergence and survival using a binomial error distribution. We used simulated annealing, a global optimization procedure, to determine the most likely parameters (i.e. the parameters that maximized the log-likelihood) given our observed data (Goffe et al., 1994). The slope of the regression (with a zero intercept) of observed on predicted pathogen abundance was used to measure bias (with an unbiased model having a slope of unity), and the R2 of the regression was used as a measure of the goodness-of-fit. We used asymptotic two-unit support intervals to assess the strength of evidence for individual parameter estimates (Edwards, 1992). All analyses were performed using software written specifically for this study employing Java (Java SE Runtime Environment v6, Sun Microsystems Inc., Santa Clara, CA, USA).