In earlier chapters I have shown that the response of the model changes with values of parameters and environmental conditions. How sensitive the model is to these changes is important information. Superficially, it might seem that the best model would be the least sensitive. But a model that was completely insensitive to parametric values could not distinguish among species - all species would respond the same way. On the other hand, a model that was very sensitive to variations in the value of a parameter would be impractical, because it is difficult to estimate the real quantity of the parameters, and a model that was too sensitive would give widely varying results with small changes in parameter estimations. As I have said before, but which bears repeating here, in my field experience, even under the best of conditions, estimations of parameters from field measurements are likely to be in error by 10 percent or more. It is therefore useful to test the sensitivity of projections against errors at 10 percent. If we cannot measure parameters more accurately than within 10 percent, we should demand no more from the model.

This discussion suggests that the most useful model would have an intermediate sensitivity. The problem is somewhat analogous to the design of aircraft. Early in the twentieth century when the first aircraft were designed, it was unclear how much sensitivity to external forces made aircraft easiest to fly. Superficially, one might think that the most stable (but therefore the least sensitive) aircraft would be the best, but such an aircraft would fly straight ahead regardless of the change in forces on it, including the controlling forces exerted by the pilot. A completely insensitive (completely stable) aircraft could not be guided, while a too-insensitive (too-stable) aircraft would require extremely strong forces to turn. On the other hand, a very sensitive (unstable) aircraft would change direction at the slightest change in the winds or in the setting of the control surfaces, and would require constant adjustment and attention. In the end, it was found from experience that airplanes flew most easily if they had intermediate sensitivity to external forces. In my experience with ecological models, the same seems to be the case. We need models that respond realistically to changes in the values of parameters and in environmental conditions, but that are not so sensitive that we cannot obtain realistic projections without absolutely precise measures of parameters.

Consider a specific parameter: the temperature response DEGDmax_i. Suppose the model gave the same results no matter what the value of this parameter. The model would be unrealistic, unable to represent the way that forest trees respond to environment. On the other hand, suppose the model were extremely sensitive, so that a change in one degree-day in DEGDMAX_i resulted in a 10 percent change in the temperature response. Since the estimates of DEGDMAX_i of a species, obtained from range maps as explained in Chapter 2, could easily vary by 10 degree-days, the model could produce projections differing by 100 percent from two estimates of the same parameter.

The ideal model would have the same sensitivity to parameter values as real trees. To know what this real sensitivity is, however, would require not only accurate estimation of parameters but analysis of the sensitivity of real trees in real forests to changes in parameters, research that is yet to be done. This discussion suggests that the sensitivity of a model to parameter estimation must lie somewhere in between extreme sensitivity and extreme insensitivity.

There are several reasons to study the sensitivity of the model. If we find that the model is not overly sensitive to errors in the estimation of parameters, then our faith in the realism of the model's projections will increase. In addition, sensitivity tests can give us additional insight into the dynamics of forest communities.

In this chapter I explore some aspects of the sensitivity of the forest model. Formally, a sensitivity test is a test that determines how great a change occurs in the value of an output variable with a change in the value of either an input variable or a parameter intrinsic to the model. A very large number of sensitivity tests could be conducted for a model with the complexity of JABOWA. Inclusion of all possible sensitivity tests would be impractical in a single volume. The availability of the model on diskettes for operation on a personal computer allows a user to conduct additional sensitivity tests. The purpose of this chapter is to explore the general issues underlying sensitivity tests and demonstrate the general level of sensitivity of the model. It is not to provide an unwieldy and unreadable volume of tables of all possible tests. Sensitivity tests can be grouped into two types: intrinsic sensitivity, which is sensitivity to parameter estimation, and external sensitivity, which is sensitivity to input or external variables, such as air temperature and precipitation.

The forest model can generate volumes of output, especially for sensitivity tests. My purpose here is to illustrate types of sensitivity tests, not conduct all of them. In this section I consider the sensitivity of the model to seven intrinsic model parameters. These are tested in a specific situation for two species. Limiting consideration to two species keeps the results to an amount I can write about in a single book and the reader can handle reasonably. The species I have chosen are two dominants of the Superior National Forest of Minnesota, balsam fir and sugar maple. Balsam fir is a dominant of mature boreal forest stands on wet soils, whereas sugar maple dominates moist, fertile, well-drained soils in the northern hardwoods forest. The site I have chosen represents a major biogeographic transition - the transition from boreal to northern hardwoods forests. In such a transition, the two species can occur together on moist sites. One expects, however, that each species, as well as the entire forest community, will be more sensitive to changes in parameters at such a transitional site than in the center of a species range (that is, midst of a major biome). In the transition, species are near the edges of their ranges and near the limits of the temperature response curve, as defined in Chapter 3 (equation 3.6). In the center of a biome, the dominant species are near the optimum of their temperature response curves. Inspection of equation (3.4) suggests that a species will be relatively insensitive to changes in parameter estimations there. The temperature response curve is steep as the function approaches its limits, and this occurs in ecologically transitional sites.

In most tests presented here, changes are made in the parameters for balsam fir, and the responses of both fir and maple are examined. The response of fir represents the response of a species to changes in its intrinsic characteristics; the response of sugar maple represents the response of a competitor to these changes. The parameters are: one affecting regeneration, (1) the parameter S_i, the species-specific maximum number of saplings that can be added to a plot in any one year; two affecting mortality probabilities; (2) maximum longevity of a species in years (AGEMAX_i) and (3) minimum diameter increment required per year for an individual tree to avoid being subjected to a higher probability of mortality (AINC_i); and four affecting individual tree growth; (4) maximum and (5) minimum number of degree-days under which the species could grow (DEGDMAX_i and DEGDMIN_i, respectively), (6) an index of soil-water saturation, which is the minimum depth to the water table that a species can withstand (DTMIN_i), and (7) an index of drought, the minimum soil moisture conditions a species can withstand (WLMAX_i). Intrinsic model parameters were varied by 10 percent above and below the standard values.

All tests were made beginning with a standard plot representing a typical old-age, undisturbed stand of balsam fir in the Superior National Forest of Minnesota. There are two steps in the sensitivity tests: (1) choosing or creating initial conditions; (2) running the model to perform the sensitivity tests. One could use observed conditions from a real plot as initial conditions, but then there would likely be a transient response of the model to these conditions, and the sensitivity would have to be interpreted against this response. It is simpler and clearer to create hypothetical old-age stands generated by the model and use these as initial conditions, and this is the approach I have taken. The initial tree population for this hypothetical but typical stand was generated by running the model for 400 years from a clearcut with 50 replicates and using the 1951-1980 Virginia, Minnesota, weather records (the nearest weather station to the southern limit of the Superior National Forest and its designated wilderness, the Boundary Waters Canoe Area) in a repeating sequence. The initial plot conditions are: 0.081 _ 0.002 (95% CI) stems/m² of balsam fir occupying 27.9 _ 1.5 cm²/m² basal area, soil depth 1 m, depth to the water table 0.8 m, soil moisture holding capacity of 250 mm water/m depth of soil, and a moderate soil fertility of 50 kg/ha. The resulting initial stand is dominated by balsam fir, but contains enough sugar maple to allow a rapid change in dominance to this species, if the parameter change results in sensitivity.

As in the creation of initial conditions, the "normal" climate for each test described below is a recurring sequence of the 1951-1980 weather records from Virginia, Minnesota. In each test, the model was run for 90 years with 50 replicates for each trial. A trial consisted of increasing or decreasing one parameter by 10 percent. The values of the parameters used in these tests are given in Table 6.1.

Results of the first tests are given in Tables (6.1) and (6.2). Basal area (Table 6.1) and density (Table 6.2) are not sensitive to 10 percent error in estimation of most parameters. Errors of 10 percent in the estimate of the maximum longevity of either species and minimum diameter increment below which mortality rates increase do not affect the abundances of either species. Nor do 10 percent errors in the estimation of soil moisture parameters affect the results significantly.

In this case, sensitivity at the 10 percent level exists only for the parameters that control degree-day limits of the species. Decreasing the maximum number of degree-days for balsam fir by 10 percent leads to 50 percent less basal area by the year 2010 than when the standard parameter value is used, and a 66-percent lower value by the year 2070 (Table 6.1). A change in the minimum degree-day parameter has no effect on the abundance of basal fir. Sensitivity to the maximum and not to the minimum degree-day parameter is due to the location of the Superior National Forest relative to the range of balsam fir, which as I explained earlier (and which should be apparent from discussions in previous chapters), is near the southern boundary of the geographic distribution of balsam fir. Changing the northern limit by 10 percent has a very small effect on the shape of the temperature response curve (equation 3.6) for balsam fir growing near Virginia, Minnesota while changing the value of the southern limit has a strong effect on balsam fir growing in that vicinity.

The response of sugar maple is the mirror image of the response of balsam fir. An increase in the minimum degree-day parameter for sugar maple effectively allows the species to grow farther north and has a large effect on the abundance of that species in the Superior National Forest, which is near its northern boundaries. On the other hand, 10 percent alteration of the maximum degree-day parameter, which determines the southern distribution of sugar maple, has no effect on the results (Table 6.1).

Basal Area (cm₂/m₂ _ 95% CI)

Year 10 Year 30 Year 60 Year 90

Balsam Fir(a)

Parameter(b) Values

Control Run

CONTROL (basal area) (degree-days: 2250-2950)

27_1 18_3 14_2 18_2

AGEMAX 200 220 29_2 19_3 15_3 20_3

AGEMAX 200 180 28_2 19_3 16_3 16_3

AINC 0.01 0.02 28_1 18_3 14_2 18_2

AINC 0.01 0.005 28_1 18_3 14_2 18_2

SAPL 2 2.2 27_2 20_3 15_2 18_3

SAPL 2 1.8 28_1 17_3 14_2 17_3

DDMIN 700 770 28_2 17_3 16_3 18_3

DDMIN 700 630 29_3 20_3 14_2 17_2

DDMAX 3700 4070 29_1 21_3 20_3 21_4

DDMAX 3700 3300 25_1 10_2 6_1 6_1

DTMIN 0.21 0.23 28_2 17_3 18_3 20_3

DTMIN 0.21 0.19 27_2 18_3 14_3 20_3

WLMAX 0.245 0.27 28_1 18_3 14_2 18_2

WLMAX 0.245 0.22 28_2 17_3 14_3 21_3

S_i 2 4 29_2 20_2 27_2 33_3

Sugar Maple

CONTROL (basal area) 3_.2 2_1 3_1 2_1

(degree-days: 3135 - 3899

AGEMAX 400 440 3_.2 3_.6 3_.9 3_.8

AGEMAX 400 360 3_.2 2_.6 3_1 3_1

AINC 0.01 0.02 3_.2 2_.5 2_1 3_1

AINC 0.01 0.005 3_.3 2_.6 3_.9 2_.8

SAPL 3 3.3 3_.2 2_.5 3_.8 3_1

SAPL 3 2.7 3_.2 2_.5 2_.8 2_.9

DDMIN 2000 2200 3_.2 .5_.3 .2_.1 2_.1

DDMIN 2000 1800 3_.2 6_.8 17_2 30_3

DDMAX 6300 6930 3_.2 2_.5 2_.8 1_.6

DDMAX 6300 5670 3_.2 3_.6 3_.7 3_.7

DTMIN 0.57 0.62 2_.3 2_.5 2_.7 2_.8

DTMIN 0.57 0.51 3_.3 2_.6 3_.9 4_1

WLMAX 0.35 0.39 3_.2 2_.6 3_.8 3_.9

WLMAX 0.35 0.32 3_.2 2_.5 2_.7 2_.6

S_i 3 6 3_.2 1_.5 2_1 3_1

(a) Initial plot conditions are: 0.081 _ 0.002 (95% C.I.) stems/m² of balsam fir occupying 27.9 _ 1.5 cm²/m² basal area; soil depth 1 m, depth to the water table 0.8 m, soil moisture-holding capacity 250 mm water/m depth of soil; available nitrogen = 50 kg/ha. The population in the initial stand is given in Appendix IV, Table A-IV.1.

(b) AGEMAX = maximum age of the species in years; AINC = minimum diameter increment per year (cm) to avoid mortality; SAPL = maximum number of saplings of a species that can enter a plot per year; DEGDMAX = maximum number of degree-days for growth; DEGDMIN = minimum number of degree-days for growth; DTMIN = minimum tolerable depth to the water table; WLMAX = maximum evapotranspiration rate permitted for tree growth (which can also be thought of as the maximum wilt tolerable by species i.) S_i is the maximum number of saplings of species i that can be added to a plot in any single year.

Table 6.2. Effects of forest model sensitivity to parameter values for balsam fir stems/100m² _95% CI, using Virginia, Minnesota, 1951-1980 weather records (a)

Results with Default Values 9_0.5 8_0.7 9_0.9 10_0.7

AGEMAX 200 220 9_0.5 7_0.6 9_0.9 10_1

AGEMAX 200 180 8_0.6 7_0.6 8_0.7 9_1

AINC 0.01 0.02 9_0.5 8_0.7 9_0.9 10_0.7

AINC 0.01 0.005 9_0.5 8_0.7 9_0.9 10_0.7

SAPL 2 2.2 9_0.6 9_0.7 10_0. 11_1

SAPL 2 1.8 8_0.5 6_0.6 7_0.7 9_0.7

SAPL 2 4 15_1 19_1 26_2 31_2

DDMIN 700 770 9_0.5 8_0.6 8_0.8 10_1

DDMIN 700 630 9_0.4 7_0.6 8_0.8 10_0.9

DDMAX 3700 4070 9_0.5 8_0.6 9_0.9 11_1

DDMAX 3700 3300 8_0.5 4_0.6 5_0.5 5_0.6

DTMIN 0.21 0.23 9_0.4 8_0.7 9_0.8 10_1

DTMIN 0.21 0.19 9_0.6 7_0.6 8_0.8 11_1

WLMAX 0.245 0.27 9_0.5 8_0.7 9_0.9 10_0.7

WLMAX 0.245 0.22 9_0.6 7_0.6 8_0.7 11_0.9

CONTROL 4_0.3 4_0.5 5_0.5 4_0.5

AGEMAX 400 440 4_0.2 4_0.6 4_0.7 4_0.6

AGEMAX 400 360 5_0.3 5_0.5 5_0.5 4_0.6

AINC 0.01 0.02 4_0.3 4_0.5 4_0.5 4_0.6

AINC 0.01 0.005 4_0.3 4_0.4 4_0.5 4_0.6

SAPL 3 3.3 5_0.3 5_0.4 5_0.6 5_0.6

SAPL 3 2.7 5_0.2 3_0.4 3_0.5 3_0.5

SAPL 3 6 9_0.6 10_0.9 11_0.8 11_0.9

DDMIN 2000 2200 4_0.3 2_0.4 2_0.4 1_0.3

DDMIN 2000 1800 5_0.4 9_0.6 12_1 16_1

DDMAX 6300 6930 4_0.2 3_0.5 3_0.5 5_0.5

DDMAX 6300 5670 5_0.3 5_0.5 5_0.6 5_0.5

DTMIN 0.56 0.62 4_0.3 4_0.6 4_0.4 4_0.5

DTMIN 0.56 0.51 5_0.3 4_0.6 5_0.5 5_0.6

WLMAX 0.35 0.39 5_0.3 4_0.5 4_0.5 4_0.6

WLMAX 0.35 0.31 4_0.3 4_0.4 4_0.6 4_0.5

(a) See Table 6.1 for initial plot conditions.

(b) See Table 6.2 for definition of parameters.

Need For Better Temperature Parameter Estimation Procedures

The parameters DEGDMAXⁱ and DEGDMINⁱ were determined by comparing maps of a species' distribution with maps of temperature isotherms. As I explained before, this method was selected simply for convenience when the model was first developed, and I expected that later users of the model would find more accurate methods. In general, they have not. In part this seems to be a result of the success of the model and its only moderate sensitivity to estimation of the temperature response parameters. However, the user should know the assumptions of this method, assumptions that could lead to errors in the accuracy of projections and the reader might at this point want to refer back to relevant sections of Chapters 2 and 3. First, this method assumes that the distribution of a species is in steady-state with the current climate. This is not likely to be the case, since over the millennia all tree species have migrated continually across the landscape in response to climate change (Davis, 1983; COHMAP 1988; Ritchie and Yarrantow, 1978; Swain, 1978; Wright, 1976). That the method has worked is probably the result of the long time required for trees to migrate, so that in terms of the estimation of this parameter, there is a reasonable correspondence between modern distributions and modern climate. In addition, the method assumes that all individuals of a species have the same temperature response - that there are no ecotypic variations. It is possible that ecotypes (local populations with distinctive genetic characteristics) near the boundary of the distribution of a species have evolved to be better adapted to limiting temperature conditions than the average for the species, and therefore they may be able to persist longer than expected from the standard estimates of the parameters. That this is possible is suggested by the observation of local differences - on the same hillside - within a species as a function of elevation (Ledig and Korbobo, 1983).

As I discussed in Chapter 3, but bears repeating here, the general shape of a species' response curve to temperature level is based on general concepts of physiological response to temperature (Kramer and Kozlowski, 1979; Kozlowski, Kramer, and Pallandy, 1991), and the maximum and minimum temperatures expressed in terms of yearly degree-days reflect the actual geographical range limits of the species (Botkin et al., 1972a and 1972b). The resulting parabolic function could be studied and verified, particularly at the extremes. The actual shape for some species may be a truncated normal curve (similar to a beta-distribution) expressing the effect of differential responses of ecotypes, such as we used in an algal community model (Lehman et al., 1975a,b).

The tests against a 10 percent change in parameter estimation are encouraging because they suggest that the model is not overly sensitive to factors whose measurement cannot be expected to be done with an accuracy better than 10 percent. However, it is useful to carry the tests a step farther and find out what percent change in a parameter will produce a significant change in the model's output. Tests of this kind were conducted for the same seven parameters and same site and initial conditions as in the previous tests. The output, given in terms of biomass, and appears in Tables 6.3, 6.4, and 6.5 and in Figures 6.1, 6.2, and 6.3.

Table 6.3. Parameter Error Required to Produce Significant Change in Biomass After 90 Years of Growth.

These tests are for biomass of sugar maple and balsam fir 90 years after parameter modification for the site described in Table 6.1. The initial forest is an old age balsam fir stand described in the text and used in Table 6.1.

_________________________________________________________________

A. S_i -- the maximum number of balsam fir saplings that can enter the plot during one year.

SAP SM biomass BF biomass

(kg/m²) (kg/m²)

1 4.8 _ 0.8 10.8 _ 2.2

2 (default) 5.6 _ 0.9 8.8 _ 1.8

3 5.6 _ 0.9 8.6 _ 1.4

--------

6 5.9 _ 1.0 11.7 _ 1.9 signif. different

12 5.5 _ 0.8 12.7 _ 1.6

24 5.0 _ 0.8 14.5 _ 1.0

B. AGEMX -- Maximum age of a balsam fir the tree in years.

AGE SM biomass BF biomass

(kg/m²) (kg/m²)

50 9.1 _ 0.9 0.1 _ 0.1

100 7.6 _ 0.9 3.5 _ 1.6 signif. difference

--------

200 (default) 5.6 + 0.9 8.8 _ 1.8

250 5.7 _ 0.8 11.4 + 1.8

--------

300 4.7 _ 0.8 13.3 _ 1.9 signif. difference

400 4.9 _ 0.6 17.2 _ 2.3

500 4.2 _ 0.7 20.6 _ 1.8

C. AINC -- Minimum diameter increment of a balsam fir tree to avoid being subject to a higher probability of mortality.

AINC SM biomass BF biomass

(kg/m²) (kg/m²)

0.005 5.5 _ 0.9 8.5 _ 1.8

0.01 (default) 5.6 _ 0.9 8.8 _ 1.8

0.02 5.5 _ 0.9 8.5 _ 1.8

0.04 6.2 _ 0.9 8.5 _ 1.8

0.06 6.1 _ 1.0 6.3 _ 1.6

--------

0.08 8.0 _ 1.0 3.8 _ 1.2 signif. difference

0.10 8.5 _ 1.1 2.0 _ 0.5

0.20 8.5 _ 1.1 0.0

_________________________________________________________________

_________________________________________________________________

D. DDMAX -- The maximum number of degree-days (above 40 ^oF) per year under which a balsam fir tree could grow.

DDMAX SM biomass BF biomass

(kg/m²) (kg/m²)

1028 8.8 _ 1.1 0.0

1400 9.4 _ 1.0 0.0

1600 9.4 _ 1.0 0.0

1800 8.7 _ 0.9 0.0

1850 7.7 _ 1.0 0.0

1900 7.6 _ 1.1 2.2 _ 0.9 signif. difference

-----------

1950 6.3 _ 1.1 7.1 _ 1.7

2000 5.4 _ 0.8 8.9 _ 1.8

2056 (default) 5.6 _ 0.9 8.8 _ 1.8

-----------

3083 5.7 _ 0.9 11.6 + 1.7 signif. difference

E. DDMIN -- The minimum number of degree-days (above 40 ^oF) per year under which a balsam fir tree could grow.

DDMIN SM biomass BF biomass

(kg/m²) (kg/m²)

98 6.1 _ 0.9 7.3 _ 1.6

195 5.0 _ 0.8 8.1 _ 1.9

390 (default) 5.6 _ 0.9 8.8 _ 1.8

585 5.1 _ 0.9 10.0 _ 1.7

790 5.8 _ 0.9 10.2 _ 1.9

1170 5.0 _ 0.8 12.5 _ 2.3 signif. difference

_________________________________________________________________

_________________________________________________________________

F. DTMIN -- Minimum depth to the water table that a balsam fir tree can stand.

DTMIN SM biomass BF biomass

(kg/m²) (kg/m²)

0.01 5.6 _ 0.9 8.8 _ 1.8

0.05 5.4 _ 0.8 10.5 _ 2.0

0.10 6.0 _ 0.8 8.3 _ 1.8

0.15 5.8 _ 1.0 9.4 _ 1.8

0.21 (default) 5.6 _ 0.9 8.8 _ 1.8

0.25 6.7 _ 1.0 7.6 _ 1.5

0.40 6.1 _ 0.9 7.1 _ 1.9

0.50 6.3 _ 0.9 6.4 _ 1.7

0.60 6.9 _ 0.9 4.8 _ 1.2 signif. difference

G. WLMAX -- Minimum soil moisture conditions that a balsam fir tree can tolerate.

WLMAX SM biomass BF biomass

(kg/m²) (kg/m²)

0.10 5.7 _ 0.8 7.8 _ 1.6

0.20 5.7 _ 0.9 9.0 _ 1.8

0.25 5.6 _ 0.9 8.8 _ 1.8

0.40 5.7 _ 0.9 9.7 _ 1.9

_________________________________________________________________

Figure 6.1 Sensitivity of Balsam Fir Biomass to Changes in Fundamental Population Dynamics Parameters of Balsam Fir. Results are shown after 90 years of simulated growth for a site in northern Minnesota representing the Superior National Forest and its designated wilderness, the Boundary Waters Canoe Area. Site conditions are given in Table 6.1. (A) Changes in the Regeneration parameter, S_i, the species-specific maximum number of saplings that can be added to a plot in any one year; (B) Changes in Maximum Age of Balsam Fir; and (C) mortality parameter 2, AINC_i, the minimum diameter increment required per year for an individual tree to avoid being subjected to a higher probability of mortality.

In the following discussions, it is useful to note that balsam fir reaches a biomass at year 90 of 8 kg/m² with default parameters. The model is not very sensitive to changes in the maximum number of saplings of balsam fir that can be added each year; the biomass of balsam fir ranges from 8 to 15 at year 90 when the maximum number of saplings ranges from 1 to 24 (Fig. 6.1[A]). For unexplained reasons, biomass of balsam fir reaches a minimum at year 90 when the maximum number of saplings is three or four; the biomass increases slightly when the maximum number of saplings is reduced to one. The increase in biomass with increasing recruitment is expected, but the model is less sensitive to changes in this parameter than I had expected.

The model is more sensitive to changes in the maximum age of balsam fir than to the maximum number of saplings that can be added in any one year. Balsam fir becomes extinct when the maximum age is set at 50 years (Fig. 6.1[B]). Biomass of this species at year 90 increases almost linearly from zero at a maximum age of 50 to 20 kg/m² at a maximum age of 500 years. Thus increasing the maximum age by 10 times increases balsam biomass by 20 kg/m², while a ten-fold increase in the maximum number of saplings increases balsam fir biomass by 7 kg/m². The connection between extinction and maximum longevity of balsam fir is of interest for questions about biological conservation, as explored by Woodby (1991).

The model is also sensitive to changes in the second mortality parameter, the one that affects trees that grow poorly. The default value (0.02) is large enough to have little effect on the response of the model. Halving and doubling this value, as indicated in Figure 6.1(C), has no effect on the biomass of balsam fir. The model responds to changes in this parameter only when the value is tripled to 0.06, which means that a tree growing 0.06 cm/year or less is subject to the higher probability of mortality. Balsam fir biomass at year 90 drops rapidly with increases in this parameter beyond 0.06, and the species becomes extinct when the parameter reaches 0.2 cm/yr. These results suggest that trees must be able to withstand considerable competition from shading and low growth rates in order to persist in a multispecies forest.

Figure 6.2 shows how a competitor is affected by changes in parameters of a species. In this case the response of sugar maple is shown for changes in balsam fir parameters. Biomass of sugar maple at year 90 declines slightly as the maximum age of balsam fir increases from the default of 200 to 500 years, and the decline approaches an asymptote of 4 kg/m² (Fig. 6.2[A]). Sugar maple biomass at year 90 increases when the maximum age of balsam fir decreases below the default, but the increase is relatively modest. At the balsam fir maximum age of 50 years, at which balsam fir becomes extinct, sugar maple biomass at year 90 reaches 9 kg/m². The relatively small rise in sugar maple biomass may be due to competition with other species.

Sugar maple is less sensitive to changes in balsam fir parameters than is balsam fir, as might be expected. For example, sugar maple biomass at year 90 increases from 5 kg/m² to 8 kg/m² (Fig 6.2[B]), while balsam fir biomass becomes extinct over this same range.

The sensitivity of the balsam fir to large changes in degree-day parameters (beyond the 10% value of the first sensitivity experiment) is shown in Figure 6.3. Remember that the maximum degree-day parameter controls the warmer (southern) limit of the species, while the minimum degree-day parameter controls the colder (northern) limit of the species. Decreasing the maximum degree-day parameter moves the southern boundary of the species northward. When that is done, balsam fir biomass declines rapidly at the test site. The default value 2,056, is near the value that occurs at the test site, which means that this site is near the southern boundary of balsam fir's range. As I explained earlier, the site was chosen purposefully with this in mind. The model projects that a species whose southern limit is 1,800 degree-days or less cannot grow at this site (Fig. 6.3[A]). Extending the southern limit (by increasing the maximum degree-day parameter) beyond 1800 allows the species to persist, but increases beyond this value have only a small effect on balsam fir biomass.

Figure 6.3C shows the effect on a competitor of a change in the warm (southern) limit of a species. Sugar maple biomass reaches an asymptote of 8 kg/m² when balsam fir is eliminated. One might expect that sugar maple would continue to increase as balsam fir becomes less and less abundant, but in a multispecies forest other species in addition to sugar maple increase as balsam fir declines. Biomass of sugar maple declines to approximately 6 kg/m² under a balsam fir maximum degree-day parameter of 1900 to 3000. At this site sugar maple appears to have two dominant states in regard to balsam fir: a high abundance of 8 kg/m² and a low abundance of approximately 6 kg/m².

Figure 6.2 Sensitivity of Sugar Maple Biomass to Changes (A) balsam fir maximum age; (B) balsam fir mortality parameter 2 (AINC); (C) balsam fir maximum degree-day parameter; and (D) minimum degree-day parameter. Results are shown for year 90 for a site in northern Minnesota representing the Superior National Forest and its designated wilderness, the Boundary Waters Canoe Area. Site conditions are given in Table 6.1.

At the test site, balsam fir is much less sensitive to changes in the colder (northern) limit defined by the minimum degree-day parameter (Fig 6.[3B]). As the northern limit is moved to the south (the value of the parameter increases), balsam fir biomass increases. This is caused by a steepening of the temperature response functions - the parabola that characterizes this response covers a smaller range so that the function increases more sharply at the extremes. Moving the cold limit northward stretches out the curve and lowers the biomass at the Virginia, Minnesota test site; moving the cold limit to the south compresses the curve and increases balsam fir biomass at this site. Sugar maple shows little response to changes in the cold degree-day limit for balsam fir (Fig. 6.3[D]), which is not surprising in light of the small effect that changes in this parameter have on balsam fir itself.

Figure 6.3 Sensitivity of Balsam Fir to Changes in Environmental Response Parameters of Balsam Fir. Results are shown after 90 years of simulated growth for a site in northern Minnesota representing the Superior National Forest and its designated wilderness, the Boundary Waters Canoe Area. Site conditions are given in Table 6.1.

A significant increase in biomass of balsam fir does not occur until the maximum number of saplings that can enter in any one year is tripled, from two to six (Table 6.3[A]; Fig. 6.1[A]). Increasing or decreasing the maximum age by 100 years significantly changes the biomass of balsam fir, but an increase of 50 years (a 25% increase) does not lead to a significant change (Table 6.3[B]). Biomass increases almost linearly over the range from a maximum age of 50 to 500 with only a slight tendency toward an asymptote. Biomass is zero at a maximum age of 50. Balsam fir with longevity restricted to 50 years cannot persist in the test site.

Figure 6.2 shows the response of sugar maple to changes in parameters for balsam fir. An increase in maximum age of balsam fir leads to a decrease in the biomass of sugar maple, but the slope is gentler than the increase in biomass of balsam fir, and a significant change occurs only at the extremes (Fig. 6.2[A]).

The sensitivity of the model to AINC_i, the second mortality parameter, is shown in Figures 6.1[C] and 6.2[B]. Remember that this parameter sets a threshold. A tree with diameter growth below that threshold for 10 years has only 1 chance in 100 to survive. The default value, 0.02 cm/yr, gives the same results as when the parameter is set at zero. This indicates that the default value does not affect the dynamics at the test site. Doubling the value to 0.04 cm/yr results in no change, but an increase beyond that level leads to a rapid drop-off in balsam fir biomass until the species becomes extinct at a value of 0.2 cm/yr. This experiment suggests that any factor or set of factors that produces a high chance of mortality in balsam fir, for all sizes and ages of individuals with diameter increments equal to or less than 0.2 cm/yr, will result in extinction of this species. This could be tested empirically by creating a stress in a test stand that reduces diameter increments below this level and observing the fate of balsam fir for 10 years.

As a wetland species, balsam fir should be able to withstand high water tables, and that is the result indicated in Table 6.3(F) and Figure 6.4(A). For the Virginia, Minnesota site, balsam fir is insensitive to changes in the soil-water saturation parameter; it can withstand a soil saturated to 10 cm below the surface, and its biomass does not show a decline until the water table depth is lowered to 60 cm. Balsam fir is also insensitive to changes in the drought tolerance parameter for a slightly drier site, shown in Table 6.4 and Figure 6.4(B), where the soil depth and the water table depth is 1 m. Examining only the response of balsam fir at this one site, one might conclude that the minimum water table depth does not affect the dynamics of the model and might be incorrectly formulated.

A more correct impression of the sensitivity of the model is obtained when we examine the response of an additional species, one characteristic of a dry site. Jack pine is such a species, and its response is shown in Figure 6.5 and Table 6.5 on a typical site for that species, a coarse sandy soil of low fertility in central Michigan. Jack pine is sensitive to changes in the drought tolerance parameter.

Remember that

WiF_i = max{0,1 - (WILT/WLMAX_i)²} (3.7)

where

WILT = (E₀ - E)/E₀ (3.8a)

E₀ is the potential evapotranspiration and E is the actual evapotranspiration.

Equation 3.8 produces a steep-shouldered curve, suggested by the steep rise in jack pine biomass at the Grayling, Michigan, site as the drought tolerance parameter increases from 0.4 to 0.6.

Figure 6.4 Sensitivity of Balsam Fir to (A) Minimum Water Table Depth; and (B) Drought Tolerance Parameter. Conditions are as in Table 6.1. Results are shown for year 90 after start of model run.

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(a) Soil depth is 1.0 m and depth to the water table is 1.0 m.

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WLMAX Jack Pine Biomass

(kg/m²)

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0.1 0.0

0.4 0.0

0.45 0.3 _ 0.1

0.46 0.4 _ 0.2

0.48 1.0 _ 0.3

0.49 0.9 _ 0.3

0.5 1.0 _ 0.3

0.53 (default) 0.9 _ 0.3

0.9 1.7 _ 0.5

1.0 1.9 _ 0.4

1.5 1.6 _ 0.5

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(a)Site is a coarse, sandy soil near Grayling, Michigan, Weather records are from Grayling, Michigan. Soil depth is 1.0 m and depth to the water table is 1.2 m. Elevation: 900 ft; water table depth 1.2 m; soil texture 50.0 mm/m; percent rock 0.0; avail. nitrogen 55.0 kg/ha. Biomass is at year 90 beginning with an established jack pine stand whose population composition is given in Appendix IV, Table A-IV.2.

Figure 6.5 Biomass of Jack Pine at Year 90 as a Function of the Drought Tolerance Parameter, WLMAX. Site Conditions are given in Table 6.5.

Another important kind of sensitivity test is the response of a model to extrinsic factors. In the case of the forest model, an important test of this kind is the sensitivity to choice of weather records. Tests of this kind have been carried out for projections of the effects of global warming on forests, which are discussed in the next chapter.

As a generalization, the forest model is moderately sensitive to errors in estimation of intrinsic parameters, which affect the fundamental population dynamics (parameters that control the maximum number of saplings that can be added in any year and the increase in mortality at low growth rates). As revealed by the examples discussed here, the location of a site relative to the range of a species affects the relative sensitivity of the output to errors in parameter estimates. This is as one would hope and expect; it indicates that the model properly responds to changes in site conditions.

The sensitivity is moderate, which is encouraging. As I discussed at the beginning of this chapter, moderate sensitivity means that the output of the model is likely to be robust against accuracy of field measurements, yet sensitive enough to show changes with an expected range of environmental conditions. The realism of the projections of the model would seem to be in part a result of its moderate sensitivity.

To the extent that one accepts JABOWA as a realistic model of forest dynamics, the sensitivity tests yield some insights into these dynamics. These can be used as hypotheses about populations of trees within forest ecosystems. Among these insights are: there seems to be a reasonable correspondence between the present geographic ranges of trees and present climate, at least sufficient to approximate the temperature boundaries of the trees, in spite of continual long-term migration of tree species across the landscape. This reinforces the use of the correlation between pollen analysis and past climates to project future geographic ranges of tree species.

Trees must be able to withstand highly suboptimal conditions and the consequent slow rate of growth, if they are to persist within a forest. Survival of existing trees is more important for persistence of a species in a forest that the quantity of individuals recruited each year to the population; thus longevity and the ability to mature trees to survive years of very poor conditions are more important than recruitment; this is based on the result that the model is more sensitive to changes in mortality probabilities than to the maximum number of saplings that can be added in any one year. Changes in genetically determined thermal boundaries for a species can change the growth of an individual tree within these boundaries except at the very center of the range; this is based on the shape of the temperature response curve. Narrowing the thermal range steepens the temperature response function, and increases the growth at any point except at the exact center. The influence is quantitatively greater at the edges of the range than near the center. These examples of interpretation of sensitivity tests show that such tests are not only valuable as part of the technical evaluation of the model, but also as a method to help us learn about the dynamics of forested ecosystems.

There are many other factors for which sensitivity tests could be made. In addition to the tests of other parameters, there are tests that concern larger assumptions of the model, including tests of the effects of plot size on the model, tests of combinations of species characteristics not present with existing species (such as great longevity in a shade-intolerant species), tests of how the model would respond if all species had the same light response function for growth but not for regeneration, and vice versa; tests in which regeneration were made deterministic while mortality remained stochastic, and vice versa. The reader can no doubt think of others. Users of the software available as a companion to this book can conduct many of these tests.

With the accuracy of the model demonstrated in Chapter 6, and the moderate sensitivity demonstrated in this chapter, we can have greater confidence in the application of the model to applied problems and to its extension as a method in fundamental research. The next chapters explore the application of the model to applied problems.