Processes in the Swiss National Park SNP

This is an example of the application of modelling techniques presented in the previous sections. The aim of model building was to reproduce the temporal pattern of succession revealed by space-for-time substitution, explained in Section 9.4.2. The description of the model follows Wildi (2002).

10.3.1 The temporal model

To keep complexity under control, species are grouped into six guilds (assemblages of species) (Wildi & Schutz 2000): these are the state variables. The basic process considered is thus colonization of plots and subsequent species interactions. The plots (the cells in the model) accord to the research grid established in the SNP for the purpose of investigation. Plot size is 20 m by 20 m and the number of plots within the unforested investigation area, Alp Stabelchod, is 286 (Achermann et al. 2000, see Figure 10.8). For simplicity, it is assumed that the total surface the species guilds occupy never exceeds 100% of the plot. The model plot is eventually overgrown by one or several species guilds, so that in the end no open soil is left.

Next, the objective is to quantify overgrowth and also replacement. In the original time series from the permanent plots it can be observed that overgrowth always starts slowly (Wildi & Schutz 2000). With increased cover of the guilds, spread accelerates. When approaching 100%, overgrowth

Figure 10.7 Overgrowth of a plot by a new guild. The white squares indicate patches inappropriate for growth, causing it to slow towards the end of the invasion process.

slows down. This finding is illustrated graphically in Figure 10.7. A function that mimics this behaviour is the logistic growth equation; in case of only one guild, it has the general form:

dt K

(Wissel 1989, see also Equation 10.6). Here, r is the growth rate of guild X and K is the carrying capacity; that is, 100% of the plot surface. As X is also measured in terms of percentage, the space not yet occupied is 100% — X. Colonization stops when X reaches 100%. The growth is regulated by X itself, as a result of intra-specific competition. It must be noted that logistic growth requires all guilds Xi to be present in a minimum quantity at the beginning of any simulation run.

Competition comes into play because of two assumptions. First, the gain in cover of guild Xi is at the expense of any other guild's lower competition power (or open ground). In order to keep the cover percentages balanced, the growth equation will have two components: a gain by population growth and a loss to better competing species. Second, 100%—Xi is the available space only for the best competing guild, i. If there is another, more successfully competing guild, Xj, then the space reduces to 100% — Xi — Xj. As will be seen in the description of the model, the mechanism has to make provision for many more competing guilds; six in the present case. Based on previous findings (Wildi & Schutz 2000) the following order of competition power was assessed:

Pinus(\) ^ Carex(2) ^ Festuca(3) ^ Trisetum(4) ^ Deschampsia(5) ^ Aconitum(6)

The logistic growth equation for Carex(X2), which is out-competed by Pinus(X1), is given by:

dt K

Carex is growing according to the logistic growth equation. But in addition a portion of the surface that Pinus (X1) is winning is subtracted. For Festuca (X3) there is additional proportional loss to Pinus and Carex:

dt K

The growth equations for all subsequent guilds are built accordingly.

The third important factor in this pasture is recurrent disturbance; that is, trampling by grazing deer (Kriisi et al. 1998). I assume that it affects all the plants within a plot similarly. The intensity will of course vary depending on animal density. Trampling is a very fast process, instantly generating open space. This causes a loss ti for guild i, which is simply proportional to its state, Xi. Re-colonization ci is also fairly fast. I assume that it happens instantly; that is, within the relative short time span of one year, the standard time-step length of the model. It is proportional to the exponential growth of each guild. Direct competition, as happens in species replacement, is not assumed. Trampling and re-colonization are balanced within the year:

This assumes that growth is sufficiently fast to colonize any gap that has occurred within one year. Furthermore, trampling leads to a yearly shift of the guilds, favouring the fast growing, provided the growth rates ri differ.

10.3.2 The spatial model

The following notation is used:

hx,y,t\i = 6; x = 1,..., 25; y = 1,..., 30; t = 1,..., 400|

where i stands for guild, x and y are the spatial coordinates and t is time, in years. The model space is a grid of 25 by 30 plots (Figure 10.8). Not only the pasture but also the adjacent forest stands fit into this rectangle. The spread of any one guild happens by spatial exchange. A portion of the content of any plot is transferred yearly to the neighbouring plots, as shown in Figure 10.6. The gains, g, and losses, I, are balanced:

d(xi,x-1,y,t + xi,x+1,y,t + xi,x,y-1,t + xi,x,y+1,t) (10.17)

From Equation 10.17 we see that the gain always comes from all four directions. The losses in all four directions (Equation 10.18) are the same as they are proportional to the composition of the central plot. The velocity of exchange is given by factor d. This is assumed constant, even though spatial

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2 Trisetum type

3 Deschampsia type

4 Festuca type

5 Carex type

6 Pinus type

Figure 10.8 Spatial design of the SNP model and the initial state (see Table 10.4 for contents of the cells).

1 Aconitum type

2 Trisetum type

3 Deschampsia type

4 Festuca type

5 Carex type

6 Pinus type

Figure 10.8 Spatial design of the SNP model and the initial state (see Table 10.4 for contents of the cells).

processes may be faster where more animals prevail. Having no measurements of exchange at hand, I keep it at the very low level of d = 0.001.

Along the edges of the system, outside the meadow, the exchange is mirrored. All these plots are covered by Pinus mugo forest, the final state of succession considered in the model.

10.3.3 Simulation results

A comparison of the different graphs in Figure 10.9 shows that, with a suitable choice of parameter values, the temporal model can reproduce the basic pattern of the time series. In the model output (graphs on the right-hand side) fluctuations are absent. The results for the 415-year and the 585-year simulations were obtained only after carefully adjusting the initial conditions (the state variables) and approximating the growth rates of the guilds by trial and error. Initial conditions for both model runs are shown in Table 10.3. The initial states of the model are within the range of the values observed in the field (Wildi & Schutz 2000). It must be noted that the observed initial cover values are themselves affected by random fluctuation, whereas in the deterministic model a fixed value is needed. Simulation runs show that the time of emergence of late successional guilds depends on the initial states.

°SS8B8?8SS8 Di8SS8iSSg8

Aconitum -o- Trisetum — Deschampsia -»- Festuca * Carex Pinus

Figure 10.9 Original (left) and simulated (right) temporal succession. Upper row: 415-year version. Lower row: 585-year version.

Aconitum -o- Trisetum — Deschampsia -»- Festuca * Carex Pinus

Figure 10.9 Original (left) and simulated (right) temporal succession. Upper row: 415-year version. Lower row: 585-year version.

Table 10.3 Initial values in the times-series data and initial state variables (percentage cover of six guilds) and growth rates used in the models.

series 415 year 585 year variable state, % state, % growth rate state, % state, % growth rate field data model model field data model model series 415 year 585 year variable state, % state, % growth rate state, % state, % growth rate field data model model field data model model

Aconitum

41.1

53.0

0.050

73.1

86.0

0.018

Trise tum

33.0

27.0

0.045

11.4

8.4

0.022

Deschampsia

10.3

10.0

0.045

11.3

4.0

0.022

Festuca

12.1

7.0

0.045

3.0

1.6

0.020

Carex

3.1

2.8

0.035

0.6

0.03

0.026

Pinus

0.3

0.2

0.030

0.6

0.01

0.022

In other words, the initial state of the guilds determines the speed of succession. This is not a realistic feature as it does not consider invasion. Even worse, in order to allow growth to occur in a solely temporal model, all species guilds have to be present in the model (i.e. within all 268 plots considered) from the very beginning of the simulation.

The approximated growth rates yielding a realistic model behaviour do not differ much between guilds. That is, growth rates do not much affect the temporal pattern the model generates (Table 10.3). However, the two time series differ in their growth rates: in the 415-year model, growth has to proceed twice as fast as in the 585-year model. If all rates are set to 0.045 (not shown here), succession will last about 400 years; when taking 0.022 this will be close to 600 years.

Including spatial extent in the model creates problems with the initial condition of the meadow; that is, the state of all 268 plots in the year 1917 (outset of succession), which is not known to us explicitly. From the present state of the meadow and the direction and rate of change observed in many similar plots we are able to suggest a simplified state using the same composition for all cells belonging to the same type as initial conditions in the year 1917 (Table 10.4, Figure 10.8). Hence, at the beginning of simulation all plots of the same succession stage have identical species scores and the system consists of a limited number of discrete states, whereas in reality the vegetation forms a continuum. As soon as simulation begins, diffusion causes differentiation of cells and all maps become continuous.

Table 10.4 Six discrete vegetation states used as initial conditions of the six state variables (guilds) in spatial modelling, cover percentage. Maps in the first column (t = 0) in Figure 10.10 are composed of these states.

Guild

no.

state 1

state 2

state 3

state 4

state 5

state 6

E

Aconitum

1

50.00

17.50

17.50

10.00

5.00

0.00

100.00

Trisetum

2

10.00

35.00

35.00

15.00

5.00

0.00

100.00

Deschampsia

3

7.00

15.00

35.00

35.00

6.00

2.00

100.00

Festuca

4

2.00

3.00

30.00

42.00

20.00

3.00

100.00

Carex

5

1.00

1.00

10.00

15.00

65.00

8.00

100.00

Pinus

6

0.00

0.00

1.00

1.00

8.00

90.00

100.00

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