This research results in models that predict stand volume after thinning and fertilisation for Douglasfir and western hemlock on the coast of British Columbia. The predictions are made from stand volume at a known age, target age of the predicted volume, site index, and thinning and fertilisation intensity. Thinning intensity is defined as volume (m^{3}/ha) removed in the thinning divided by the time since thinning. Fertilisation intensity is similarly defined as the amount of urea fertiliser (kg/ha) applied divided by the number of years since the application. Therefore, the effect of both thinning and fertilisation on volume growth decreases over time. The models that were developed are dynamic in the sense of DiéguezAranda et al. (2006). That is, model predictions are based on a known volume/age point (V_{i1}, t_{i1} in Equation 3), which is then projected forwards or backwards in time. The models are path invariant which means that the projections from age t_{i1} to age t_{1} can be made in a single time step or multiple time steps and the same predicted volume will be obtained (DiéguezAranda et al. 2006).
This work closely follows that of Nishizono (2010), the major difference being the definitions of thinning intensity and the addition of fertilisation. Analyzing the models using a thinning intensity defined as the volume removed in a previous thinning, as was used by Nishizono (2010) resulted in a poorer fit. The definition of thinning intensity that was implemented (volume removed divided by time since thinning) results in the effect of the thinning in the model dwindling over time. The stands in the data set were only thinned once; hence only one thinning was modelled. It is not intuitive how to modify the implemented thinning intensity definition for multiple entries. Two options are to either accumulate the thinning intensities or to calculate the thinning intensity as if a previous thinning had not been done. The two methods would give similar results if the thinnings are far apart in time.
Similarly for fertilisation, the variable F (fertilisation intensity) is the amount of fertiliser applied (kg/ha) divided by the time (years) since fertilisation. This allows the growth rate to increase rapidly after fertilisation but over time the effect of fertilisation dissipates.
Unlike Nishizono (2010), who analyzed the thinning model as a nonlinear mixedeffects model, the model was fitted as a nonlinear populationaverage model. The response (volume) of the population average was of more concern than predicting the response of individuals, making the populationaverage model more appropriate (Diggle et al. 2002, p. 130). Furthermore, in application it is unlikely that there will be enough measurements to adequately calibrate a mixedeffects model. Unlike linear models, a nonlinear mixedeffects model with the random parameters set to their expected value of 0 does not give unbiased estimates of the population mean (Meng and Huang 2009, p. 245). Hence, in the situation where the model will not be calibrated for individuals, it is preferable to fit the model as a populationaverage model.
In the data analysis, the models project the volume for one measurement only resulting in short projection periods. Consequently, evaluating the residuals from the analysis does not give a good indication of model performance over a long period of time, as would be useful in a real application. A more pragmatic test was performed by projecting the volumes with the first posttreatment observation (or first observations for the control plots) and examining the residuals from these projections. This was done for the control plots and plots that only had one treatment in order to isolate the model behaviour by treatment type. The residual variance increased as the length of the projection period increased as is typical in yield projections. The control plots did not show any bias over the projection period, nor did the fertilised western hemlock plots. The thinned Douglasfir plots and the fertilised Douglasfir plots each showed a bias of about 1 m^{3}/ha/yr and the thinned western hemlock plots had a bias of less than 2 m^{3}/ha/yr. These biases are small but nevertheless could be an area of future refinement to the models. In practice, periodic stand volume measurements could be taken and used as input to the model to reduce the length of future stand volume projections.
The starting parameters for the site index, start and finish ages, and thinning levels in the simulations (Table 2, Figures 1, 2, 3 and 4) were chosen to cover the range of the data. The initial volumes were obtained from runs of the Tree And Stand Simulator (TASS) growth and yield model (Mitchell 1975; Mitchell and Cameron 1985). Although the 100year end time for the simulations is longer than typical rotation ages for Douglasfir and western hemlock in coastal British Columbia (Brown 2003; Spittlehouse 2003), it may be informative to investigate the response to thinning and/or fertilisation over this longer time horizon.
Interpreting the effect of parameters b_{0} and b_{2} on model behaviour is somewhat complicated by the formulation. This is an artefact of parameter b_{1} in Equation 1 being a function of parameters b_{0} and b_{2}. Parameter b_{0} in Equations 1 and 3 is the asymptote; hence larger values of b_{0} indicate a greater potential volume from the site. Parameter b_{2} is indicative of the growth rate of the stand with higher values indicating faster growth. However, a graphical approach has to be taken to fully assess the behaviour of the models because of the interaction between parameters b_{0} and b_{2} in Equations 3/4.
Snowdon (2002) classifies growth responses to silvicultural treatments as being a type 1 or type 2 response. Type 1 responses are characterized by a temporary increase in growth rate that reduces the time needed to reach a given stage of stand development. Type 2 responses are characterized by a real and sustained change in stand productivity. The response to fertilisation can be either type 1 or type 2, depending on the amount of fertiliser and the nutrient deficiency of the site (Snowdon 2002). Thinning will most likely be a type 1 response since the volume of a thinned stand rarely exceeds the volume of an unthinned stand, although the response to thinning may be sustained for quite a while. Given the model formulation in Equations 3/4, the response to both fertilisation and thinning dwindles over time making the response a type 1 response. Simulations using average start values in Table 2 confirm that both thinning and fertilisation are type 1 responses for western hemlock and Douglasfir. Initially, fertilisation increases growth and thinning decreases stand growth but over time the stand growth for both treatments converge to the growth of the untreated stand. Note, however, that an extreme thinning will reduce the volume to a point where the growth of the treated stand diverges from the growth of the untreated stand over a typical rotation period.
The response of Douglasfir and western hemlock to thinning and fertilisation can now be compared to published responses using these new models. This is done in part to assess the integrity of the model and to evaluate specific responses. The assessment is done analytically by examining the parameters of the model (Equation 4) but mostly by using graphical techniques.
More productive sites (higher site index) have increased growth and yield
Site index is a measure of site productivity; better sites produce a greater volume of timber and the trees on the site are faster growing (Carmean 1975). Parameter b_{0} increases and parameter b_{2} decreases with site index for both species. As gleaned from the value of parameter b_{0}, the potential yield (asymptote) increases as site index increases. A graphical analysis (Figure 1) shows that growth also increases as site index increases even though parameter b_{2} decreases. The models behave as expected with respect to site index.
The response to thinning depends on the thinning intensity
On a per tree basis, a heavy thinning will reduce competition more than a lighter thinning and hence the growth response is expected to increase as the thinning intensity increases (Pukkala et al. 2002). However, on a stand level the response is less clear because there are fewer trees to contribute to stand growth. Cañellas et al. (2004) found that Quercus pyrenaica growth increased with increasing thinning intensity but declined with the heaviest thinning. D’Amato et al. (2011) did not find any standlevel growth response to thinning of Picea glauca stands. For Pinus sylvestris stands, stand volume increment decreased as thinning intensity increased and the difference increased over time (del Río et al. 2008). To evaluate the fitted models, volume was plotted against age at 4 levels of thinning that occurred at age 30 assuming a site of medium quality (site index = 29 m) for both species (Figure 2). In absolute terms, the volume in the thinned stands did not approach the volume in the unthinned stands. That is, the volume lost in the thinning was not recovered through increased growth of the trees left on the site. However, on a relative basis, the thinned stands had proportionally more volume at the end of the 100 year simulation than was removed. That is, for Douglas fir, the volume lost when compared to the unthinned stands was 4%, 12%, and 23% for the 20%, 35%, and 50% removals, respectively. The volume lost for western hemlock was 9%, 19%, and 32% for the same levels of removal.
The effect of fertilisation on tree growth is relatively shortlived
Nitrogen (N) is the most limiting nutrient in Douglasfir and western hemlock sites (Bennett et al. 2003; Chappell et al. 1991). Consequently, fertilisation with N is more common than fertilising with other nutrients. The period of time for which tree growth is elevated after fertilisation is variable but relatively short, e.g. 5 – 10 years (Bennett et al. 2003; Binkley and Reid 1985), although exceptions can be found (Binkley and Reid 1985). When fertiliser is applied at the time of thinning the maximum response is observed about 5 years after treatment (Snowdon 2002). It has been hypothesized that the length of time that a fertilisation effect is observed depends on the amount of fertilisation applied in relation to the amount of the nutrient on site before fertilisation (Miller 1981). These hypotheses can be evaluated with the models. For each species, sites at three levels of site index were fertilised with 450 kg/ha of N fertiliser at age 50. This level of fertiliser is generally the maximum that is economically feasible (Barclay and Brix 1985). The percent change in growth as compared to an unfertilised site with the same site index was plotted (part a of Figure 3 is for Douglasfir; part b is for western hemlock).
The response to fertilisation is much greater for Douglasfir than it is for western hemlock, and the response lasts substantially longer for Douglasfir. The response and the period of time over which the response is observed decreases as the site index increases. This is likely due to the better sites (as indicated by a higher site index) having a higher store of N on site at the time of the fertilisation, hence N is less limiting in growth. A similar response was observed by Heath and Chappell (1989) for Douglasfir. A few points need to be made with this analysis:

1.
Confidence intervals for the predicted percent change in volume growth are not available, so it cannot be determined when the change in volume growth is not significantly different from zero.

2.
There is a significant interaction term between fertiliser amount and site index for Douglasfir (but not western hemlock) and the coefficient for this term is negative (Table 3, parameter b_{06}). This results in a predicted loss of volume growth at high levels of fertilisation application and site index. This is more likely an artifact of the modelling than a real response since the predicted loss of growth is small and may not be significantly different from zero.

3.
As with all models, users have to be wary when extrapolating these models. The response to fertilisation is extrapolated in terms of the length of the response.
Interaction between fertilisation and thinning
It may be of interest to determine whether the effects of thinning and fertilisation are additive or if there is an interaction between the two treatments (Miller et al. 1979; Zhang et al. 2005). This can be investigated with the models. The models for both Douglasfir and western hemlock have significant interaction terms (parameters b_{04} and b_{24} for Douglasfir, parameter b_{04} only for western hemlock, Table 3). In a much smaller experiment, McWilliams and Thérien (1996) did not find any interaction between thinning and fertilisation in Douglasfir. Simulation was used to investigate how this interaction term affects volume. Three site index levels were assumed with corresponding volumes at age 20 (Table 2) for each species. Thinning and fertilisation treatments (35% removal with 450 kg/ha urea fertiliser) were applied at age 30. The same scenario was applied with the interaction term removed from the model. The difference in volume is then due to the thinning/fertilisation interaction. The volumeage trajectories are plotted in Figure 4, parts a (Douglasfir) and b (western hemlock). The effect of the interaction between fertilisation and thinning is slight for Douglasfir (approximately 20 – 25 m^{3}/ha at age 100 for low and medium sites, practically no difference for high sites) and virtually nonexistent for western hemlock. Interestingly, though, the very slight effect of the thinning × fertilisation interaction for western hemlock is negative which is due to the negative estimated value of parameter b_{04} in Equation 4. The lack of a significant interaction response for western hemlock (even though the parameter for this term is statistically significant) can be attributed to the small estimated value for parameter b_{04} in Equation 4. When this parameter is multiplied by the thinning and fertilisation effects (T and F, respectively, in Equation 4), the term does not have a large effect on the asymptote, particularly as the thinning and fertilisation effects diminish over time.
Comparison to published yield tables
In this section, the models developed here (Equations 3/4 with appropriate parameters from Table 3) are compared to other published yield tables. For western hemlock, the comparison is made to yield tables published by Barnes (1962) and Taylor (1934). These tables are for fully stocked stands with a minimum diameter limit of 1.5 cm, making them approximately the same as the EP703 hemlock plots. However, both of these yield tables are for mixed species stands, and no thinning or fertilisation information is available. Barnes (1962) published yield tables for Oregon/Washington, Alaska, and British Columbia; the British Columbia tables were used in this comparison. At the same starting conditions at age 20, both the Taylor and Barnes yield curves are approximately the same for site index 25 and 29 stands. The western hemlock model predicts greater yields than these two yield tables at site index 25 and 29, but at site index 29 the yields are similar below age 60. At site index 34, the yields from Equations 3/4 and from Barnes’ yield table are almost identical (Taylor did not publish yield tables for site indexes greater than 32).
The fitted Douglasfir model was compared graphically to published yield tables calculated using the Douglasfir simulator (DFSIM) (Curtis et al. 1982). The DFSIM is a wholestand growth and yield model developed specifically for Douglasfir in the Pacific Northwest region of North America (Curtis et al. 1981). It is able to simulate natural and planted stands with precommercial and commercial thinning, as well as fertilisation. Four stand types were compared (natural, planted, planted and commercially thinned, natural with precommercial thinning and fertilisation) at three levels of site index (25.9, 32.0, and 38.1). These correspond to Tables 1, 2, 7, and 11, parts A, B, and C in Curtis et al. (1982). The simulations started at age 30. The starting volumes for the fitted model were taken from the corresponding tables in Curtis et al. (1982). A utilisation limit of 4 cm (1.6 inches) was used in the comparison when more than one utilisation limit was published. The commercial thinning and fertilisation treatments were applied at the same rate and age as in the published tables (Curtis et al. 1982). The thinning removed approximately 30% of the volume and the fertilisation was done with 225 kg/ha of N. The precommercial thinning in the fourth scenario was not modelled because the thinning was done too early for the fitted model to have any validity.
Figure 5 parts a – d show the results of the comparison. For natural stands (part a), there is good correspondence between the two models for the first 20 to 30 years after which the fitted model estimates higher volumes than those estimated by the DFSIM. The deviations between the two models increase with increasing site index. Curtis et al. (1981) had very little data for total ages greater than 60 years even in untreated natural stands. The plantation yield curves (part b) show similar yield projections over the longer term, but again the DFSIM yields are extrapolations beyond about total age 45 (Curtis et al. 1981). Unlike the natural stand yield curves, the differences between the two models decreases as site index increases. Part c shows that use of the DFSIM and Equations 3/4 lead to dissimilar volume predictions for thinning regimes. Both before and after thinning, volume growth rates are greater using the DFSIM model than for Equations 3/4 at all levels of site index, Figure 5c. As noted previously, a slight bias may be present in Equations 3/4 for thinned stands, but not to the extent that it would explain the differences in yield indicated in part c of Figure 5. The two models may be projecting different populations as there are no data for trees over age 45 in the DFSIM development data set. The fertilisation scenario (part d) shows good agreement between the two models, particularly at higher site indexes. The models developed here do not show as much fertiliser response as the DFSIM at the lower site index level.