ISSN 20790864, Biology Bulletin Reviews, 2015, Vol. 5, No. 1, pp. 28–35. © Pleiades Publishing, Ltd., 2015. Original Russian Text © D.O. Logofet, O.I. Evstigneev, A.A. Aleinikov, A.O. Morozova, 2014, published in Zhurnal Obshchei Biologii, 2014, Vol. 75, No. 2, pp. 95–103.
Succession Caused by Beaver (Castor fiber L.) Activity: I. What Is Learnt from the Calibration of a Simple Markov Model D. O. Logofeta, O. I. Evstigneevb, A. A. Aleinikovc, and A. O. Morozovad a
Laboratory of Mathematical Ecology, Institute of Atmospheric Physics, Russian Academy of Sciences, per. Pyzhevskii 3, Moscow, 119017 Russia email:
[email protected] b State Nature Biosphere Reserve “Bryanskii Les,” st. Nerussa, Bryansk oblast, 242170 Russia email:
[email protected] c Centre for Problems of Ecology and Productivity of Forests, Russian Academy of Sciences, ul. Profsoyuznaya 84/32, Moscow, 117810 Russia email:
[email protected] d Department of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia email:
[email protected] Received May 27, 2013
Abstract—A homogeneous Markov chain of three aggregated states, “pond–swamp—forest,” is proposed as a model of cyclic zoogenic successions caused by beaver (Castor fiber L.) activity in a forest biogeocoenosis. The data obtained from field studies undertaken in the “Bryanskii Les” Reserve in 2002–2008 proved to be sufficient for calibrating the chain transition matrix. With the use of the formulas of finite homogeneous Markov chain theory, the main results for calibration of a model were obtained: the stationary probability dis tribution of chain states, the matrix (T) of mean first passage times, and the mean durations (Mj) of succes sional stages. The former illustrates the distribution of relative areas under succession stages if the current trends and transition rates of succession are conserved in the long term—it appeared to be close to the observed distribution. Matrix T provides the quantitative characteristics of the cyclical process by specifying the ranges proposed by the experts for the duration of stages in the conceptual scheme of succession. The cal culated values of Mj detect potential discrepancies between empirical data, the expert knowledge that sum marizes the data, and the postulates accepted in the mathematical model. The calculated M2 value falls out side the expert range, which gives a reason to doubt the validity of the proposed expert estimation, the aggre gation mode chosen for chain states, and/or the accuracy of data available, that is, to draw certain “lessons” from partially successful calibration. Disaggregation of aggregated vegetation states, refusal to postulate the time homogeneity, and refusal of the Markov property of the chain are also discussed among possible ways to improve the model. DOI: 10.1134/S2079086415010053
mosaic (Vostochnoevropeiskie…, 1994; Popadyuk et al., 1995; Smirnova, 1998); (2) the use of the territory according to the “fallow” system, in which beavers leave their settlements for a while and go to other places. This system causes cyclical development of communities and their spatial redistribution along the river (Evstigneev and Belyakov, 1997; Evstigneev et al., 1999; Aleinikov, 2010, 2011).
INTRODUCTION According to modern concepts, the driving force behind successional transformations of communities in the forest biocoenosis cover may consist in various groups of animals: ants, birds, small rodents, ungu lates, and others (Vera, 2000; Lindeman, 2004; Rubashko et al., 2010; Severtsov, 2012; Evstigneev and Voevodin, 2013). In an undisturbed biogeocenotical cover of small river valleys, successional transforma tions are caused by river beaver (Castor fiber L.) activ ity (Sinitcin and Rusanov, 1989; Evstigneev and Bely akov, 1997; Vostochnoevropeiskie…, 2004; Zav’yalov et al., 2005; Danilov et al., 2007; Rechnoi…, 2012). The following processes are most important for SCBA in valleys of small rivers: (1) the construction of dams, which changes the soil and hydrological regime of the territory (Morgan, 1868; Sinitsyn and Rusanov, 1989; Zav’yalov, 1999) and creates the largest vegetation
Markov chains, as a simple type of stochastic pro cesses, serve as a means to describe formally the course of succession (Horn, 1975; Jeffers, 1978; Logofet, 2010). It is assumed that one has defined the stages of succession under study and knows a scheme of transi tions between them—a conceptual scheme of succes sion. It contains a finite number of specified vegeta tion types or plant associations and indications of their order (or alternative orders) in the succession series. The states of the chain are identified with the stages of 28
SUCCESSION CAUSED BY BEAVER (Castor fiber L.) ACTIVITY
succession, and the conceptual scheme determines the pattern of the transition probability matrix for the chain (for one time step), or the transition matrix, and the task to construct an adequate model is reduced to the calibration of this matrix using empirical data. The calibrated model provides a quantitative character to the expert knowledge formalized by this model and, in particular, enables us to estimate the average durations of the individual stages, the mean time of achieving the climax state starting from any other stage, and the mean time of returning to the past stage (when it comes to the cyclic succession). In this paper we construct a simple Markov model of cyclic SCBA in the valleys of small rivers in the “Bryanskii Les” Reserve and attempt to calibrate it using data from 2004–2008 case studies. The outcome of calibration has detected a discrepancy between the model, the empirical data, and the expert estimates of the stage durations, while the analysis of potential causes of discrepancies made it possible to formulate a number of methodic inferences—“calibration les sons”—that have determined the directions to further improvement of the model. MATERIALS AND METHODS Objects. The objects of study were forest and non forest complexes of communities in the flood plains of small rivers in the “Bryanskii Les” Reserve “mas tered” by river beavers (Castor fiber L.). By small riv ers, we mean watercourses that beavers can dam to create a pond. The river length ranges from 5 to 20 km, and the floodplain width in the middle course ranges from 50 to 400 m. The rivers have mixed inflow: groundwater and atmospheric. The Reserve is located in NerussaDesna Polesie in the southeast of the Bry ansk oblast (Russia). In botanical and geographical terms, the area belongs to Polesie Subprovince of the Eastern European Deciduous Province (Rastitel’nost’…, 1980). In zoogeographic terms, the study area is part of the Central Russian region of the mixed forest prov ince in the borealforest subregion of the Palearctic region (Kuznetsov, 1950). Methods. There are three stages in the develop ment of communities in the ecotopes of small river floodplains caused by beaver activity: aqueous, herba ceousswampy, and woody. The first stage, with a predominance of an aquatic community (pond). As long as the dam operates, the communities are flooded over an area up to 20 ha. It is dominated by aquatic cenoses. Thus, deepwater areas of the beaver pond are covered with freefloating plants (Lemna minor, L. trisulca, Spirodela polyrhiza, etc.), while shallow sites near the bank are covered by amphiphytes (Oenanthe aquatic, Phragmites australis, Carex riparia, etc.). Over 3–10 years, beavers com pletely or partially destroy arboreous plants around the settlements and leave for other places, and the dam gets destroyed by spring floods. BIOLOGY BULLETIN REVIEWS
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Table 1. Percentage of areas under succession stages Years of observation Stage of succession Stage no. 2002
2004
2008
Pond
1
16.2
14.4
13.0
Herbaceous swamp
2
4.4
9.1
13.0
Forest
3
79.4
76.5
74.0
The second stage, with a predominance of herba ceous swamps. After beavers' departure and water downturn in most of the abandoned pond, a herba ceous swamp forms in its central part. In the periphery of the former pond, a middle floodplain meadow is formed; a small wellheated backwater arises in the dam depression that beavers dug when shoveling the soil for the construction of dams. It is dominated by Spirodela polyrrhiza, Potamogeton gramineus, P. tri choides, Hottonia palustris, Hydrocharis morsusranae, etc. The maximum duration of the herbaceous swampy stage is determined by the time needed for arboreous plants to form a closed canopy (usually about 30 years under the Reserve conditions). The third stage, with a predominance of forest. Over three decades, a closedcanopy black alder (Alnus glu tinosa) forest forms at the place of the abandoned pond. The herbaceous cover is dominated by alder black species: Urtica dioica, Scirpus sylvaticus, Lycopus europaeus, etc. By 60–70th year, senile alders fall out, forming “gaps.” The younger generation of alder and elm takes root on deadwood. Accumulation of the humusaccumulative horizon facilitates the introduc tion of the ash tree. As a result, an elmalder ash forest forms eventually in abandoned beaver ponds. The unidirectional development of cenoses is interrupted by the “habit” of beavers to use the terri tory according to the “fallow” system. Beavers can come back at the herbaceous swampy stage before its completion, for example, after ten years if woody plants are partially restored. In this case, shortened cycles of community development (figure) lasting 13– 40 years are formed. When beavers renew their activity at the forest stage, longer cycles arise, lasting 43–140 years or more. According to surveys carried out in 2002, 2004, and 2008 (Aleinikov, 2011), territories of the surveyed floodplains were distributed among the types of vege tation described above as it is presented in Table 1. The area of the bottoms of small river valleys was taken as 100%. In 2002–2004, field studies of watercourses were carried out in the Reserve and its buffer zone: all of the lands transformed by beavers were mapped and described. The total length of the surveyed water courses was more than 100 km. On the basis of these data, typification was carried out for the communities of the floodplain ecosystems. In 2008, the stages of
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LOGOFET et al. 1. Stage with a predominance of aquatic communities (pond), 3–10 years
A
B
2. Stage with a predominance of herbaceousswampy communities, 10–30 years
3. Stage with a predominance of forest communities, 30–100 years or more Cyclic successions in the vegetation cover of small river floodplains caused by beaver activities. Thin arrows are the directions of community development determined by the vegetation itself; thick arrows are the directions caused by beavers; A denotes the shortened cycles of vegetation development when beavers come back at the herbaceous swampy stage of community development; B denotes the elongated cycles of vegetation development when beavers come back at the forest stage of community development. Years show the expert estimation of the duration range for each stage.
community development were identified using images from the satellite Alos (resolution of 2.5 m, ScanEks, 2012). The field data were processed in ArcGIS v. 9.1 (ESRI, 2012). In 2002, the percentage of ponds accounted for 16.2, the percentage of the herbaceous swampy stage was 4.4, and the percentage of the forest stage was 79.4. By 2004, beavers had transformed part of the forests, constituting 2.9% of the total area, into ponds. Some of the ponds, constituting 4.7% of the total area, were abandoned, and, as a result of succes sion, those ponds were transformed into herbaceous swamps. By 2008, beavers transformed part of forests, constituting 2.5% of the total area, into ponds. During the same period, the beavers left some of the ponds constituting 3.9% of the total area, and the herba ceousswampy communities were formed in their place. In terms of the model, the distribution of propor tions at the year t is expressed by a column vector x(t) = [x1(t), x2(t), x3(t)]T, where T denotes transposition, and the component xj(t) of the vector is the probability of the state j at the time t and is interpreted as the relative area (or per centage) of the territory occupied by vegetation of the jth stage (j = 1, 2, 3) at the year t. It is convenient to choose the model time step Δt as two years, and then,
according to the method of Markov models of succes sion (Logofet, 1999, 2010), the time dynamics of the vector x(t) obeys the equation x(t + Δt) = P(t)x(t), (1) where the structure of the transition matrix, P(t), cor responds to the conceptual scheme of succession. Generally speaking, the notation P(t) = [pij(t)] suggests that nonzero elements pij of matrix P, meaning the probability of transition from state j into state i for time Δt (i, j = 1, 2, 3) and indicating the rate of succession processes, depend on the time t, but the hypothesis of (time) homogeneity denies such a dependence. If the hypothesis is true, then equation (1), with due regard to the scheme shown in Figure, takes the form p 11 p 12 p 13
x(t + Δt) = p 21 p 22 0 x(t),
(2)
0 p 32 p 33 where the values of pij do not change in time and should sum to one in each column according to their probabilistic sense. Quantitative values of pij are deter mined as a result of calibration of the transition matrix according to the observation data (see below). Once calibrated, the transition matrix enables the calcula tion of certain characteristics of the model process, T such as the stationary distribution x* = [ x *1 , x *2 , x *3 ] , of probabilities for the chain state, that is, a positive and normalized (by the condition x *1 + x *2 + x *3 = 1 or 100%) solution of the equation x* = Px*, (3) and the mean first passage times (tij) of state i starting from state j. It is known (Kemeni and Snell, 1970) that (under mild technical constraints, which matrix (2) does meet) t
lim P = X*,
(4)
t→∞
where X* is the matrix composed of identical columns x*, and the values of tij are the elements of the matrix T = D(I – Z + ZdgE), (5) where D = diag{djj} is the diagonal matrix with ele ments djj = 1/ x *j (j = 1, 2, 3) on the diagonal, I = diag{1, …, 1} is the identity matrix, E is the matrix whose elements are all equal to one, Z = (I – P + X*)–1 is the fundamental matrix of the chain, and Zdg is its principal diagonal (Kemeni and Snell, 1976). A fundamental relationship is also known to exist between the mathematical expectation (Mj) of a ran dom number of steps after which the state j does change and the corresponding diagonal element pjj of the transition matrix, that is, the probability that the chain remains in state j for one step more: pjj = 1 – 1/Mj, j = 1, …, n. (6) BIOLOGY BULLETIN REVIEWS
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Table 2. Calculated characteristics of homogeneous Markov chain for calibrated transition matrix (7) Mean first passage times tij, (steps) roundoff years Stage
x*, %
j i
Pond Herbaceous swamp Forest
12.53 14.83 72.64
1 2 3
1
2
3
(7.98) 16 (2.66) 5 (7.51) 15
(18.53) 37 (6.74) 13 (4.86) 10
(19.95) 40 (22.60) 45 (1.38) 3
Therefore, the formal calibration of the transition matrix enables one to compare (with regard to the actual length of the time step Δt) the obtained values of Mj (j = 1, 2, 3) with the expert estimates of the duration range for each stage (Figure; Logofet and Lesnaya, 2000; Logofet and Korotkov, 2002). In other words, this technique makes it possible to find out how the summarized expert knowledge matches the empirical data within the conceptual scheme accepted under the hypothesis of homogeneity. Calibration method. In general, the calibration problem is to select the values of model parameters such that deviations of the model outcome from the empirical data are minimal. In our particular case, the problem reduces to finding those values for the seven nonzero elements of matrix P for which model (2) reproduces the observed data (Table 1) with a minimal error. The mathematical details of the calibration problem and a method to solve it are presented in Appendix A. RESULTS Formal calibration of the transition probability matrix gives it quantitative certainty (with an accuracy up to 10–6 and approximation up to 10–2): 0.623436 0.072668 0.050133 P = 0.376564 0.681742 0 0 0.245590 0.949867
(7)
0.62 0.07 0.05 ≈ 0.38 0.68 0 , 0 0.25 0.95 which is qualitatively consistent with the conceptual scheme of succession (Figure). The eigenvector x* of matrix (7) corresponding to its dominant eigenvalue 1 (that is, the proper solution to equation (3)) is presented in Table 2. It is seen that the downward trend in the forest percentage observed in 2002–2008, which was due to waterlogging of the areas (Table 1), is close to its completion. Table 2 contains both spatial and temporal charac teristics of the model succession (matrix T calculated by formula (5)), which reflect the various rates and BIOLOGY BULLETIN REVIEWS
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Mean duration Mi of stage i, (steps) roundoff years (2.66) 5 (3.14) 6 (19.95) 40
cyclicity of the corresponding processes. Where the scheme suggests only one version of transition from the current state (1 → 2 and 3 → 1 in the Figure), the corresponding elements tij (t21 and t13 in Table 2) logi cally coincide—both in their meanings and calcula tion results—with the mean durations of the corre sponding stages. While the expert estimates of the cycle durations provide very broad ranges (see page 29), the vector [M1, M2, M3]T and matrix T offer certain aver age indicators: if, for instance, the pond turns into a swamp and then returns to the pond state without affor estation (route 1 → 2 → 1), it takes (t21 ≡ M1) + M2 = 5 + 6 = 11 years on the average; a forest disturbed by bea vers will recover (that is, route 3 → 1 → 2 → 3 with probable “delays” 2 → 1 → 2) in t13 + t21 + t32 = 40 + 5 + 10 = 55 years on the average. Comparing the calculated values of the mean stage duration (the last column in Table 2) with the expert estimates of the duration ranges for each stage (Figure), we see that the calculated duration of stage 2 (“herba ceous swamp”) does not fall into the range. This apparent discrepancy can be regarded as a failure in the formal calibration of the proposed model. The next section discusses what conclusions can be drawn from this failure. DISCUSSION The formalization of expert knowledge via a simple homogeneous Markov chain as a model of cyclic SCBA succession and the calibration of this model with the data of field studies in the “Bryanskii Les” Reserve in 2002–2008 revealed a certain inconsistency between the empirical data and expert knowledge (see above). Generally speaking, the responsibility for the failure of calibration should be shared among the three components of the modeling process: the empirical data, the summarized expert knowledge (that is, a conceptual scheme of succession), and the adopted mode of formalization (that is, accepted postulates of the mathematical model). Therefore, we can note the following potential causes of the failed calibration. (1) Inaccuracy in empirical data. If we change a given area ratio up to 1% (Table 1) in favor of a forest or herbaceous swamp in the data of only one of the field study years, the calculated durations of all the stages will occur in the given expert ranges. The results
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LOGOFET et al.
Table 3. Results of computational experiment 1 Perturbed data, % Stages Pond Herbaceous swamp Forest
pii
[MiΔt], years**
0.7368… 0.8719… 0.9751…
8 16 81
i 1 2 3
2002
2004
2008
16.2 4.4 79.4
14.4 –8.1* +77.5*
13.0 13.0 74.0
* The sign before the number indicates the direction of change. ** Rounded up to integers.
of the corresponding numerical experiments with the model are presented in Appendix B. We observe the mean duration sensitivity to be unexpectedly high for the “forest” stage (the mean doubles!) after such a small variation in the original data. Dozens of such experiments have been carried out, and not all of them provided the desired result, either in the existence of a feasible solution to the system of calibration equations or in the average duration of stages. The exact solutions have demonstrated perma nently high sensitivities of model results to variations in the original data. This means that the proposed “simple” model is not robust in the mathematical sense; that is, it lacks the quality of practical impor tance in the presence of errors in the original data. The mathematical reason for the model being not robust seems to be hidden in the search for an exact solution to the system of calibration equations. The latter can hardly make sense for inaccurate data, and one might be content with an approximate solution. (2) Imperfection of the conceptual scheme of suc cession and/or subjectivity in expert estimates for the duration ranges of stages. Expert estimates are always subjective to an extent, and the extent increases when a defined stage of succession combines (as in the pro posed scheme, see the Figure) several specific vegeta tion states observed in the field. If the expert accepts the timehomogeneity postulate, that is, believes that, over the years of observation, there were no significant changes in the key factors that determine the rate of succession transitions, then it is the homogeneous model that should be an objective tool for estimating the duration of stages. The application of homoge neous Markov models is a promising way to determine the duration of forest successions in young reserves whose history does not span even a single stage of for est succession, as well as in protected areas lacking longterm observations of succession. One way to reduce subjectivity in the estimates leads to disaggregating, perhaps, excessively aggre gated states. For example, our scheme has been aggre gated from a more detailed scheme of shifts in cenoses, which was provided by A.A. Aleinikov (2011, Fig. 1): flooded forests, waterlogged forests, and ponds are combined into a single state called “a pond”; herba ceous swamps and swamps with bushes and under
growth trees are combined into “a herbaceous swamp”; wet (fresh) forests become “a forest.” If the field research data are sufficiently diversified, the con struction and calibration of the homogeneous model with a 6 × 6 transition matrix is just a technically more complex task, the solution of which would give a more adequate model. (3) The hypothesis of homogeneity. Mathemati cally, the hypothesis of homogeneity implies the invariance of the transition matrix in time and, in essence, invariability in key environmental factors that determine the rate of succession transitions. The nature of zoogenic succession under concern suggests at least that the beaver population has reached the environment carrying capacity for beavers and will not deviate from this level in the future. It is clear that, for foreseeable time intervals, the expert may assert the invariability of factors or take it on faith, but, if the task is a far more longterm forecast of the area develop ment, then the model will have to be linked to a proper scenario of climate change and/or other factors that may influence the course of succession. Technically, this means that the transition probabilities cannot remain constant and must depend on the key environ mental factors that accelerate or slow the course of suc cession. If the values/states of these factors change over time, the transition matrix also changes, thus generating an inhomogeneous Markov chain (Knyaz’kov et al., 1992; Logofet et al., 1997; Logofet and Denisenko, 1999; Logofet et al., 2005). The elegant algebraic for mulas of the homogeneous chain give way to simula tion experiments under certain factor scenarios as the basic tool to analyze the inhomogeneous model and to obtain a model forecast. (4) The Markov property. Historically, Markov chains appeared in probability theory as a result of transition from the canonical Bernoulli scheme of independent random trials to a sequence of dependent trials “linked into a chain” (Markov, 1910, p. 1). A.A. Markov considered the outcome of each trial to be dependent on that of the preceding trial. In the modern language of Markov chains, this means that the current state does determine all possible transi tions from it at the next step and their probabilities. Taking the conceptual scheme of succession with all of the stages and all the possible transitions, such as those BIOLOGY BULLETIN REVIEWS
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Table 4. Results of computational experiment 2 Perturbed data, % Stages
i
Pond Herbaceous swamp Forest
1 2 3
2002
2004
2008
16.2 4.4 79.4
14.4 8.1 77.5
–12.5* +13.5* 74
pii
[MiΔt], years**
0.7503… 0.9218… 0.9752…
8 26 81
* The sign before the number indicates the direction of change. ** Rounded up to integers.
in the Figure, without any additional terms, the researcher essentially takes the postulate of the Markov property. As a result, all of the probabilities of state transition/retention are organized into the tran sition matrix, which determines all other properties of the chain as a random process. “Additional terms” may violate the Markov chain property. For example, there are data that indicate the influence of the extent to which beavers have mastered a river valley in the course of cyclic succession (Alein ikov, 2011). In terms of the model, “the mastering extent” is expressed by the sum of the first two compo nents of the vector x(t): x1(t) + x2(t), and its impact on the course of succession means that the transition matrix does depend on x(t), the current distribution of all state probabilities: P = P[x(t)]. This dependence violates the Markov property, turning the transition matrix P into a nonlinear matrix operator P[x(t)], which implies nonstandard methods of mathematical analysis in the model study. Nonstandard methods may also lead to nonstandard results, that is, to model effects that are not reproducible by means of a Markov model. This may be, for example, the climax nonat tainability effect in a forest ecosystem under present conditions, which was shown in a nonMarkov chain with an absorbing state (Korotkov et al., 2001). We are not aware of any studies of nonMarkov effects in models of cyclic successions.
The first attempt to formalize the expert knowledge on SCBA (that is, successions caused by the activity of the beaver Castor fiber) in terms of Markov chains has led to a homogeneous chain consisting of three aggre gate states “pond–swamp–forest” with a certain pat tern of transitions between them (Figure), which reflects the cyclical nature of processes. To calibrate the chain transition matrix, the data gained from the field studies undertaken in the “Bryanskii Les” Reserve in 2004–2008 have appeared to be sufficient. According to the formulas of the theory of finite homogeneous Markov chains, the main results of the calibrated model are obtained: the steadystate distri bution of chain state probabilities, the matrix (T) of mean first passage times, and the mean durations (Mj) Vol. 5
APPENDIX A Calibration Problem for the Transition Matrix According to equation (1), the available data (Table 1), and the choice of Δt = 2, the calibration of the homo geneous model reduces to the solution of the following system of two vectormatrix equations: ⎧ x ( 2004 ) = Px ( 2002 ), ⎨ 2 ⎩ x ( 2008 ) = P x ( 2004 ),
CONCLUSIONS
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of succession stages. The first shows the distribution of the relative areas for successional stages under the cur rent trends and rates of successional transitions pre served in the long term, which are close to those observed. The second provides quantitative character istics of the cyclical process, specifying the ranges for stage durations suggested by the experts in the conceptual scheme of succession. Finally, the calculated values of Mj identify potential discrepancies between the empirical data, the summarized expert knowledge, and the accepted postulates of the mathematical model. The calculated value of M2 has proved to be outside the expert range, and it gives a reason to doubt the validity of the respective expert estimate, the selected method of aggregation of chain states, and/or the accuracy of the data, that is, to draw some “lessons” from partially suc cessful calibration. Rejection of the postulate of homoge neity in time or the Markov chain property are also among the possible ways to improve the model.
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(A1)
where the matrix P is unknown and the vectors x are taken from Table 1. With regard to equality (2), the system of equations (A1) becomes more specific, namely: ⎧ p 11 ⎪ x ( 2004 ) = p 21 ⎪ ⎪ 0 ⎪ ⎨ ⎪ p 11 ⎪ ⎪ x ( 2008 ) = p 21 ⎪ ⎩ 0
p 12 p 13
p 22 0 x ( 2002 ), p 32 p 33
(A2)
2
p 12 p 13 p 22 0 p 32 p 33
x ( 2004 ).
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System (A2) contains seven unknown elements pij and consists of six componentwise equations in which the components of vectors x represent the coef ficients at the unknowns and the free terms. Since the unknowns have to meet three more columnstochas ticity conditions for matrix P: p1j + p2j + p3j = 1 (j = 1, 2, 3), (A3) there are only four independent unknowns of seven dependent ones. For the same reason, there are only four independent equations among the six in system (A2), so the system may have an exact solution (which is atypical for calibration problems; Logofet and Korotkov, 2002; Logofet, 2013). This solution will be the result of the formal calibration of the transition matrix using the observational data if it is feasible, that is, if the values found for nonzero pij belong to the interval (0, 1) and they satisfy the normalization con dition (A3). APPENDIX B Calibrating the Transition Matrix from Perturbed Data Experiment 1. Let us increase the percentage of forest in 2004 by 1%, reducing respectively the per centage of herbaceous swamps by 1%, while keeping the other data unchanged. Then calibration of matrix P by the method described in Appendix A gives the diagonal elements presented in Table 3. Experiment 2. Let us increase the percentage of herbaceous swamp in 2008 by 0.5%, reducing respec tively the percentage of ponds, while keeping the other data unchanged. Then the calibration of matrix P gives the diagonal elements presented in Table 4. REFERENCES Aleinikov, A.A., Status of population and environment transforming activity of the European beaver in Bryan skii Les Nature Reserve and its protected area, Extended Abstract of Cand. Sci. (Biol.) Dissertation, Tolyatti: Inst. Ekol. Volzh. Basseina, Ross. Akad. Nauk, 2010. Aleinikov, A.A., Role of beaver in formation of coenotic diversity in NerussoDesnyanskoe marshy woodland, in Izuchenie i okhrana biologicheskogo raznoobraziya Bryanskoi oblasti. Materialy po vedeniyu Krasnoi knigi Bryanskoi oblasti (Study and Protection of Biological Diversity in Bryansk Oblast: The Data for the Red Book of the Bryansk Oblast), Bryansk: Desyatochka, 2011, no. 6, pp. 82–92. Danilov, P.I., Kan’shiev, V.Ya., and Fedorov, F.V., Rechnye bobry Evropeiskogo Severa Rossii (River Beavers of European North of Russia), Moscow: Nauka, 2007. ESRI, 2012. http://www.esri.com/software/arcgis/arcgis fordesktop. Evstigneev, O.I. and Belyakov, K.V., Influence of beaver’s activity on dynamics of vegetation of the small rivers in Bryanskii Les Nature Reserve, Byull. Mosk. Ova. Ispyt. Prir., Otd. Biol., 1997, vol. 102, no. 6, pp. 34–41.
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Translated by K. Lazarev corrected by D. Logofet
