scrub, Pinus-Rhododendron forest and Crassocephalum-Erechtites community have been identified, and these show cyclic succession under the influence of ...
Ecol. Res. 5: 163-171, 1990 ECOLOGICAL RESEARCH 9 by the Ecological Society of Japan 1990
A Markov Approach for Describing Post-fire Succession of Vegetation Yuji ISAGI, Laboratory of Silviculture, Kansai Research Center, Forestry and Forest Products Research lnstitite, Kyoto, 612 Japan and Nobukazu NAKAGOSHI, Department of Environmental Studies, Faculty of Integrated Arts
and Sciences, Hiroshima University, Hiroshima, 730 Japan Abstract
Vegetation dynamics in the coastal area of the Seto Inland Sea region in Japan, where wild fires occur frequently, were described using a stationary Markov model. In this region, vegetation types of Miscanthus-Pleioblastus grassland, Lespedeza-Mallotus scrub, Pinus-Rhododendronforest and Crassocephalum-Erechtitescommunity have been identified, and these show cyclic succession under the influence of fires. The model uses parameters determining fire frequency and rate of successional change to analyze the effect of variation in these parameters on the areal ratio of each vegetation type at equilibrium and on the time taken for one vegetation type to succeed another (elapsed successional time). The effect of fire frequency differs between hypothetical habitats with high and low productivity. A policy for vegetation management in areas of high and low productivity is proposed. The advantages and limitations of applying Markov models to studies of vegetation succession are also discussed. Key words: Areal ratio; Markov model; Pinus densiflora forest; Vegetation succession; Wild fire. Introduction
The coastal region of the Seto Inland Sea, Japan, is a district with relatively low rainfall, being 1000-1200 mm annually. In this region, Japanese red pine (Pinus densiflora Sieb. et Zucc.) forests dominate. These trees provide rich fuel for wild fires, which occur more frequently than in any other region of Japan, according for about 30~ of all outbreaks (Nakagoshi et al., 1987). Red pine forests disturbed by fire recover to the original forests through succession. In this region, fire frequency varies greatly from one location to another (Hada and Tanino, 1988) and among different vegetation types (Nakagoshi, 1987). Furthermore, land productivity varies with soil and microtopographical conditions, affecting the rate of successional change (Nakagoshi, 1984, 1987). It is often observed that the areal ratio of vegetation types differs among locations as a result of differences in fire frequency and land productivity. Therefore, to understand the trends in vegetation succession, it is important to assess the influences of the above two factors. Few studies have examined this problem, although there have been many descriptive reports on post-fire succession of vegetation or fire-history. Accepted 19 December 1989
164
Y. Isagi & N. Nakagoshi
This paper presents the results of a Markov approach to the description, analysis and prediction of the effects of fire frequency and land productivity on post-fire succession, based on community-to-community replacement. Methods Patterns of post-fire succession
Fires destroy portions of vegetation and produce a mosaic (multi-phase) vegetation structure. Nakagoshi et al. (1987) described four vegetation types in the coastal region of the Seto Inland Sea of Japan; M i s c a n t h u s - P l e i o b l a s t u s grassland (abbreviation of vegetation type: MP), L e s p e d e z a - M a l l o t u s scrub (LM), Pinus-Rhododendron forest (PR) and CrassocephalumErechtites community (CE). The detailed species compositions of these vegetation types have been given in previous reports (Nakagoshi, 1987; Nakagoshi et al., 1987). In the course of succession, MP develops to LM and is followed by PR. In the absence of fire, PR would be succeeded by a climatic climax vegetation of evergreen broad-leaved forest. However, fire destroys MP, LM and PR, reducing them to CE, and thus the climax stage hardly develops. Hence, under the influence of fire, PR is the sub-climax vegetation in this region. The relationships among the four vegetation types are shown in Fig. 1. CE derived from PR changes quickly to LM through the germination of surviving seeds buried in soil and through the development of stump sprouts originating in PR. On the other hand, CE derived from MP or LM has few buried seeds or stump sprouts and changes to MP rather than LM. Therefore, in this study, the CE vegetation type was divided into two sub-types, CE1 which is derived from PR, and CE2 which is derived from MP or LM (Fig. 1).
| Pin us Rhododendron
s55(1-b51)
forest (PR) ~
~ b51
re !
(~Crassocephalum-[ Erechtites [
community (CE1) [ Sll
s45(1-b42)
@Lespedeza Ma l l ot us
s41(1- b42)
scrub (LM)
s34(1- 32)
b42 Fire !
|
Erechtites
II
community(CE 2) ] s22
| Miscanth us Pleioblastus
$33(1-b32)
grassland (MP)
Fig. 1. Patterns of forest successionfollowingfire in the coastal region of the Seto Inland Sea, Japan (modified from Nakagoshi et al., 1987). Numbers in boxes (1-5) are assigned to each vegetation type. sij indicates the probability of transition from vegetation type i to j by successionalchange, bty indicates the probability of transition from i to j by fire destruction.
A Markov approach to post-fire succession
165
The model The transitions between vegetation types are shown in Fig. 1. The probability o f transition from a vegetation type i at time t to another type j at time t + 1 by successional change is denoted by sij. W h e n a vegetation type i at time t still remains i at time t-k 1, the transition probability is expressed as si~, Then,
sii+sij---- 1
(1).
Similarly, the probability o f transition f r o m i to j by fire destruction is denoted by b~j. The model is formulated in discrete time periods o f five years. Consider a row vector
vt =[P~t P2~ P3t P4~Pst]
(2),
whose elements are the areal ratios o f each vegetation type to the whole study area at time t. The vector v~+z can be obtained by the equation v~+l = vtT
(3),
where T is the transition matrix (Table 1). There is little fuel in CE and it rarely suffers from wild fires. Therefore, it is assumed that all o f the CE develops into M P or L M during one time period o f five years (Nakagoshi et al., 1984). Then, we have si4 Sil sza s22
: : = =
1 0 1 0
(4).
Calculations Since the transitions b e t w e n vegetation types in the present model are irreducible and aperiodic, by repeating the procedure represented by Eq. (3), the vector v converges to a steady-state vetor u
= [Pl* P2*/03* P4* P5*]
(5),
whose elements are the areal ratios o f each vegetation type at equilibrium, and obvisouly 5
pi* ---- 1
(6).
Table 1. The transition matrix T with probabilities of successional change sij and fire probabilities b~y from time t to t + l . At time t Vegetation types CE1 : Crassocephalum-Erechtites community type 1 CE2: Crassocephalum-Erechtites community type 2 MP: Miscanthus-Pteiobtastus grassland LM: Lespedeza-Mallotus scrub PR: Pinus-Rhododendron forest
At time t § MP
CE1
CE2
LM
PR
sli
0
0
s14
0
0
s22
s~a
0
0
0 0 b51
bz~ b4z 0
szz(1--b32) 0 0
s34(1--b3s)
0
$44(1--b42) $45(1--bgz) 0
s55(1--b51)
166
Y. Isagi & N. Nakagoshi
The vector v* is the unique probability vector that satisfies the equation v* = v*T
(7),
By solving the linear equations given by (6) and (7), we obtain the areal ratios of vegetation types at equilibrium. The ratio is Pl* : P2* : Ps* : P4* : Ps* = bs~ ss~ s45 (1--b~) (1--b4~) : --b4z b~(bs~ s34--b~--s~4) : b4~ b~ : bs~ 6'34(1--bz2) : ss, s45(1--b~2)(1--b42)
(8).
With Eq. (8), we can evaluate the effect of fire frequency and rates of successional change on the areal ratio of each vegetation type at equilibrium. To estimate the elapsed successional time until PR, we treated PR as an absorbing state. Then we calculated the average time for each vegetation type to reach the absorbing state. The average time is equivalent to the elapsed successional time until PR. Replacing the fifth row of the transition matrix T (Table 1) by (0 0 0 0 1), we defined a matrix T' which treated PR as an absorbing state. Then we partitioned T' as 0
Q
0 0
T~
s45(1--b42) 0
0
0
0
(9).
The fundamental matrix M is given by M -= ( I - Q ) -1
(10),
where I is a 4 • 4 identity matrix. The element of M, miy, is an average time for which the system started in vegetation type i will be in the vegetation type j before being absorbed (Morimura and Takahashi, 1979). Hence the elapsed successional time from a vegetation type i to PR, hi, is calculated as 4
hi = Y], mij
(11).
j=l
The land productivity affects the probability of transition from a vegetation type i to j. In order to analyze the effects of fire frequency on succession for different land productivities, we consider two hypothetical situations; one with relatively high rates of successional change (sa~ = 0.8 and s45 = 0.8) and another with low rates of successional change (sa4 = 0.3 and s4~ -- 0.3). Typical examples of these parameter sets were obtained from contrasted sites differing from each other in microtopography (Nakagoshi et al., 1984), annual precipitation and nutrient conditions. Results and Discussion Areal ratio of vegetation types at equilibrium Figure 2 shows the effect of fire frequency, b~l and b4~, on the areal ratio of Pinus-Rhododendron forest at equilibrium (Ps*) when the rate of successional change is high (Fig. 2a) and low
A Markov approach to post-fire succession
167
b
O"
.8
,8
.6
~6
,4
~4
92
~2
0
~0
Fig. 2. Three-dimensional representation of the effects of fire frequencies, b42 and b51, on the ratio of PinusRhododendron forest at equilibrium (ps*) in an area of (a) high and (b) low productivity. Fire frequency of Miscanthus-Pleioblastus grassland, b32, is assumed to be 0.05.
(Fig. 2b). It is evident that Ps* decreases as the fire frequency increases, and Ps* is always higher in areas of high productivity. In the Seto Inland Sea region, the realistic ranges of b51 and b42 for a period of five years are less than 0.5. Within these ranges, an important difference inps* was shown between the two situations. In the neighborhood of b51 = 0, a small change in b51 has a more marked effect on the changes in Ps* than that of b42 in both situations, but the absolute value of ap~*/Ob~t in less productive areas is two or three times higher than that in productive areas. The effect of a change in b42 o n 1o5" increases with increasing baa in the ordinary range of bs~, which is less than 0.2. However, the effect is more significant in less productive areas. Such trends in the effect onps* can provide useful information for vegetation management. For example, if a policy aims to increase the ratio of PR in a region of high productivity, effort should be made to reduce b51 rather than b42, because the ratio of PR is mainly determined by the former than by the latter in the commonly observed range of fire frequency, i.e. b5~
