Journal of Environmental Management (1990) 30, 11 l-l 30

The Semi-Markov Process: a Useful Tool in the Analysis of Vegetation Dynamics for Management

Andrew D. Moore

Ecosystem Dynamics Group, Research School of Biological Sciences, Australian National University, G.P.O. Box 475, Canberra 2601, Australia

Received 28 July 1988

A general mathematical model of vegetation dynamics is presented which is based on the semi-Markov process (an extension of the Markov process). When biological models of succession and natural disturbance are used in conjunction with it to define the community states, the semi-Markov process overcomes the well-known problems of the Markov process as a model of vegetation change. The semi-Markov model may also be used to analyse plant-by-plant replacement processes. Analysis of the semi-Markov model produces equations giving quantities of interest to land managers: (1) the probability that a vegetation stand will be in a particular state at a given time, (2) mean times to local extinction of vulnerable species and (3) optimal use of a prescribed disturbance, given an assessment of the value of each community state. The analyses are illustrated with an example from forests of western Montana. In order to construct the example, the vital attributes scheme of Noble and Slatyer (1980) has been both corrected in order to make it dynamically sufficient and extended to accommodate multiple disturbance types.

Keyworcls: semi-Markov process, mean extinction time, vital attributes. prescribed disturbance.

1. Introduction

It is now two decades since the simple Markov process was first used in vegetation science (Anderson, 1966; Stephens and Waggoner, 1970). In that time, it has come to be applied in two main contexts: in the analysis of plant-by-plant replacement processes (Hobbs and Legg, 1983) and in the investigation of the dynamics of stands of vegetation (Stephens and Waggoner, 1970; Shugart et al., 1973). The distinction between the two contexts was made clearly by van Hulst (1979), but has been blurred on occasion for two reasons: first, the Markov processes describing them are formally identical, and, second, the mathematical fact that the stationary time distribution of a regular plant-by-plant replacement process is the same as the equilibrium distribution of occupancies in a stand made up of a mosaic of individuals.

111

03014797/90/020111+20$03.00/0 0 1990 Academic Press Limited

II2 Vegetation and the semi-Markov process

The appeal of the Markov process to ecologists lies in its ability to model systems with a stochastic component, and in its extreme simplicity. It has been claimed to possess predictive utility by some (e.g. Horn, 1975); at other times, it has been presented as a useful heuristic device for formalising concepts in the study of succession (Usher, 198 1).

Markov models of succession have been reviewed critically by van Hulst (1979) and Usher (198 I). Van Hulst (I 979) considered that their major drawback was the necessity of assuming that vegetation processes were “memoryless”: that is, that the probabilities of transition depended only on the current state and not on the state at previous times. He quite rightly considered this assumption to be biologically unlikely. (The alternative of making the state description more detailed results in the state space “blowing out” to enormous size.) Usher (198 1) restated this criticism, together with a number of others; his major new point was that defining the states of a process was vitally important, but very difficult. The difficulty is not apparent, however. in the study of plant-by-plant replacement, where the state may be readily defined by the species occupying a point.

This paper applies an extension of the simple Markov process. known as the semi- Markov process, to the analysis of vegetation stand dynamics. The semi-Markov process is capable of dealing with historical effects such as are caused by non-equilibrium age structures. The question of state definition is resolved by returning it from mathematics to biology; I assume that there exists a model, based on biological processes, which defines the states and also constrains the possible transitions between states to manageable numbers. At least two such models exist: the vital attributes scheme of Noble and Slatyer (1980). and the FATE model of vegetation dynamics (Moore and Noble, 1990). Both these biological models incorporate the effects of natural disturbance as well as of succession. Various conceptual models of vegetation change, for example those proposed by Childes and Walker (1987) or Marrs (I 988) might also be adapted to the purpose. As Noble and Slatyer (198 1) have observed, Markov models are generally fitted to data rather than being used to explain them; the use of a biological model to generate a stochastic process obviates this.

The semi-Markov process has appeared only once before in the ecological literature. Henderson and Wilkins (1975) used it to analyse the equilibrium distribution of community types in forests of southwestern Tasmania. They used a set of states provided by the biological work of Jackson (1968). Contrary to their conclusions, their semi- Markov model actually predicted the proportions of communities in the landscape of southwestern Tasmania reasonably well (see Brown and Podger, 1982). Henderson and Wilkins (1975) mentioned that the process has wider application in vegetation science, but it has not been developed further until now.

There is a body of mathematical results regarding the semi-Markov process (Howard, 1971), many of which are analogous to results from the theory of Markov processes. These results become available for the analysis of vegetation dynamics once vegetation change has been described in semi-Markov form. In particular, the equi- librium properties of a semi-Markov process are simple extensions of those of a simple Markov model and are not especially interesting. Other features of a semi-Markov process description of vegetation dynamics, however, lead to results of interest to those concerned with the conservation of vegetation.

2. The semi-Markov model

The description of the semi-Markov process and the notation used in this paper are

A. Il. Moore II3

based on Howard (1971). I will concern myself only with the discrete formulation of the model; a continuous-time analogue does exist.

As in the Markov process, a semi-Markov process moves between a number of states in a stochastic manner. The difference between the two processes lies in the nature of the transitions. In the semi-Markov process, transitions need not occur at all time steps, or epochs; instead, when the process enters a state, a waiting time is randomly chosen from a distribution which is specific to that state. At the end of the waiting time, the process undergoes a transition. The probabilities of transitions depend not only on the initial and destination states, but also on the waiting time.

Some basic quantities are:

w,(m)= the probability that the waiting time in state i will be exactly m time steps. p$m) = the probability of a transition from state i to statej given that the waiting time in

state i was m time steps (known as a conditional transition probability). c,,(m) = the probability of a transition from state i to state j exactly m time steps after

entry to state i. The matrix of functions C(m) with elements c,(m) completely specifies the process; it is known as the core matrix.

P,, = the probability that a transition out of state i will be to state j. This may be calculated by summing the values of c&m) over all possible epochs m.

6,= I if i = j; 6, = 0 otherwise.

3. A model of vegetation change using the semi-Markov process

As stated above, I assume that there exists a model based on biological processes which defines the states (in terms of species composition and/or age structure) into which a stand of vegetation may be classified. I further assume that transitions from state to state may occur only as a result of two processes:

(i) Successional change in community structure. This may be thought of as causing a transition from one state to another at a fixed time after the vegetation enters the original state. The destination state is always the same for each initial state.

(ii) Naturaf disturbances occur stochastically. The probability that a disturbance of a particular type occurs in a community is assumed to be a function of the community composition and the time since the vegetation entered the state; this assumption allows for such biological phenomena as the buildup of fuel in fire-prone vegetation. I assume that the biological model predicts only a single possible outcome from the disturbance of a particular community; cases such as fires of different severity may be treated by considering them as a separate disturbance types. (Alternative formulations are entirely possible.) It is assumed that only a single disturbance may occur in a time step.

The above assumptions could be relaxed somewhat; I have chosen them as the simplest which possess a reasonable degree of reality. At the spatial scale of interest (a “landscape unit”: Naveh and Liebermann, 1985), epochs of one year are usual, but this is not necessary, as will be seen below.

The specification of a semi-Markov process to describe this system is as follows:

Let there be N states (i.e. different community compositions) and D distinct disturbance types.

114 Vegetation and the semi-Merkov process

For each state i, define the following:

s(i) = the state to which the vegetation will change as a result of succession; L;= the length of time from entry into state i to exit into state s(i) in the absence of

natural disturbance. Some vegetation states, corresponding to a vegetation “climax” or climaxes, will be self-perpetuating in the absence of disturbance; in this case, Li may safely be set to infinity.

d(i,q) = the state to which the vegetation will change if disturbance q (q = I . . . D) occurs while it is in state i;

g,(n)= the probability that disturbance q will occur if the vegetation is in year n of an occupancy of state i (q=I . . . D, n=l . . . L,).For climatically induced natural disturbances such as cyclones or floods, g,(n) is likely to be constant for all states and times. For fires, it is likely to increase with time as fuel builds up, and communities are differentially flammable. One might use a fuel buildup model such as that of Kessell et af. (1982) together with a fire ignition model to approximate the changes in fire probability.

G,(n) = the probability of any disturbance in a community during year n of its occupancy of state i. This is simply calculated as

Gi(n)= g g,,(n) Q i=l...N, (1) lr= I n=l . . . Li

The waiting time distribution for the semi-Markov process describing this vegetation is given by

m-l wi(m)=G;(m) n [l - Gi(n)l Q m=l . ..L.-1

I n= I

L,- I

Wi(Li)= n [l -Gi(n)l n= I

Wi(m)= 0

and the conditional transition probabilities by

Q m>L,

Q m=l. q=l.

Pi,Ni) (LJ = 1 - GjLJ P,(n) =0

This gives a “core matrix” C with elements

m- I

LiP D

1

Ci,d(i,q>(lfZ)=giq(m) n [l- GAnll n= I

otherwise.

Qi=l . . . N, q=l . . . D,

m=l . . . Li I

. .

Ci,s(i)(Li)= 6 [l- Gi(n)l ll= I

c,(m) =o otherwise.

(2)

(3)

(4)

A. D. Moore 115

I will use the notation Ct4 (n) for ci,d(i,qj (n) and S, for c,,~~) (L,). In the case where two different disturbance types result in the same transition, i.e.

d(i.q) = d(i,r) for some i and q # r, this formation is not strictly correct: the true values of C,y will be sums of the two (or more) values. This will not cause any problems, however, as every use made below of the C,,s involves summing them over the disturbances.

4. Interval transition probabilities

The calculation of interval transition probabilities uses the following reasoning. There are two possible paths by which the vegetation may reach one state (sayj) from another (say i). First, if the two states are the same, then it is possible that no transition has occurred. Second, the vegetation may have undergone a first transition from state i to some other state (say k) at time m, and then made its way to statejin the remaining (n-m) time steps. The probability of going from state i to state k at time m is cik (m); the probability of then moving from state k to state j in (n-m) steps is another interval transition probability.

So, if we calculate the probability > wi (n) that the waiting time is greater than n, i.e. that the vegetation does not change state up to time n:

>w,(O)= I

.> Wi(rn) = fi [l-Gi(n)]=~lm,=;; v m=l . ..L.-1 I

(5) n= I ,

>w, (m)=O V m2Li

We can then sum the probabilities of all the possible ways of reaching statej at time n to arrive at the interval transition probability.

C,k (4 P&, (n - ml V i=l...N, k=lm=l j=l . . . N. (6)

n>O

This expression allows the recursive evaluation of any p value. If we evaluate it for the general vegetation model (Equations 4), we obtain two different expressions, one for n<L,andoneforn2L;

V n<L,,

pti (n)=bij wicn+ l) G (n+ *)+ 5 i c;q (m)%(i.y,Jn- m) I fJ=lm=l

\ (7) V nZ=L,, I

D Ll rpj,~n)=~i~.7~,,,j (n-L;)+ C C Ciqtm)V)d(i.q),j(n-m)

q=l m=l I

The first term is missing from the second of these equations because a first transition out of a state is certain once we pass the time required for succession.

116 Vegetation and the semi-Markov pracess

If the process is monoaksmic (i.e. has at most one trapping state or set of states), then the interval transition probabilities will tend to equilibrium values which are indepen- dent of the starting state. The equilibrium distribution of cp is given by the equation

(8)

where

tj= the mean time spent in state i, given by

L

Tjj= i n wj (n); (9) ?I= 1

rrj= the equilibrium probability of occurrence of statej in the Markov process defined by considering only the points at which the semi-Markov process makes a transition. (This Markov process is known as the embedded Markov process.) The values of zj are obtained by solving the set of simultaneous equations

N

rj= c 7Tipii

i= I

(10)

Equation (8) tells us that, at equilibrium, the proportion of time that the vegetation is in a particular state varies with both the relative number of times it enters the state, and to the mean time that it remains in that state. This result was stated intuitively by Horn (1975, p. 200).

5. An example: a forest community in Montana

Figure 1 shows a replacement sequence for a forest community of western Montana (Kessell, 1979) derived using a slightly extended version of the vital attributes scheme. In this extension, communities which have all species in the same life stages may nevertheless be distinguished if they differ in the time from the last period of high resource availability (and consequent recruitment of intolerant species). This time is shown where necessary on Figure 1 in square brackets. For the present, I consider only a single disturbance type, namely wildfire; for simplicity, I shall use an epoch of 10 years in the example analyses.

For a disturbance model, I assume that the probability of a wildfire remains constant at a value ai within each community state i; the values for each state are given in Table 1. Since there is only one disturbance type, Equations (2) and (3) simplify to:

v i=l . ..N.

wi (m) = a, (1 - a,)“- ’ V m=l . . . L,-I

wi (Li)=(l --Ui)Li_ 1

wi (m)=O V m>Li I

A. D. Moore 117

V i=l...iV,

V m=l . . . L,-1 \

Pi.d(i.1) Wi) = a;

P,,.v(r) (4) = 1 - ai

p,(m) =O otherwise

The elements of the resulting core matrix are shown in Table 1. It is worth noting that, apart from the parameters required for the vital attributes description of the vegetation (three to five per species), only one parameter per community state (a,) is required to set up this predictive model. Further, a, has a clear biological meaning.

Table 2 gives values of the interval transition probabilities for the example com- munity described above, assuming that it started in state 1. It also shows the equilibrium values of the interval transition probabilities.

Species 0 20 130 150 180? 300 _

Popuhs trembides (AS) VI -m-k

Pinus confofta (LO) GI nl~l---8

Lark midmfalis (LA) DI -nl 1-e

Rcea gfauca (SP) DT --m k

/--------- f \

1. AS,+LO,+LA,+SP, 20 2. AS+LO+lAtSP 110 3. LO+LA+SP 20 4 LOp+LA+SP ) - * IO1 1201 I [I301

/

j 11501

__----

J

,- _____/

I t 110

10. Llq+L++SP, M -11. LO+LA+SP 30

Figure 1. Dynamics of a forest community of northwestern Montana derived using an augmented version of the vital attributes scheme (see text). The only disturbance type incorporated is severe wildfire. Successional transitions are shown with solid arrows while transitions due to wildfire are shown by dashed arrows. The subscript “j” denotes the immature population stage; the subscript “p” denotes the propagule population stage. Numerals by arrows denoting succession give the length of the successional stage. Note that this convention differs from that of Noble and Slatyer (1980), who placed times since the last disturbance in the same positions on their replacement sequences. Numerals in square brackets give the time since the last period of high resource availability.

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120 Vegetation and the semi-Markov process

6. Extinction probabilities

Many conservation reserves are created, at least in part, in order to preserve particular rare species. It is therefore important to manage such reserves so as to minimise the probability that these species become locally extinct, that is to maximise the mean time to the extinction of the species. We can use the semi-Markov process model to estimate the mean time to local extinction of a vulnerable species under a given disturbance regime and so compare different disturbance regimes.

To do this, we first alter the semi-Markov process described above by lumping all states in which the species is extinct into a single state e. If E is the set of states in which the species is extinct, then the new process has a core matrix C’ (m) with elements

Cfie (n) = c jtE

clJ (4

cfij (n) = cv (n)

V i#E, n>O (11)

V i#E, jEE, n>O

It will become necessary to calculate the embedded transition probabilities ptij from the core matrix. Extinction corresponds to entry into state e. If we assume that the vulnerable species cannot recolonise the site, then e is a trapping state; once the vegetation enters state e it never leaves it. (If the species can recolonise, then the following analysis still applies to the mean time to an episode of extinction.) The time to extinction is known mathematically as thejirst passage time (fl,,) from an initial state i to state e. Note that the first passage time is calculated from the time of entry to state i.

Usually, e will be the only trapping state; biologically speaking, this is equivalent to saying that the species of interest remains vulnerable no matter what happens to the rest of the vegetation. A hypothetical exception is where competition from another species is necessary to cause the extinction, but the second species is also vulnerable; in this case, the mean time to extinction of the first species becomes infinite, as it will persist indefinitely if the second species precedes it to extinction. In such a case, the corre- sponding semi-Markov process is polydesmic (has more than one distinct set of trapping states) and rather less amenable to analysis.

If e is in fact the only trapping state, then the mean extinction time 8, can be calculated using the following equations (10.10.43 of Howard, 1971):

(12)

The first passage time 13, is arrived at by adding two times: (1) the time spent in the state before a transition, and (2) the time spent getting from the destination state to e (zero if the first transition is to e). The mean first passage time is in fact the mean time spent in the state plus a weighted average of the mean times to get from each possible next state to e.

For the general vegetation model, we may substitute Equations (9) and (11) into Equation (12) to arrive at

e,,=o n L‘ n L, \ (13)

e,,= siLj + C C ’ ‘cq Cn) + sj es(,).e +C ll C ‘iq Cn>18,i,q).e ‘q=ln=l q= I “= I

V i#E J

A. D. Moore 12.1

Equations (13) form a set of simultaneous equations which can be solved using standard techniques. It is also possible to calculate a variance for the time to extinction; the equations are given in section 10.10 of Howard (1971).

Recolonisation of a site by a locally extinct species is a stochastic event and is formally identical to a disturbance. Such a “disturbance” would have zero probability in communities where the species was not extinct; its probability in other states would depend upon the competitive relations between the disseminule and the existing vegetation.

7. Extinction times in the Montanan forest example

I have calculated the mean time to extinction of lodgepole pine in the Montanan forest described above. Here, the set E= {5,6,7,8,9,12,13}. The simultaneous Equations (13) and their solution are shown in Figure 2. For example, the mean time to extinction of lodgepole pine when the vegetation has just entered state 1 is 456 years.

19.5

70.0

18.2

24-8

I9 5

70-O

t

0.905

0.669 0.331

0.668 0,332

0.455

0905

0,331 0.669

8: [456,482, 325,232,456,462]

Figure 2. Calculation of the mean extinction times for lodgepole pine in the Montanan forest community of Figure 1. Zero entries have been left blank.

m

,o E too-

F:

6 f / I I I I I 1 I I I I I I I I I / / I I I I I I

0 05 IO I.5 2.0 2.5

Relative wlldfre frequency

Figure 3. The Montanan forest example: the effect of varying the wildfire frequencies shown in Table 1 on the mean extinction time of lodgepole pine. The frequencies shown in Table I correspond to a relative tire

frequency of 1.

122 Vegetation and the semi-Markov process

Figure 3 shows the effect on the mean extinction time from state 1 of multiplying all fire probabilities by a constant factor. Because lodgepole pine can become locally extinct as a result of inter-fire periods which are either too long or too short, the mean time to extinction has a maximum at intermediate values of general disturbance frequency. It is intriguing to note that this optimum appears to lie roughly at the fire frequency prevailing in the Montanan forests.

8. Controlling the dynamics of the vegetation using prescribed disturbance

I will now consider the situation where a manager has assigned a score or value to each vegetation type of a community (based on its perceived conservation and/or economic worth) and wishes to use an additional disturbance type to manage the vegetation of an area. The classic case is the use of prescribed burning in the management of fire-prone vegetation. The question is how this prescribed disturbance should be used to maximise the value of the vegetation to the manager over time; the mathematical techniques of stochastic process control (Howard, 1971) provide the means to answer it.

The theory of stochastic process control is based around the idea of associating rewards with events in the life of a stochastic process, and then seeking a set of decisions which maximise the expected total value of the rewards. This optimisation problem can be cast in a number of ways: it can consider either a finite or an infinite time horizon, and the rewards associated with future time steps may be discounted relative to those associated with the present time step. Because the human and physical environment of any piece of vegetation will inevitably change in a relatively short interval, I have used a finite time horizon. Also, because the notion of discounting future values is alien to the philosophy of nature conservation, I have used the case with no discounting. The appropriate technique is a form of dynamic programming known as value iteration.

The first step is to formalise the decision and reward structures for the problem. For each state i, I will consider L,+ 1 possible decisions (notionally made at the time

the vegetation enters the state). Decision 0 is not to apply the prescribed disturbance, while decisions 1 . . . Li are to apply the prescribed disturbance-in the corresponding epoch after the vegetation enters state i (assuming that an uncontrolled disturbance has not already taken place). Note that, once the decision is made, the prescribed disturbance is a deterministic event similar to succession.

Howard (1971) presents a reward structure for semi-Markov processes with three components:

vi(O) = the value of having theprocess in state i with no time remaining. I will take this to be uniformly zero.

yk&n) = the value accruing during an epoch when the process has been in state i for n epochs, given that decision k has been made and the process will move to statej. I have assumed that this yield component is independent of time, destination state and decision; that is, the manager has assigned a conservation value Yi to each community on the basis of local priorities.

bkti (n) = the bonus accruing when a transition is made from state i to state j at time n given that decision k has been made. Here these bonuses measure costs in money and other resources associated with disturbances (prescribed or uncon- trolled), and so will be negative. The bonus associated with succession, bki,sCr7 (Li) is zero. The cost of a disturbance event is likely to vary with time since entry to a state. An example is the buildup of fuel in fire-prone vegetation: it will mean that wildfires occurring after long intervals will be more intense, and so more

A. D. Moore 123

costly to control, than those occurring after shorter periods. These costs will not, however, vary with the decision (except as it affects the time of a management action) or the destination state (which is fixed in any case).

Let

a(i) = be the state to which vegetation in state i will move (with probability one) when the prescribed disturbance is applied.

4, (n) = &.4l,y) (n) be the value of an uncontrolled disturbance event of type q at time n in state i. For a wildfire this would include the average cost of controlling the fire, compensation to those affected adversely by it etc.

B,, (n) = h,.,,,) (n) be the value of implementing the prescribed disturbance at time n in state i.

Given these values, we can write the elements of the core matrices of the process given each decision:

‘d i=l...N.

c” r.r,,, w = s,

Coi.d,l.Y, (n) = c,y (n)

coy (n)= 0

v q=l . ..D.

n=l . . . L,

otherwise

V i=l..N, k=l . . L,

C ki.a(i)(k) = fi ‘[I- G, (n)l n= I

k v q=l . ..D.

n=l . . . k

otherwise

(14)

Now, we can simply use the recursive equation for the solution of the value iteration problem (equation 15.2.1 of Howard, 1971); it can be rewritten for the case with no discounting as

vi (n) = Max v“; (n) k

(15)

where

j=lm=l Y k,, (0 + bk, (m) + v, (n - ml 1

(16)

124 Vegetation amI the semi-Markov process

In English, V, (n) is the expected maximum of the value the process can attain if it has just entered state i with n time periods remaining. The first term in square brackets is the reward accruing in a case where no transition occurs in n time steps, while the second term in square brackets gives (recursively) the reward accruing in a case where a transition does occur. The expected reward is just a weighted sum of these. Finding the solution depends upon our obtaining expressions for the vk, (n).

For decision 0,

voi(n)=nYi [ s,+ 5 q= I

S ciq Cm)] + m=n+ 1

D n

C C G,W[m Y, + 4,(m) + vd(i,q) (n - m)l q=lm=l

voi (n) = si [

L,Y, + V,(,) (n - L;) 1

+

5 ? Ci,(m)bYi+ Big(m) + vd(i,q) (n - 41 q=lm=l

For decision k (k= 1 . . . LJ,

'i Cnjcn yi [c,,ac,l + qg, m =c+ , ciq Cm)] -t

5 i ciq (m)bri + 4,(m) + vd(i,q) (n - 41 q=lm=l

vk, (n) = Si [k Y, + B, (k) + vpcfi (n - k)] + V n>k

2 i Ciq(m)[m yi ’ Biq(m) + vd (i,q) (n - m)] q=lm=l

v n<L,

v rl>Li

V n<k

Note that the second term is the same in each of these except for the second summation index.

9. A hypothetical value iteration for Montanan forests

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126 Vegetation and the semi-Markov ~NKXZZS

sequence which is summarised in Table 3. Also shown in Table 3 are the (constant) probabilities of wildfire LX,, and conservation values Y, for each state. A strong emphasis has been placed on the high-diversity state 2.

For the costs B, associated with each fire type, I have assumed that fuel buildup and fuel characteristics are more or less independent of community composition and depend only on the time since the last fire. This is at least consistent with the work of Kessell et al. (1978). I have constructed an arbitrary sigmoid function for the cost based on this assumption:

- 4,,,, (4) Biq (‘) = 1 + exp{ - 0.3 [n + T, - T,, (q)l} 118)

B,,, (q) is the maximum cost of disturbance q (1 = wildfire and 2 = prescribed fire), T,, (q) is the time at which the cost of disturbance q reaches half the maximum value and T, is the time since the last fire at entry into state i. All times in Equation (18) are in units of IO-year epochs. The values of Ti are given in Table 3; the remaining parameters are B,,,,, (1) = 500, T,, (1) = 80 years, B,,,,, (2) = 50 and T,, (2) = 20 years. The variations in fire costs with time are shown graphically in Figure 4. The scaling of the yields and bonuses is such that the maximum cost of implementing a prescribed fire is, for instance, reckoned to be equivalent to the value of a stand remaining in state 2 for 50 years.

It should be stressed that this reward structure is purely hypothetical and implies nothing about actual management priorities.

Table 4 shows how the optimum decision in each state varies as the horizon of decision-making is increased. All optimal decisions here are either 0 (do not apply prescribed fire) or 1 (apply prescribed fire immediately); in several states, the optimal decision changes from laissez-faire to immediate burning as the horizon increases and the increased risk of wildfire over the long term is taken into account. The switch in optimum decision for state 2 at a horizon of 90 years suggests that decision-making with respect to prescribed burning should use a planning horizon at ,least as long as the timescale of fuel buildup.

Time since lost fire lyeors)

Figure 4. The arbitrary cost function used in the value iteration example. The upper line shows the cost of wildfires and the lower line shows the cost of prescribed fires as a function of the time since the last fire.

A. D. Moore 127

TABLE 4. Variation in the optimal prescribed burning policy as the planning horizon changes in the hypothetical Montanan forest system specified in Table 3

Optimal decision [k*(n)] for state Horizon n (years) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 I8

10 0 0 0 0 0 0 0 0 0 0 0 0 000000 20 000011001 0 0 0 0 0 0 0 0 0 30 000111001 0 0 0 1 0 0 0 0 0 40 0011110010101 0 0 0 0 u 50 0011110010101 0 0 0 0 0 60 0011110110101 0 0 0 0 0 70 00011101101 0 1 0 0 0 0 (I 80 00111101101010 0 0 0 0 90 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 0 0 0

100 0 111110 110 10 10 0 0 0 (J 250 0 1111 10 110 10 1 0 0 0 0 0

TABLE 5. Variation in the optimal prescribed burning policy as the relative importance of the conservation values YI changes in the hypothetical Montanan forest system specified in Table 3

Optimal decision [k*(n)] for state Scaling factor 1 2 3 4 5 6 7 8 9 10 I1 12 13 14 15 16 17 IP

-- 0.0 0 11lllO 110 10 10 0 0 0 0 I.0 0 ill 1 IO 110 10 1 0 0 0 0 0 1.1 0 11 1110 110 10 1 0 0 0 0 0 1.2 0 I I1 110 110 10 I 0 0 0 0 0 I.3 0711110110101 0 0 0 0 0 I.4 0711110110101 0 0 0 0 0 1.5 0 8 11110 110 10 1 0 0 0 0 0 I.6 0901110110101 0 0 0 0 0 1.7 0110 1110 110 10 1 0 0 0 0 0 1.8 0110 1110 11 0 1 0 1 0 0 0 0 0 1.9 0110 1110 110 10 I 0 0 0 0 0 2.0 0110 1110 110 10 1 0 0 0 0 0 3.0 0110 11 IO 110 10 10 0 0 0 0

In Table 5, I show the effect on the optimum burning policy of varying all the

conservation values Y, by a constant factor. A horizon of 150 years has been used; by this time, the optimum policies appear to be stable. The optimal decisions change for states 2 and 3 (i.e. the high-diversity state and one in which aspen is locally extinct but the other species are not immediately vulnerable). In both cases, increasing the conservation values decreases the role of prescribed burning. The variation in the optimal decision for state 2 is especially interesting. It first changes abruptly from

immediate prescribed burning to burning after 70 years, when the cost of wildfires is rising rapidly; the optimal time for prescribed burning then becomes later and later until it reaches the last epoch before the extinction of the aspen, and there it remains.

128 Vegetation and tke semi-Markov process

10. Discussion

10.1. ANALYSIS OF STOCHASTIC MODELS OF VEGETATION DYNAMICS

The limitations of the Markov process as a model for the dynamics of vegetation are well known, as discussed in the Introduction. Horn (1975) produced the major theoretical, as opposed to statistical, analysis of Markovian vegetation processes. Horn (1975) switches between scales of resolution or, more precisely, aggregates the results of many plant-by- plant replacement processes to arrive at a model for stand composition. His theoretical analysis focuses predominantly on the equilibrium composition of the vegetation stand; for him, the only interesting feature in the rest of a stand’s successional pattern is the rate at which it approaches this equilibrium. For predictive purposes, this is worse than useless; most vegetation stands, even in forests, are away from compositional equi- librium for most of the time because of large-scale disturbances (White and Pickett, 1985), and therefore this concentration on equilibrium properties is divorced from reality. The question of whether equilibrium compositions are reached at the landscape spatial scale (the “shifting mosaic” of Bormann and Likens, 1979) is a separate one.

The approach I have taken differs from that of Horn (1975) and from the empirical and statistical work of Usher (198 1) in two ways. First, the semi-Markov process can deal successfully with the dynamic complexities which arise in nature because of non- equilibrium age structures. Second, I have used biological transitions to limit the possible transitions to a fraction of the entries in the transition matrix (some of Horn’s (1975) theoretical constructs also exhibit this property). My analysis also differs from that of Horn (1975) in focusing on the non-equilibrium dynamics which actually take place in vegetation stands.

One limitation of the analyses presented here which should be borne in mind is that the stand being modelled is treated as isolated, and in particular that disturbance probabilities do not change with the composition of nearby stands.

Finally, the semi-Markov process model and the uses I have demonstrated here encourage a strategic view of vegetation dynamics, in which the centre of attention is the system composed of the vegetation together with both the successional processes and the stochastic disturbance regime which drive its dynamics. “System” here should be understood to mean a set of organisms, their environment and especially the relation- ships between them; it definitely should not carry any Odumesque overtones. A happy consequence of this viewpoint is that the old debates about climax concepts, successional convergence etc. (e.g. Drury and Nisbet, 1973) reduce to non-questions.

The semi-Markov process analyses presented here were all carried out on microcom- puter. In order to bring them into the hands of managers, it is intended to link them in a software package together with vegetation models such as the vital attributes scheme.

10.2. THE SEMI-MARKOV PROCESS AND PLANT-BY-PLANT REPLACEMENT PROCESSES

It is possible to describe plant-by-plant replacement or “gap dynamics” by means of a semi-Markov model. Each state would represent occupation of a site by a plant of a particular species, the waiting time function wi would correspond to the mortality schedule of the species, and the transition probabilities P, (m) would describe the recruitment of plants into the gaps created by the death of established plants. A starting point for modelling the P, (m) values might be

A. I). Moore 129

p,, (ml K -y

P,, (m) x6,,y,+x,

V m-CM, 1 j

(19) V m3M,

where M, is the maturation time of species i; xj is a term describing the invasive ability of speciesj, and y, is the advantage accruing to the previous incumbent of a site due to higher propagule abundances or root suckering.

10.3. THE USE OF OPTIMISATION TECHNIQUES IN LAND MANAGEMENT

Using ecological models to discover optimal management strategies has been an aim of modellers since at least the advent of electronic computers (Swartzmann and van Dyne. 1972). In fact, dynamic optimisation techniques are very much part of the toolkit available to resource managers in forestry (Weintraub and Navon, 1986), pest manage- ment and fisheries (Walters and Hilborn, 1978). Williams (1985) coupled an infinite-time horizon variant of stochastic dynamic programming with a simulation model of shrub growth to explore optimal policies for grazing management in north-western Utah; the stochastic element was introduced by year-to-year variation in weather. So far, however. there seems to have been no use of dynamic optimisation techniques in management of vegetation for conservation purposes.

One very good reason for this omission is that conservation goals, unlike those of forestry. fisheries or rangeland management, are difficult or impossible to express on a common scale with the (largely economic) costs of management actions or natural disturbances. In the hypothetical value iterations described above, I blithely assumed that conservation values (and other non-monetary factors such as risk to human lives) could be assessed in monetary terms. Of course, governments do something very similar informally every year when they hand down their budgets for land management; however, it is difficult and indeed undesirable for land managers to quantify conserva- tion values exactly in this way.

This difficulty does not render the value iteration technique presented here useless, however. It should be possible to assess relative conservation values for the various vegetation states and then to conduct an analysis similar to that presented in Table 5; even if the scaling factor between conservation and monetary values cannot be specified, it should be possible to locate it within a relatively broad interval. Seeing the changes (or lack of changes) in the optimal policy caused by varying the relative importance of conservation values should be an enlightening experience for managers. In short, the value iteration procedure should be used for decision support, not decision-making.

It should be noted that situations do exist where vegetation states may be assigned economic values directly. An example is the management of catchments in order to minimise the sediment loads entering a water storage system: in this case, the yield of each vegetation type would be the economic cost of the erosion rate associated with it.

References

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vegetation types in south-west Tasmania. Ausrralion Journal of Ecology 7, 203-205.

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Childes, S. L. and Walker, B. H. (1987). Ecology and dynamics of the woody vegetation on the Kalahari Sands in Hwange National Park, Zimbabwe. Vegetatio 72, 111-128.

Drury, W. H. and Nisbet, I. C. (1973). Succession. Journal of the Arnold Arboretum of Harvard University 54, 331-368.

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