W O R K I N G P A P E R
PREDICTIVE FOREST ECOSYSTEM MODELS AM)
IMPLICATIONS FOR 1NTM;RATED MONITORING
M.Ya. Antonovsky M.D. I a r z u k h i n
J u l y 1986 Tm-~6-036
I n t e r n a t i o n a l I n s t i t u t e for Applied Systems Analysis
NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR
PREDICTIVE FOREST ECOSYSTEM MODELS AND IMPLICATIONS FOR INTEGRATED MONITORING
M.Ya. Antonovsky M.D. Korzukhin
July 1986 WP-86-36
Working Papers are interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have r e c e i v e d only lim- ited review. Views o r opinions e x p r e s s e d h e r e i n do not neces- s a r i l y r e p r e s e n t t h o s e of t h e Institute o r of i t s National Member Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria
FOREWORD
For many y e a r s Soviet scientists have provided leadership in t h e design of integrated environmental monitoring system. The a t t a c h e d p a p e r by M. Ja. Antonovsky and M.D. Korzukhin i s a good example of t h e a p p r o a c h taken, in which a working ecological model i s developed and t h e n applied t o t h e p r a c t i c a l problem of designing a n a p p r o p r i a t e environmental monitoring system.
The p a p e r w a s p r e s e n t e d at a Symposium held in Moscow 13-16 March, 1985, cosponsored by IIASA and t h e USSR National Committee f o r IIASA.
During t h e same week a Memorandum of Agreement was d r a f t e d and subse- quently signed by Academician Yu.A. Izrael and Dr. T. Lee, assuring Soviet collaboration in t h e IIASA P r o j e c t , ' ~ c o l o g i c a l l y Sustainable Development of t h e Biosphere".
One outcome of t h e Memorandum i s t h e a r r i v a l of P r o f e s s o r Antonovsky at IIASA in May 1986. In his capacity as Chief Scientist in t h e Environment Program, Dr. Antonovsky will continue his r e s e a r c h in t h e fields of ecologi- cal modeling and t h e design of integrated monitoring systems.
R.E. Munn
Head, ENV Program
SUMMARY
This work i s devoted t o a method of modeling f o r e s t ecosystem dynamics t h a t c a n b e used in integrated global monitoring of t h e biosphere. A s i s w e l l known, attempts t o calculate global ( o r regional, zonal, etc.) variation of, .for example, f o r e s t productivity under varying global t e m p e r a t u r e but with a n a v e r a g e warming t r e n d with corresponding variations in productivity of a single tree and a number of trees in a given t e r r i t o r y will n e v e r b e suc- cessful: t h e r e s u l t will b e quantitatively i n c o r r e c t even with r e s p e c t t o sign, because t h e multilevel f o r e s t s t r u c t u r e and a g r e a t number of possible impacts of t e m p e r a t u r e on f o r e s t ecosystem are not t a k e n into account.
Indeed, C02 growth stimulation in a fixed density stand f i r s t leads t o a n i n c r e a s e in t h e accumulated biomass and t h e n t o a reduction, because of competitive mortality ( a t t h e individual and group levels). The associated drying up and reduction of t h e area of swamps, at t h e same time, stimulates f i r e s , t h u s shortening t h e a v e r a g e a g e of f o r e s t s . On t h e whole, f o r e s t biomass, despite a n i n c r e a s e in a r e a , may e i t h e r i n c r e a s e o r d e c r e a s e ( a t t h e landscape and regional levels). For t e r r i t o r i e s with low t e m p e r a t u r e , warming will stimulate individual growth, f u r t h e r reducing t h e biomass at t h e fitocenous level as well as at t h e landscape level due t o t h e stimulation by f i r e and phytophagas; t h e n e t e f f e c t of t h e s e variations may b e e i t h e r positive o r negative.
Generally speaking, we use a well-known a p p r o a c h , viz., from a con- c r e t e n a t u r a l problem t o a h i e r a r c h y of mathematical models, followed by a computing experiment and i n t e r p r e t a t i o n of t h e result. The complexity of a model h a s t o correspond t o t h e study goal, t h e preciseness of experimental d a t a , t h e level of detail of existing methods and algorithms.
P a s t e x p e r i e n c e h a s shown t h a t many complex nonlinear phenomena could b e p r e s c r i b e d with t h e aid of relatively simple models. S e e , f o r exam- ple, t h e work of A. Turing on modeling of morphogenesis and analogous modeling of chemical r e a c t i o n s by I. Prigogin.
We wish t o e x p r e s s o u r a p p r e c i a t i o n and g r a t i t u d e t o P r o f e s s o r R.E.
Munn, Leader, Environment Program at IIASA, f o r his suggestions.
PREDICTIVE FOREST ECOSYSTEX MODELS
AND
IMPLICATIONS FOR INTEGRATED MONITORING
M.Ya. Antonovsky* and M.D. Konukhin**
I t i s convenient t o divide t h e f a c t o r s influencing t h e s t a t e of an ecosys- tem into t h o s e d i r e c t l y affecting i t ("local", o r explicit ones), and indirect ones, whose e f f e c t s depend on t h e s t a t e of adjacent ecosystems ("terriro- tial" f a c t o r s , with implicit effects).
In t h e following analysis, we shall discuss f o r e s t ecology of t h e taiga zone. The local f a c t o r s of importance are as follows: microclimate, bog- ging, f o r e s t pests, o t h e r biotic f a c t o r s and windfalls. Regional f a c t o r s include: f o r e s t f i r e s , bogging, regional-scale p e s t o u t b r e a k s and o t h e r biotic f a c t o r s (birds, fungus diseases) [24].
In a c c o r d a n c e with t h i s view, t h e r e are two a s p e c t s t o predictions of changes in t h e state of t h e taiga f o r e s t s under hypothesized climate varia- tions o r p h y s i c m h e m i c a l changes in t h e atmosphere:
*currently Chief S c i e n t i s t in t h e Environment Program a t IIASA.
..
Natural Environment and Climate Monitoring Laboratory COSKOMCIDROW, Moscow.1. I t is n e c e s s a r y t o p r e d i c t t h e d i r e c t ecological impacts due t o changes in e x t e r n a l i t i e s , i.e. t o assess changes in biomass, t h e number of t r e e s , etc. Taking into account t h e fact t h a t t h e r e are s t r o n g eco- logical effects d u e t o competition in t h e b o r e a l f o r e s t s 1203, t h e r e s p o n s e of t h e ecosystem as a whole would not be, generally speaking, t h e sum t o t a l of r e s p o n s e s observed in t h e case of non-interacting indi- viduals. An a d e q u a t e ecological model is essential in o r d e r t o p r e d i c t t h e r e s p o n s e of t h e t o t a l system.
2. I t is a l s o n e c e s s a r y t o p r e d i c t i n d i r e c t ecological impacts due t o large-scale c h a n g e s in externalities. These predictions are obtained from measurements and models of ecological s t r u c t u r a l changes within e a c h vegetation zone. A region i s conceived as a "mosaic" of a multi- t u d e of ecosystems existing in various h a b i t a t s and at d i f f e r e n t succes- sional s t a g e s . In t h e case of t h e taiga f o r e s t , t h i s mosaic i s largely due t o f i r e
[Ill.
H e r e i t should b e noted t h a t methods f o r modelling ecosystem dynamics are relatively developed; however, o u r under- standing of ecosystem s t r u c t u r a l changes o v e r l a r g e regions i s still relatively p o o r , and only l i n e a r models are used ( s e e below).A primary input into regional models is a n ecological classification of t h e study a r e a , including t h e identification of primary h a b i t a t types. Then specific succession c l a s s e s must b e identified, since they emerge in e v e r y h a b i t a t following most common t y p e s of initial shocks (past investations, f o r e s t f i r e s , etc.).
A t t h e ecosystem level, forecasting f o r e a c h f o r e s t t y p e i s c a r r i e d out using a n ecological model describing t h e integrated dynamics of t h e dom- inating species. The model should b e sufficiently universal t h a t i t is possi- ble t o d e s c r i b e long-term ecosystem behaviour following any set of initial conditions c r e a t e d by exogenic f a c t o r s under study, f o r a l l h a b i t a t types.
A principal f e a t u r e of f o r e s t vegetation is t h a t i t s succession dynamics
2 3
is r a t h e r long-term (10 -10 y e a r s ) as compared t o t h e observation period a c c e s s i b l e t o a n individual r e s e a r c h e r (10'-10' y e a r s ) . The d i f f e r e n c e in t h e s e time s c a l e s calls f o r t h e employment of a n indirect method f o r study of t h e development of f o r e s t vegetation:
(1) a number of areas with approximately similar ecological conditions but with d i f f e r e n t s t a g e s of vegetation are selected;
(2) a n assumption i s made t h a t t h e s e areas r e f l e c t d i f f e r e n t develop- ment s t a g e s or o n e arrd t h e same area shifted in time;
(3) based on s e l e c t e d sets of c h a r a c t e r i s t i c s , t h e s e f o r e s t s are a r r a n g e d in o r d e r , r e p r e s e n t i n g t h e i r temporal sequence.
Although t h i s i n d i r e c t method i s t h e only method available and is e a s y to use, extrapolations forward in time are uncertain. For a v a s t t e r r i t o r y with c o n t r a s t i n g ecological conditions subjected to various e f f e c t s , t h e number of observed and t h e o r e t i c a l l y possible genetic lines of development i s extremely high. Long-term observation of succession dynamics at "typical"
s i t e s i s impossible. Thus, w e believe t h a t t h e only method of f o r e c a s t i n g i s to c o n s t r u c t a dynamic, multi-species, age-distributed quantitative model, c o r r e l a t e d with t h e ecological conditions of t h e t e r r i t o r y insofar as f a r as d a t a permit. The model i s considered not only as a means of making projec- tions, but a l s o as a n instrument f o r ecological r e s e a r c h .
One such model which h a s been verified by field d a t a [2,14] i s d e s c r i b e d below.
A N EXAMPLE
OF
AECOSYSTEM
MODELW e p r o p o s e a dynamic-demographic a p p r o a c h [7], which implies t h a t t h e form of t h e frequency distribution of tree a g e s i s t h e r e s u l t of b i r t h and d e a t h p r o c e s s e s . In t h e following, t h e analysis is made in terms of competi- tion p r o p e r t i e s , s e e d production intensities, mortality probabilities f o r e a c h a g e , s e e d germination probabilities and so on. S e v e r a l demographic models of population dynamics have been d e s c r i b e d in a number of p a p e r s . Some of t h e models are linear [3,5], i.e. t h e tree mortality probabilities are density-independent; evidently such models are a d e q u a t e only f o r s h o r t p e r i o d s of time. The proposed non-linear models (e.g. [4]) r e q u i r e l a r g e volumes of ecological d a t a and they a r e still not completely r e a l i s t i c in t h e ways t h a t t h e y d e s c r i b e ecological mechanisms.
W e have attempted t o formulate a relatively simple model, which r e q u i r e s only d a t a obtained during conventional-type f o r e s t surveys. A f e a t u r e of o u r a p p r o a c h is t h a t some of t h e hard-to-measure ecological p a r a m e t e r s are estimated by adjusting t h e o r e t i c a l a g e distributions (obtained f r o m a model) to empirical ones. Then t h e s e p a r a m e t e r s are used in model projections.
Below w e shall d e s c r i b e s o m e p a r t i c u l a r f e a t u r e s of a g e distributions, a question t h a t promoted t h e construction of t h i s model. A typical a g e dis- tribution form f o r f o r e s t vegetation with a relatively high density of mature individuals i s p r e s e n t e d qualitatively in Figure 1. This form is typical of f o r e s t stands of d a r k coniferous [10,16,20], beech [8], and oak [8]. Similar distributions have been observed in various s t a g e s of a deciduous
-
d a r k coniferous succession in t h e Central Ob region. The non-linearity and non- monotonicity c u r v e shows t h a t t h e situation i s non-stationary, i.e. t h e sys- t e m i s f a r from reaching a climax. The t y p e of c u r v e p r e s e n t e d in Figure 1 i s associated with a situation in which a major p a r t of t h e forest w a s des- t r o y e d (e.g. by c l e a r cutting, f o r e s t f i r e ) T y e a r s ago. The "package" of s e n i o r a g e s o n t h e right-hand side of Figure 1 r e p r e s e n t s individuals t h a t occupied t h e forest during t h e f i r s t decades a f t e r t h e shock. These s p e c i e s suppresed f u r t h e r r e g e n e r a t i o n , which explains qualitatively t h e e m e r - gence of t h e "valley" in t h e medium-age group. A quantitative description of t h i s "valley" r e q u i r e s t h e employment of a n essentially non-linear model.Of g r e a t i n t e r e s t i s t h e t a s k of estimating t h e conditions under which t h e
"package" of mature individuals o c c u r s and of t r a c i n g t h e dynamics, s u b j e c t t o t h e ecologically i n t e r p r e t e d model p a r a m e t e r s .
Another task is estimation of t h e success of r e g e n e r a t i o n . Since juvenile dynamics depends primarily on t h e state of t h e p a r e n t stand, a f o r e c a s t of r e g e n e r a t i o n can b e accomplished only within t h e framework of a g e n e r a l f o r e c a s t of t h e dynamics f o r individuals of a l l a g e groups.
Finally, t h e r e i s a r a n g e of problems concerning a change of species in t h e c o u r s e of succession dynamics. The description of such changes a l s o r e q u i r e s age-distributed models
-
in t h i s c a s e , though, f o r e a c h species.A s a f i r s t attempt of t h i s kind, w e shall use o u r model t o d e s c r i b e a two- s p e c i e s succession.
Front C
The "package" of mture individuals
min
Figare 1: A qualitative p i c t u r e of t h e frequency distribution of tree a g e s observed in a s p e c i e s following destruction of t h e f o r e s t stand T y e a r s ago. n (T, T)
-
t h e number of T-aged in- dividuals at t h e Tth moment, n (T,o )-
t h e intensity of inva- sion.When formulating t h e model, w e a d h e r e d t o t h e principle of "the least number of d e s c r i p t i v e variables", i.e. w e made use of a v e r y limited set of variables, sufficient f o r adequate description of t h e empirical d a t a . Addi- tional input v a r i a b l e s would b e a p p r o p r i a t e only in cases where t h e e f f e c t s cannot b e explained by t h e "standard" set of variables. The hasty u s e of a n
"excessive" set of v a r i a b l e s inevitably leads to r a t h e r speculative model projections, owing t o t h e usual lack of field data.
The simplest possible set of variables f o r a n age-distribution model includes values ni (t ,T), where i i s t h e s p e c i e s number, and n is t h e quan- tity of individuals at a g e T at time t . W e consider t h e a g e dependencies of tree heights and diameters t o b e fixed, i.e. they d o not depend on t h e i r c u r r e n t numbers ni ( t , T). t h e introduction of more complex growth p r o c e s s e s would at least double t h e number of v a r i a b l e s and make the model
more complicated. Introduction of distrlbution by dimensions d in e v e r y given a g e , i.e. operating with ni (t ,r,d) values would make i t even more complicated.
Assume t h a t lifetimes of two coexisting s p e c i e s are P and Q. Then w e divide t h e life cycle of individuals into yearly intervals. The numbers of individuals f o r each such interval i s $ ( i = 1 ,
. . .
, P) f o r t h e f i r s t s p e c i e s and Yk (k=
1 ,. . .
, Q ) f o r t h e second one; a unit area of one h e c t a r e i s con- sidered. For simplicity t h e v a r i a b l e t is omitted. Since w e a r e t o d e s c r i b e t h e whole s p a n of t h e a g e c u r v e , t h e f i r s t classes i=
k=
1 give t h e r e s p e c t i v e quantities of s e e d sX1
and Y1. Let us introduce t h e following notations:ai and
Pk
r e p r e s e n t fertilities, i.e. t h e a v e r a g e number of s e e d s pro- duced annually by t h e f i r s t species at a g e i and t h e second one at a g e k ;ai and bk stand f o r survivability coefficients from a g e i t o a g e i
+
1(and, respectively, from k t o k
+
1);Obviously a * , bk S 1 and 1
-
a i , 1-
bk are annual death r a t e coeffi- cients;j' and f 2 a r e equivalent t o t h e annual seed immigration r a t e s of t h e f i r s t and t h e second s p e c i e s respectively, p e r unit h e c t a r e of t h e study area.
Let us divide t h e ecological f a c t o r s influencing t h e life cycle into two groups
-
density-independent and density-dependent ones and l e t us p r e s e n t f e r t i l i t i e s and survivabilities in t h e form of a multiplication of density- independent ahd density-dependent f a c t o r s :In keeping with t h e view t h a t a r e s t r i c t e d s e t of model variables should b e used, all f a c t o r s of t h e f i r s t t y p e are supposed t o b e r e p r e s e n t e d by t h e p a r a m e t e r s of t h e model.
The dependence of t h e introduced values a,*,
oak,
a , * , bok on t h e den- s i t i e s X , Y is t o b e found with t h e help of p a r t i c u l a r models describing com- petition and r e g e n e r a t i o n of t h e species under consideration. We have t a k e n up t h e method proposed in [I51 as a competition model, which consists of t h e following. I t i s assumed t h a t sunlight i s t h e principal density- dependent f a c t o r inhibiting species growth in ecosystems; t h e crown s h a p e of a n individual tree is modelled in t h e form of a horizontal o r v e r t i c a l s c r e e n , partially absorbing light; t h e a v e r a g e amount of light falling on a n individual tree of a given size is computed. Let us denote t h e amount of light f o r a n individual t r e e a t a g e s i and k as:Values (2) depend in general on a l l t h e variables of t h e system, i.e. on
q ,
Y k . In o u r model, w e have chosen t h e simplest way of r e p r e s e n t i n g indi- vidual trees-
by horizontal s c r e e n s having area Sm at height hm f o r a g e m . In t h i s case according t o [7], t h e values (2) are computed by t h e formu- las:H e r e R, is t h e intensity of t h e initial light flux;
S i , Vk a r e s u r f a c e a r e a s of individuals (in o u r p a r t i c u l a r model
-
t h e a r e a of horizontal s c r e e n s ) ;p l , pz
-
light absorption coefficients f o r s c r e e n s of t h e f i r s t and t h e second type;qi
-
t h e minimal a g e j when s c r e e n j of t h e second t y p e becomes h i g h e r than s c r e e n i of t h e f i r s t type;rk
-
t h e minimal a g e j, when s c r e e n j of t h e f i r s t t y p e becomes higher than s c r e e n k of t h e second type.Through indices qi and r k , o u r model c a p t u r e s t h e simple f a c t t h a t each s c r e e n i s in t h e shadow of higher ones. I t should b e noted t h a t a simi- l a r idealization i s used in a group of closely r e l a t e d models of f o r e s t dynam- i c s t h a t originate from [3,22]. (A model in which individuals are vertical s c r e e n s would b e more r e a l i s t i c b u t would r e q u i r e much more complex for- mulas f o r R l i , Rzk .)
For p r a c t i c a l application, o u r model must be modified from a n e x a c t one f o r t h e s c r e e n populations into a semi-empirical model f o r r e a l popula- tions. Let us assume t h a t S , V a r e effective a r e a s of individuals, and t h a t they a r e proportional t o t h e s q u a r e s of heights S
-
h' ,
V-
g from dimen- sional considerations. F u r t h e r , instead of absorption coefficients P 1 , P 2 , we introduce empirical coefficientswhich are equal t o t h e "effective" absorption coefficients, i.e. ones taking into account t h e volume and shape of t r e e crowns, t h e density and orienta- tion of t h e i r phytoelements, i.e. all t h e i r deviations from ideally homogene- ous f l a t s c r e e n s , and a l s o including proportionality coefficients between S and h and between V and g
' .
The model of r e g e n e r a t i o n (ontogenesis) should b e constructed s o as t o d e s c r i b e t h e dependence of f e r t i l i t i e s and survivabilities on t h e r e s o u r c e quantities (2) available f o r t h e individuals, i.e. t h e functions should b e of t h e form
The i n t e r p r e t a t i o n of t h i s c a s e i s a difficult biological-mathematical t a s k , which does not y e t have a n a c c e p t a b l e solution. T h e r e f o r e , w e used t h e sim- plest and biologically adequate assumptions t o estimate functions (5). I t w a s assumed t h a t in e v e r y a g e group t h e s e functions are equal t o t h e value of t h e r e s o u r c e s available p e r unit area of a n individual's s u r f a c e
A power-law dependency on p a r a m e t e r s 61, 62 i s used because of o u r d e s i r e t o obtain from t h e s e p a r a m e t e r s t h e simplest form of density-dependent s u r - vivabilities and f e r t i l i t i e s
where
a r e coefficients of i n t r a s p e c i e s (yll , yZ2) and interspecies (ylz , yZ1) com- petition, t h a t t a k e into account both t h e different absorption capacities (coefficients d j ) , and d i f f e r e n t shade t o l e r a n c e capacities (coefficients 6l ).
The r e q u i r e d model, whioh is a system of dynamic equations, i s con- s t r u c t e d by t h e usual balance method. Let t h e numbers of individuals at time t be equal t o ( t ), Yk (t ); then at time t
+
1 we havewhere a, and b, are coefficients of seed adaptability and dispersion t a k e n as density-independent; t h e dependency of f e r t i l i t i e s and survivabilities on X , Y is defined by formulas (6) and (7);
L
, M a r e t h e a g e s when species become f e r t i l e .Competition coefficients y y as well as seed immigration intensities of d a r k coniferous and deciduous species f 1 , f 2 were estimated by adjusting t h e t h e o r e t i c a l by t h e empirical age-distribution c u r v e s .
A s a measure u f o r finding f i and Ytk, w e took t h e t o t a l (by species and ages) RMS r e l a t i v e deviation of model distributions from t h e empirical ones p e r annum
.- . -
where T is t h e a g e of succession, and z (T , T), Y ( T , T) and n l ( T , T), n 2 ( T , T) a r e t h e o r e t i c a l and empirical a g e distributions of d a r k conferous and deciduous species, respectively. Optimal values of f i and Yik,
fFt
and7zt
t h a t yield t h e minimum f o r u were obtained by a v a r i a n t of t h e gradient method [18].Values of t h e o t h e r p a r a m e t e r s were t a k e n from t h e l i t e r a t u r e (fertil- ity c u r v e s ) o r estimated during preliminary computations (density- independent mortality curves).
The resulting optimal values of t h e p a r a m e t e r s (in t h e s e n s e of minimal u [18]) are p r e s e n t e d in Table 1. These r e s u l t s are discussed below.
1. The constants of inter- and i n t r a s p e c i e s competition yil: r e v e a l a grouping f o r e a c h type of interaction (according t o t h e rows of t h e table). The clustering is shown in a n o t h e r way on a numerical a x i s using a logarithmic scale.
h he
Y12 constant turned out t o be so small t h a t t h e gradient method yielded Y12=
0.Thus, t h e accuracy of fi Id data proved t o be i n s u f f i c i e n t t o determine t h i s constant; be- ginning with Y12
=
1 0 -8
, t h e value U (10) ceases t o respond t o changes of Y12; hence an y p e r limit i s presented in t h e table.Constant here i s higher than i t s real value, since there are no Juveniles in t h e deci- duous stands on t h e t e s t s i t e s , owing t o competitive suppression by t h e dark conifers, as well as by t h e development of a moss cover. Since t h e model oes not contain a variable t h a t describes t h e s e dynamics, t h e constant turned out t o be too high.
The groups are distributed as might b e e x p e c t e d from a con- sideration of s p e c i e s distributions and shade-tolerances. This' grouping seems to confirm t h a t a l l f o u r test s i t e s , which have been used to adjust t h e model, belong to t h e same succession line, i.e.
ecological conditions at t h e s e s i t e s are similar.
The grouping of uniform yik implies t h a t mean values of parame- ters yu should give suitable projections of behaviour at e a c h test site. Actually, w e have o n e of t h e possible verification variants:
assuming t h a t t h e s i t e s belong to t h e s a m e succession line, w e find t h a t t h e yik grouping t e s t i f i e s to t h e adequacy of t h e succession model used; if t h e model i s adequate, t h e n t h e s i t e s should belong to t h e s a m e succesion line.
2. The smallness of t h e values of p a r a m e t e r yI2
-
t h e competitive influence of t h e deciduous s p e c i e s on t h e d a r k coniferous one-
p r o v e s t h a t t h e latter i s actually "autonomous", and t h a t t h e f o r m e r h a s no e f f e c t on it. On t h e c o n t r a r y , t h e deciduous s p e c i e s i s "the follower" h e r e , i.e. t h e influence of t h e d a r k coniferous s p e c i e s on t h e s h o r t e r deciduous trees i s much s t r o n g e r t h a n t h e self -induced e f f e c t s of t h e latter. The "package" of deciduous individuals, having "escaped" from t h e influence of t h e d a r k coni- f e r o u s s p e c i e s i s a f f e c t e d only by i n t r a s p e c i e s competition.
3. W e h a v e examined variations of values yu f o r e a c h area in o r d e r to c l a r i f y t h e model's sensitivity to changes of t h e s e c e n t r a l p a r a m e t e r s . F o r t h i s purpose, cr isolines w e r e found correspond- ing to a 20% deviation from
#"";
taking into account t h e , relatively l o w a c c u r a c y of t h e quantitative description w e believe t h a t a 20%-
12-
Table 1: Results of model adjustment t o empirical age-distributions
I
-
180
-
0
10 -5
0.07
0.00021
2000
1000000
0.025
46
0.53 Succession
Constants of competitive i n t e r a c t i o n s
Intensity of s e e d immigration number p e r h e c t a r e Juvenile survivability of d a r k c o n i f e r o u s s p e c i e s The a g e s i n c e C*
=
1RMS deviation values
a g e T y e a r s
y ll ( d a r k coniferous- d a r k c o n i f e r o u s ) yi2 (deciduous-dark coniferous)*
yzl ( d a r k coniferous- deciduous)**
yZ2 (deciduous- deciduous)
f
1 ( d a r k c o n i f e r o u s )f 2
(deciduous)
N lJ ck k =1 K = l ( s e e Eq. 1) N y e a r s
Pi"
100
0
<lo5
0.1
0.00036
1645
100000
0.069
20
0.53
120
0
U O - ~
0.13
0.00069
1550
75000
0.059
17
0.56
160
0
<lo
-50.08
0.00041
1550
100000
0.047
50
0.51
value (but not lower) is reasonable. W e assume t h a t within t h e domain
a n y value of ytk gives a r e a s o n a b l e system dynamics projection.
F o r b r e v i t y , w e consider domains o l , w2, w3, w4 corresponding to areas with T
=
100, 120, 160, 180 y e a r s . Out of a l l possible two- dimensional projections, w e shall c i t e a p a r t i c u l a r one, namely (yll , yZ2); see Figure 2. (Note t h a t p a r a m e t e r yll h a s a s t r o n g e r effect t h a n t h e o t h e r s on t h e f o r m of t h e t h e o r e t i c a l c u r v e x ( T , T) and t h e d e g r e e of adjustment, i.e. t h e value). G r e a t e r extension of a l l domains wl along coordinates yZ2 compared to yll obviously c o r r e s p o n d s t o g r e a t e r c o n t r o l of t h e d a r k coniferous as compared t o t h e deciduous species, since t h e system r e s p o n d s by s t r o n g changes of a g e distributions t o t h e variation of yll and by weak o n e s-
to t h e variation of yZ2. for T=
100, t h e domain i s extended along yz2 much l e s s t h a n along t h e o t h e r s-
t h i s i s obvi- ously due to a relatively underdeveloped d a r k coniferous popula- tion. So, changes of t h e f o r m from wl to w2 r e f l e c t t h e rapidity with which t h e d a r k coniferous s p e c i e s t a k e c o n t r o l of t h e succes- sion dynamics.APPLICATION OF THE MODEL
The model c a n b e used f o r quantitative projections of deciduous
-
d a r kconiferous succession u n d e r various scenarios:
1. having determined c e r t a i n initial distributions of x (o
,
T ) , y (o , r ) , one c a n d e s c r i b e t h e e f f e c t of a p a r t i a l burning of t h e f o r e s t with subse- quent r e g e n e r a t i o n ;2. by reducing t h e population numbers in s o m e a g e g r o u p s at a given time, one can simulate s e l e c t i v e cutting;
Figure 2; Isolines of a which c o r r e s p o n d to a 20% deviation from pin;
oi, 02, u3, uq c o r r e s p o n d to t h e s i t e s with succession a g e s T
=
100, 120, 160, 180, respectively.3. by slightly modifying t h e system p a r a m e t e r s , one c a n estimate t h e f o r e s t ' s r e s p o n s e t o weak "background" impacts and t h u s use t h e pred- ictions f o r t h e p u r p o s e of vegetation monitoring [12].
Simulation of e f f e c t s caused by climate change w a s c a r r i e d out by introduc- ing a n e x t e r n a l f a c t o r , which inhibited o r stimulated t h e i n c r e a s e of tree height and diameter, according t o a time-exponential l a w . W e h a v e con- s i d e r e d in g r e a t e r d e t a i l t h e c o u r s e of succession with t h e p a r a m e t e r s given in Table 2 and with
where r a t e s of increment changes A, p may b e of e i t h e r sign. Of all possible c h a r a c t e r i s t i c s of succession, w e have studied two f e a t u r e s of primary i n t e r e s t , viz., t h e number of mature individuals, and t h e i r biomasses The impact intensities (the values of A, p ) are supposed t o b e weak, s o a l t e r a t i o n s of N and M may b e found in t h e form
where ai are r e s p e c t i v e logarithmic derivatives t a k e n in t h e u n p e r t u r b e d state X
=
p=
0. Note t h a t all values in (13), e x c e p t f o r X and p depend on t h e a g e of succession T.The biomass of e a c h s p e c i e s w a s found by t h e formula
where i, i s a n initial time, viz., t h e time when t h e individual t r e e s became mature. (During t h e adjustment of t h e model, it w a s established t h a t t h e b e s t coincidence of
~i~~~
and N z X P i s achieved when i,=
tmin-
15 (Figure 1 ) f o r t h e deciduous species.)i, was set equal t o t on t h e c u r v e y, (t ); s e e Figure 1 ;
Y i are t h e c o r r e c t i v e coefficients f o r calculation of s t e m volume;
Hi
=
hi , g i are tree t o p heights f o r t h e two species, determined from (12);Di
=
d l i , d 2i are t h e s t e m diameters of t h e two species defined by f o r - mulas, similar t o (12)where
dial,
diO2 are u n p e r t u r b e d c u r v e s of diameter growth, obtained from test s i t e data. S o i t i s assumed t h a t t h e growth p r o c e s s of each species con- s e r v e s t r e e shape. The p a r t i a l derivatives ai in (13) were estimated at t h e point X , p=
1 0 -3.Table 2: P a r a m e t e r values of a deciduous
-
d a r k c o n i f e r o u s succession model f o r o n e of t h e test s i t e sAge of succession T 1 2 0
Constants of competitive i n t e r a c t i o n
Intensity of s e e d
immigration number p e r h e c t a r e p e r annum
P a r a m e t e r s of t h e c u r v e r e p r e s e n t i n g density- independent juvenile survivability of d a r k - c o n i f e r o u s s p e c i e s Initial f e r t i l e a g e S e e d s '
survivability c o e f f i c i e n t Maximal f e r t i l i t y c o e f f i c i e n t
yll ( d a r k coniferous-dark c o n i f e r o u s ) (deciduous-dark c o n i f e r o u s ) ( d a r k coniferous-deciduous) (deciduous-deciduous) j' ( d a r k c o n i f e r o u s ) j' (deciduous)
N
t h e a g e s i n c e Cf=
1 (s e e Eq. 1 )L ( d a r k c o n i f e r o u s )
M
(deciduous) a, ( d a r k c o n i f e r o u s ),
(deciduous)tP
( d a r k c o n i f e r o u sNq
(deciduous)The r e s u l t s are p r e s e n t e d f o r t h e test s i t e with c o n s t a n t s f r o m Table 2 . Values af w e r e found f o r 1 0 0 S T 5 160.
Values a l, a 2 , a s , a 6 , a7 have proved t o b e almost independent of succesion a g e T; values as, a g show a weak dependence on T and are given in (16) f o r T
=
120. Due to t h e weak influence of t h e deciduous s p e c i e s on t h e d a r k - coniferous one, a 2 and a 6 are v e r y small.Note t h e influence of growth rate changes of e a c h s p e c i e s on t h e numbers and t h e biomasses:
1 ) when growth is s u p p r e s s e d (A, p <0), t h e population numbers i n c r e a s e while t h e i r biomasses d e c r e a s e ;
2) when growth i s stimulated, t h e e f f e c t i s r e v e r s e d ; h e r e w e h a v e a purely ecological ( o r competitional) compensation e f f e c t , s i n c e in t h e a b s e n c e of competition, t h e biomass change would h a v e been much g r e a t e r .
Let us give a n example, a change in t e m p e r a t u r e T'. A s is known [17], a 1' i n c r e a s e in
T'
(all o t h e r p a r a m e t e r s remaining in t h e optimum zone) leads t o a 10% biomass i n c r e a s e , which c o r r e s p o n d s to a 3% change in tree heights and diameters. Let u s assume t h a t t h e change of 1' inT'
t a k e s p l a c e during 30 y e a r s . Then t h e annual i n c r e a s e in l i n e a r dimensions i s equal to(such values give a 10% change in tree heights and diameters o v e r 100 y e a r s ) .
For t h e s e A , values and f o r t h e study test s i t e , some r e s u l t s are p r e s e n t e d in Figure 3 and 4. In p a r t i c u l a r , f o r a succession which began in 1980, t h e change in population numbers in t h e dark-coniferous s p e c i e s would b e -17%, while by t h e y e a r 2100 t h e i n c r e a s e in biomass would b e
+20%.
Figure 3: Dependence of t h e population numbers of mature individuals on succession a g e '.2
Nol, NO2
are t h e u n p e r t u r b e d (X=
p=
0) numbers of t h e dark-coniferous and t h e decidu- ous species, whileN1, N 2
are those p e r t u r b e d , withx = =
F o r f a c t o r s having non-equal influences on t h e two species, t h e impacts c a n be examined in a similar way. (For example, a reduction of light flux due t o a n increased atmospheric a e r o s o l loading would e x e r t a h e a v i e r impact on t h e light-requiring deciduous species than on t h e coniferous one.)
AN EXAMPLE OF A REGIONAL-SCALE FORECAST
The mathematical technique f o r prognosis is as follows. First t h e region being studied is subdivided into habitats of types j
=
1 ,. . .
,R ,
e a c h having i t s own intrinsic successional dynamics following exogenous dis- turbances. In i t s t u r n , each succession line is subdivided into d i s c r e t e s t a t e s ( o r stages) n
=
1 ,. . .
, Q including all t h e s t a t e s t h a t have a p p e a r e d due t o exogenous disturbances and endogenous dynamics. Let us introduce t h e quantities pjn, which r e p r e s e n t f r a c t i o n s of t h e t e r r i t o r y , r e l a t e d t o habitats with ecosystems at state n . W e shall limit o u r discussion t o t h e c a s e of ecosystems t h a t d o not i n t e r a c t , i.e. they have t h e i r own endogenous dynamics and d o not influence t h e i r neighbours a c r o s s t h e boundary. Then t h e dynamics of f r a c t i o n a l a r e a s are described by a l i n e a r Markovian sys- t e m [22]where ajn,
V
) are t h e transition probabilities from succession s t a g e n into s t a g e s ; t h e s e values d e s c r i b e both endogenous f a c t o r s and t h e r e p l a c e - ment of s t a g e s due t o exogenous disturbances; 'j r e p r e s e n t s climate param- e t e r s and t h e physico-chemical s t a t e of t h e atmosphere; t i s time.To simplify o u r t a s k , we shall not d e a l with population dynamics, but will d e s c r i b e t h e state of t h e ecosystem by only one variable f o r e a c h s p e c i e s i , i.e. by i t s biomass:
where i i s t h e s p e c i e s number, j t h e t y p e of h a b i t a t , n t h e s t a g e of succes- sions q are p a r a m e t e r s , and
I
i s as defined e a r l i e r . For t h e s a k e of simpli- city it i s assumed t h a t all successional s t a g e s last t h e same length of time, viz., t h e time s t e p in (17).The essence of t h e proposed f o r e c a s t scheme i s t h e simultaneous use of models (17) and (18); t h e latter i s based on t h e s e v e r a l exogenous effect- mechanisms, which could b e "soft" o r "hard". The f o r m e r include f a c t o r s t h a t weakly change t h e p a r a m e t e r s of t h e vegetation environment. A l l t h e explicit background f a c t o r s along with some implicit ones (e.g. bogging) are
Figure 4: Dependence of mature individuals' biomasses on t h e succes- sion a g e T. Mol, MO2 are t h e u n p e r t u r b e d (A
=
p = 0 ) bio- masses of t h e dark-coniferous and deciduous s p e c i e s , andM I ,
M2 are t h o s e with a X=
p=
p e r t u r b a t i o n .in t h i s group. The second t y p e comprises f a c t o r s t h a t change t h e a f f e c t e d ecosystem r a t h e r rapidly into a n o t h e r state
-
t h e s e f a c t o r s include f o r e s t f i r e s , s t r o n g windfalls a n d p e s t infestations; such e f f e c t s d o not change t h e"state" of t h e Markovian system (17). For t h i s r e a s o n , background f a c t o r s (weak by definition) are not included in transition probabilities a d e s c r i b - ing endogenous changes, but t h e y are included in probabilities of intermit- t e n t transitions t h a t o c c u r under t h e influence of "hard" f a c t o r s . Changes of productivity, f e r t i l i t y and survivability, a r i s i n g from t h e influence of
"soft" background f a c t o r s are t a k e n into account in t h e model (18). The e f f e c t of h a r d f a c t o r s is not d e s c r i b e d by this model.
Let u s have a habitat of a given type, occupied by a single dominant species; t h e r e f o r e w e c a n d r o p t h e index k for p, a values in (17) as well as i and k indices for m in (18). Let t h e t e r r i t o r y b e in a state of equilibrium as a whole, i.e. t h e f r a c t i o n s of areas
Pn
c o v e r e d by e a c h of t h e succes- sional s t a g e s are constant. Suppose t h e r e i s a weak change in t h e value of o n e exogenous p a r a m e t e r ,4P/f
S 1. This e f f e c t , which i s s o f t at t h e ecosystem level, would change t h e mean values of t h e biomasses a t e a c h suc- cessional s t a g eA t t h e regional s c a l e , f o r e s t f i r e burn-out probabilities would change slightly f o r e a c h succession s t a g e
along with equilibrium f r a c t i o n s of a r e a s
The p a r a m e t e r s p , a, n are coefficients of susceptibility to f a c t o r f
.
The t o t a l biomass of t h e s p e c i e s u n d e r consideration p e r unit regional area i s
Let u s t r y to assess t h e values p ( n , f ) , nn U ) . Suppose t h e f a c t o r f i s t e m p e r a t u r e , and
4P =
lo. F o r b o r e a l f o r e s t s , values of p(n , f ) , are in t h e o r d e r of 0.05-0.1 l / d e g[Ill.
Assessment of ) 'n, j ( i s more difficult. F o r taiga f o r e s t s , t h e burn-out probability (k + I ) transition is approximately equal tol o 9
l / p e r annum[Ill.
When t h e climate warms, t h i s value changes (mainly d u e to t h e variation of frequency of d r o u g h t y e a r s ) . F o r t h e Euro- pean p a r t of t h e USSR during t h e n e x t s e v e r a l decades, t h e p r o j e c t e d change of t h i s value d u e to greenhouse warming will b e within t h e r a n g e of from 0.3 t o 0.4 l / p e r annum. Assuming t h a t t h e burnout probabilityi n c r e a s e s in t h e same proportion, i t follows t h a t a n l 0.3, and a, l(f
+ 41
)=
0.01(1+ 0 . 3 4 1 ). By taking reasonable steady state valuesP,
(f), i t is easy t o show t h a t t h e change ofP,
V ) h a s t h e s a m e o r d e r of magnitude as a, (f), t h e calculation being c a r r i e d out in a c c o r d a n c e with (17), i.e. t h a t n , V ) 2 -0.3. S o values of t h e summands within e v e r y p a i r of b r a c k e t s in (19) d i f f e r slightly but have opposite signs, i.e. t h e biomass i n c r e a s e e f f e c t at t h e ecosystem level would b e compensated by i t s d e c r e a s e at t h e regional level. The a u t h o r s believe t h a t t h i s preliminary assessment indicates t h a t a more detailed study should b e undertaken.IMPLICATIONS OR ECOSYSTEM MONJTORING
T h e r e is growing recognition of t h e need t o establish monitoring sys- t e m s t o provide e a r l y indications of ecosystem change; s e e , f o r example,
[ Z l l .
I t i s o u r belief t h a t specialized information s e r v i c e s should b e esta- blished f o r t h i s p u r p o s e at t h e regional level.
The technological basis f o r t h e information s e r v i c e s should b e pro- vided by remote sensing and thematic i n t e r p r e t a t i o n of imagery, one of t h e f i r s t t a s k s being t o undertake a regional ecological survey. This would include:
d a t a on t h e f r a c t i o n s of land s u r f a c e s occupied by e a c h of t h e pri- mary vegetation types (forests, meadows, bogs) at c i r c a 1:106 s c a l e imagery;
descriptions of t h e ecosystems within e a c h vegetation t y p e (a listing of dominants, t h e number of individuals) by means of c i r c a
l:lo5
s c a l e imagery and supporting information f r o m ground-truth sites;information on c u r r e n t exogenous stresses (climate anomalies, r e c e n t forest f i r e s , p s t outbreaks, etc.).
Prediction of f u t u r e regional ecosystems states would b e based on t h e models described in [9],
[I81
and [17], values of t h e input p a r a m e t e r s being obtained f r o m t h e ecological survey d a t a described above. At r e g u l a r intervals, t h e survey should b e r e p e a t e d t o update t h e predictions, and t oobtain a growing file of time-series data.
Summarizing, it i s t h e opinion of t h e a u t h o r s t h a t highly aggregated f o r e s t r y information will not b e particularly useful in predicting t h e eco- logical impact of a n exogenous stress such as a f o r e s t f i r e , climate anomaly o r p e s t o u t b r e a k . Instead, it will b e necessary t o collect detailed informa- tion on ecosystem s t r u c t u r e (dominant species, a g e distributions, suscepti- bility t o damage by p e s t s , etc.). This information should b e obtained in a n integrated way o v e r a n e n t i r e region.
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