Markov models

In this section a model process capable of reproducing simple cases of mul-tivariate patterns of change is presented. Very much like linear regression, it is elementary by nature in frequently fitting observed patterns locally. It abstracts from noise, complexity and nonlinearity and therefore fails to fit patterns when the rules in systems change, such as the competitional hierarchy of species. However, the process remains fundamental as its application may frequently be the first step in the evaluation of temporal patterns. The functioning and use of Markov models for vegetation surveys is explained below (see also Wildi 2001).

Changes in permanent plots can be interpreted as replacement processes. Several plant populations occupy the same resource. In the context of vegetation ecology, the primary resource is frequently the physical space. Because space can be defined as a fundamental resource occupied by plants, cover percentage is a logical (even though it is two-dimensional only) surrogate for measuring resource consumption. If gains and losses in space are in balance, so that any state of the system can be derived from the previous one, then screening for a Markov process is worthwhile (Usher 1981).

Win and loss of every species is defined in a transition matrix P. This allows derivation of the state of the releves x at time t from the preceding step:

In an ordinary permanent plot survey the observation vectors x are vegetation releves. Unfortunately, this observation does not allow measurement of the elements of the transition matrix: the wins and losses of the species (and this is the main reason why Markov models are not used routinely): if a species wins space then we do not know which other species has lost it, and if a species loses ground, we do not know which one will profit from it. A Markov process, if present, remains undetected. Two plausible assumptions may help to overcome this situation in a method devised by Orloci et al. (1993):

1 If a species loses part of the main resource then any other dominating species will most likely profit (i.e. profit is proportional to the species cover).

2 If a species increases its partition of the resource then the remaining dominant species will lose in proportion to its cover.

Both of these assumptions can be questioned. In succession, a change in abundance of a species may be caused by colonization of space by a new species, but a Markov model in its basic form does not foresee invasion. Similarly, a colonizing species may expand its cover at the expense of rare species. This is one indication why Markov chains cannot be applied in isolation when invasion occurs.

The resource space is not always entirely occupied by the vegetation. For this reason it may be important to add one more variable to the species list: quantifying the open soil (see Figure 9.5 for an example). Formally, this functions like any ordinary species. It is important that the sum of all cover values, including open space, exactly amounts to 100%. This is achieved by the appropriate transformation of the initial state:

Figure 9.5 A Markov model of the Lippe et al. (1985) data set. Upper graph: field data. Lower graph: simulated data.

In this equation, vector x contains the cover values of species i, t is the present time and n is the number of species. In the artificial example used for further explanations (Table 9.1), the sum of cover values is already 100%.

First the transition matrix for time step 1 to time step 2 is calculated (Orloci etal. 1993). For each species i a difference results, expressing change in time:

Positive values of Diff(i) signify a gain, negative ones a loss. The transition matrix contains all the losses of species i in row i, and the gains of the same

Table 9.1 Numerical example for demonstrating a Markov process (raw data).

Xt=1

Xt=2

Xt =3

Species 1

60

40

10

Species 2

25

35

55

Species 3

15

25

35

E

100

100

where | is gain and ^ is loss of species i. The diagonal elements contain the proportions each species covers at the end of the time step: xi,t+i. The gains of species i at the expense of species h, as well as the losses of species i from species h, are given by the equation:

This means that gains and losses occur in proportion to the resource (i.e. the space) each species occupies at the end of the actual time step. When processing the first species in our example, it can be seen that it loses 20% of the total ground from t = 1 to t = 2 (Diff(1) = -20). The new diagonal element is 40. The loss of the second element, a portion of 20% of the 35% cover, is 7%. The third element is 20% of the 25% cover -5% - completing the first row:

40

7

5

P(t 1; 12; spec.1) =

0

0

0

0

0

0

Species 2 exhibits a win of 10% (a factor of 0.1) to be added in column 2. This is again proportional to the covers of the species at time t + 1:

The procedure is completed by applying it to species 3: P(t 1; 12) =

After normalizing the rows (the sum adjusted to 1) the transition matrix is: P'(t1; 12) =

For each following time step the procedure is resumed to yield one more transition matrix, as for time steps t = 2 to t = 3, where it is:

10

10.5

11.5

P(t2; t3) =

0

55

5.5

0

7.0

35

For all time steps, all transition matrices, P, are averaged. The new transition matrix, P, is assumed to hold for the entire time series. This also means that it is kept constant over time and it will therefore outbalance fluctuations. In our example, after normalizing by rows, we get:

0.500 0.2950 0.2050 0 0.9091 0.0909 0 0.1367 0.8633

Through simple matrix multiplication according to Formula 9.1 the simulated releves are derived (Table 9.2).

Here the first releve is identical to the field data (Table 9.1). It represents the initial state of the dynamic system; all subsequent states are merely

Table 9.2 Numerical example demonstrating a Markov process (simulated data).

Species 1 60 30 15

Species 2 25 42.5 51.23

Species 3 15 27.52 33.77

approximations. This becomes obvious from the examples below. OrlOci et al. (1993) published a time series documenting recovery of a heath-land after fire, using data from an investigation by Lippe et al. (1985). The example is presented here as a case where a linear Markov process successfully reproduces the temporal pattern found.

The raw and the simulated data ('Lipperaw' and 'Lippesim' in Appendix B) are shown in Figure 9.5. From the upper graph it can be seen that in the first few years there is a directed change. After about eight years (^ 1970), an equilibrium state is reached in which merely random oscillation occurs. One objective of the analysis is to determine the equilibrium state for which the Markov model is derived. The transition matrix is calculated as shown above; that is, from the 19 states of the system. It is the mean of 18 matrices calculated for each time step. Then, beginning with the first field observation, 18 Markov releves are derived. They are shown in the lower graph in Figure 9.5. After 19 years, the model has almost reached an equilibrium state. In the present example, the model fits the field data almost perfectly. However, only the deterministic part of the variation is reflected by the simulated time series; the temporal fluctuation is completely suppressed.

The Markov model perfectly explains multispecies change as long as this change is linear. But why is it called linear? The response curves are not linear, but curved and monotone. However, the similarity pattern is linear and when the data is ordinated it can be seen that the time trajectory is an almost straight line (Figure 9.6, upper graph) and the Markov model

Figure 9.6 PCOA ordination of the Lippe succession data. Upper graph: field data used. Lower graph: Markov data used. Arrow pointing in direction of time.

generates a perfectly linear pattern (Figure 9.6, lower graph). This illustrates that a linear process yields a linear pattern, even if the underlying species response curves are bent!

But when does the linear Markov model fail to reproduce succession? This is simply the case if the 'rules of the game' change with time. In terms of the vegetation process, it takes place when the relative competitional power of species changes. Ordinations will reveal such situations in presenting time series as horseshoe-shaped trajectories (see Figure 9.16 for an example). While the data in Figure 9.5 shows a successful application, I also add one demonstrating a failure in Figure 9.7. This is the successional series from the Swiss National Park, presented in more detail in Figure 9.13. There is a fairly good match during the first few of the 81 time steps but the shape of

Figure 9.7 A Markov model (Lower graph) of the time series of the Swiss National Park (upper graph), taken from Figure 9.13. y-axis shows relative cover.

the real and simulated curves start deviating fundamentally as soon as new species invade and the equilibrium state in Figure 9.7 is far from reality.

A Markov chain of the type shown above simulates a classical succession towards a monoclimax: the vegetation reaches an equilibrium state inherent in the model, from which it does not escape. In systems theory, the final state is called a point attractor. In other systems, however, it can happen that a cycle is reached, as proposed by Watt (1947) (meaning that the system has a cyclic attractor), or there can be apparent random fluctuations like in the data of Lippe et al. (1985).

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