Supplementary Material
Impact assessment revisited – improving the theoretical basis for management of invasive alien plant species
Jan Thiele, Johannes Kollmann, Bo Markussen and Annette Otte
S1 Mathematical basis
Let A be the abundance, and let E be the effect per individual (per-capita effect). The effectdepends on the abundance via the possibly non-linear function f,
E = f(A),
and the abundance follows a probability distribution φ. From this it follows
that the mean abundance Ā, the mean effect Ē, and the mean local per-area
effect are given by:
S2 Discrepancy of eqn (1) and the basic linear formula
The discrepancy between the basic linear formula (Ā×Ē; Parker et al.,1999) and the local per-area effect is:
, which equals cov(A, E).
Constant f:
For constant f we have ΔP = 0. Thus, the basic linear formula is exact.
Linear f:
For linearf, i.e. f(A) = γ0 + γ1·A, we have
Ā= mean(φ),
Ē= γ0 + γ1·mean(φ),
= γ0· mean(φ) + γ1·(mean(φ)2 + variance(φ)),
and hence
= γ1· variance(φ).
Thus, the discrepancy depends on the slope of f and the variance of the abundancedistribution.
Sigmoid f:
For sigmoid f, it is not possible to find a general mathematical expression describing the discrepancy between the two formulas. Rather the discrepancy depends on the characteristics of the abundance distribution. Thus, calculations need to be based on a particular (hypothetical) distribution, such as in the example below.
S3 Impact computations for sigmoid f
S3.1 Abundance distributions
To demonstratehowfor a sigmoid fthe discrepancy between eqn (1) and the basic linear formula proposed by Parker et al. (1999) may depend on the distribution of invasive species abundance, we computed impacts for different hypothetical distribution patterns. For this purpose, we used Betadistributions with different means and variances to model varying patterns of abundance distribution (Fig.S1). In these examples, abundance A is understood as a number between 0 and 1, where 1 corresponds to maximum abundance. The beta-distributions on the interval [0,1] used for the abundanceA are parameterized by two numbers and have a probability density function given by
.
Here is the Beta function, which normalizes the probability density function such that. If,or and, then the Beta-distribution has modus in 0. If,or and, then the Betadistribution has modus in 1.Ifand, then the Beta-distribution has modus in .The mean and the variance of the Beta-distribution with parameters are given by:
,
.
In Figure 2, the graphs are displayed as functions of the variance for different instances of the mean. Since the mean and the variance in a Beta-distribution satisfy the inequality 0 < variancemean · (1-mean), these graphs end at the upper bound for variance given bymean · (1-mean).
S3.2Sigmoid function
The sigmoid relationship fused between abundance A and per-capita effect E in the panels in Figure 2 is given by
(seeFig.S2).
Figure S1. Beta distributions with different means used for impact computations shown in Fig. 2. In addition to the distributions shown in this figure, we also used mean=0.7 and mean=0.9 which mirror mean = 0.3 and 0.1, respectively.
Figure S2. Sigmoid function relating per-capita Eeffect to abundanceA.