We estimated invasion speed using Neubert & Caswell's (2000) integrodifference equation approach for a stage-structured population. We developed code in the computer program R (R Core Development Team. 2004). A life-cycle diagram for P. nigra is shown in Fig. 2 and the corresponding stage-structured population projection matrix, A, is shown below (equation 1); parameter estimates and their sources are given in Table 1.
Figure 2. Life-history diagram and transition parameters for P. nigra. Juvenile 3 and Juvenile 4 stages are not shown. Transitions between all juvenile stages not shown are given by sj.
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Table 1. Parameter values for the life-history matrix, A, and dispersal parameters for dispersal matrix, K
| ||Point estimate||Uncertainty range, minimum–maximum or distribution||Description|
| s a ||0·983||0·971–0·988||Adult survival|
| f 1 ||2500||460–5920||Fecundity for adult 1 trees|
| f 2 ||14160||2500–44280||Fecundity for adult 2 trees|
| z ||0||0, 0·5, 0·9||Grazing intensity, 0 = no grazing|
| p ||0·1||0·001–0·2||Proportion of seeds in long-distance dispersal kernel|
| q ||0·62||0·11–0·78||Probability of remaining in final juvenile class, delay in reproductive maturity|
| r ||0·931||0·898–0·956||Probability of remaining in adult 1 class|
| b ||0·188||SD = 0·043||Seedling establishment for grassland|
|0·058||SD = 0·035||Seedling establishment for shrubland|
| s j ||0·976||0·952–0·999||Seedling to juvenile, for grassland|
|0·63||0·063–0·999||Seedling to juvenile, for shrubland|
| v ||0·5|| ||Seed viability for adult 1 trees|
|0·75|| ||Seed viability for adult 2 trees|
|β||1287||1148–1888||Long-distance dispersal kernel shape parameter|
|σ||27·39||24·37–31·1||Standard deviation of short-distance dispersal|
- ( eqn 1)
The parameters are: z, intensity of grazing (0, no grazing, up to 1, where grazing completely prevents establishment); b, probability of a seed establishing; f1, fecundity of the adult 1 class; f2, fecundity of the adult 2 class; sj, survival of juveniles; q, reproductive delay parameter (0, immediate progression, up to 1, where all trees are retained in the juvenile 5 class); sa, adult survival; r, retention parameter for the adult 1 class (0, immediate progression, up to 1, where all trees are retained in the Adult 1 class) (Table 1). The matrix A describes life-stage transitions with a yearly time-step, including one juvenile stage class where residence is dependent on grazing, z, and three further classes where, if juveniles survive, they progress yearly. The final juvenile stage class has a reproductive delay parameter, q; individuals can be retained in that class before moving on to the first reproductive adult class (Fig. 2). Yearly progression through the juvenile classes is introduced to ensure that no trees reproduce at ages younger than an observed minimum. Parameters are assumed to be typical of low-density populations and both classes of adults have the same dispersal kernels. The dispersal kernels are contained in the matrix K of the same dimensions as A. If there is no dispersal the kernel is the Dirac delta function. Where dispersal does occur, the kernel is a mixture of short-distance (Gaussian) and long-distance (negative exponential) dispersal (see for a similar approach Clark 1998):
- (eqn 2)
where P(d) is the probability of travelling a distance d, p is the probability of a seed entering the long-distance dispersal kernel, Pshort(d) is the Gaussian short-distance dispersal kernel with a mean of 0 and a standard deviation of σ:
- (eqn 3)
and Plong(d) is the negative exponential long-distance dispersal kernel with a mean of β:
- (eqn 4)
Moment-generating functions of the dispersal kernels are needed for the analysis of the Neubert & Caswell (2000) model and for P(d) are given by:
- (eqn 5)
The moment-generating function of a kernel only exists for some finite interval around s= 0 and i= 7 or 8 (where the subscripts of m represent movement of seeds from either of the two adult stages to the seedling stage). The minimum value of the wave speed is calculated as c* = min0<s<ŝ[1/s ln ρ1(s)], where ρ1 is the dominant eigenvalue of the matrix H, which includes both the stage-structured demography and dispersal parameters. H =A°K, where ° is the element by element or Hadamard product (for details see Neubert & Caswell 2000).
Sensitivity of wave speed to model parameters measures how a small additive change in the parameter affects wave speed, whereas elasticity measures how a proportional change in the parameter affects wave speed. Some transitions were functions of underlying parameters (equation 1) and some underlying parameters occurred in more than one transition (e.g. sj). We therefore used the following formulae to decompose sensitivity of λ, population growth rate, and c*, wave speed, to particular stage transitions (ak.l) into sensitivity of λ and c* to the underlying demographic parameters (x), using methods in Caswell (2001):
where k,l give the subscripts of each transition, a. ∂λ/∂ak,l is the sensitivity of λ to the transition ak,l, which is the conventional sensitivity of population growth rate to changes in the transition value calculated as per Caswell (2001). Similarly elasticities of λ and c* to the underlying demographic parameters are given by:
and are easily worked out given terms for ∂λ/∂x or ∂c*/∂x (see Appendix S1 for ∂λ/∂x).
Sensitivities and elasticities of c* to demographic transitions ∂c*/∂ak,l are calculated according to equations 26 and 27 in Neubert & Caswell (2000), and sensitivity and elasticity of c* to dispersal parameters from equation 28 in Neubert & Caswell (2000) and equation 8 in Caswell, Lensink & Neubert (2003), which corrects the error in Neubert & Caswell (2000) (see Appendix S1 material for the sensitivity of the dispersal kernel moment generating functions, ∂m/∂x). Sensitivities and elasticities of c* to the underlying parameters in the transitions are calculated in the same way as outlined above and in Appendix S1 for λ.
Sensitivity and elasticity analyses indicate which parameters, if perturbed, lead to the greatest additive or proportional changes in λ or c*. Elasticities are commonly used to rank parameters in a management context to allow comparison of changes in parameters with very different scales. For parameters close to 0, however, such as p or b, a very large proportional change in the parameter is needed in order to achieve a change in the wave speed relative to another parameter slightly greater in magnitude. In reality, changes in different parameters are not equally easy or costly to achieve; sensitivity and elasticity analyses therefore only provide part of the answer to how best to manage populations (de Kroon, van Groenendael & Ehrlén 2000).
Our data were insufficient to partition the variance in parameters systematically according to measurable covariates. Therefore, we defined distributions for each parameter based on published and unpublished data and, where data were not available, distributions were estimated with high uncertainty from personal observations or indications in the literature. Distributions were obtained either by calculating a best-fit distribution from existing data or, if insufficient data were available to specify a best-fit distribution, a triangular distribution was specified with a median point estimate, minimum and maximum values, where probability decays towards the maximum and minimum.
Given a distribution of values for each parameter, we sampled a value at random from each distribution and used these random parameters to populate the demographic matrix A and the dispersal matrix K. Each matrix contained a random value for each parameter, there was no correlation between parameters and 100 000 different matrices were created. Empirical 95% CI were constructed around median values of λ, c* and their sensitivities and elasticities, ensuring that uncertainty in the parameters was translated into uncertainty in model outputs.