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MODELING STRATEGIES FORPROJECTING VEGETATION TRENDS
IN AN AREA OF NORTHEASTERNMINNESOTA
May, 1977
! •
MODELING STRATEGIES FORPROJECTING VEGETATION TRENDS
IN AN AREA OF NORTHEASTERNMINNESOTA
written by
Reed Sloss
for the
Regional Copper-Nickel Study
and submitted to
Dr. E.J. CushingDept. Ecol. and Behav. BioI.College of Biological SciencesUniversity of Minnesota
for academic credit
in May, 1977
- -c ontents _.-
Introduvv.J..on
Successional models
Succession in the Mine Site
The model of Shugart ~ ale (1973)(description and problems)
The cover-state replacement process(introduction of an alternate
model)
Applying the alternate model
Modeling management in the MlNESlTE Area
Modeling natural succession in theMlNESlTE Area.
Fire, drought, and epidemics
Conclusion,.'
Literature cited
Appendix: Solving a system of lineardifferential equations
1-8
3-5
5-8
9-15
16-20
21-32.
22-29
29-32
31
33-34
35-36
37-40
In 1974 at the request of the Minnesota Environmental.
Quali-~ Council, the State Legislat~re established the Regional
Copper-Nickel St~dy to conduct a regional study of the possible
impacts resulting from potential copper~nickel development in
northeastern Minnesota. A major objective of this study is
~o characterize terrestrial life as it exists now and,will
exist in the future prio,r to development. The accomplishment
of this objective is dependent upon the development of a process
that will accurately predict trends in vegetation--i.e. a math
ematical model that simulates succession. In this report I
attempt to explore the possibilities of modifying proposed
successional models found in the literature to suit this purpose.
Simplifying the task of 'characterizing terrestrial life
in a 2000-square-mile study area, the Copper-Nickel Study team
has focused its attention on a 560-square-mile area where mining
development is most probable and for which data are immediately
available (Fig. 1). Such data include the land-ownership, soil
type, and vegetation cover type assigned to each one-hectare
unit within this area. It is this MlNESITE Area for which
successional modeling will be examined (MlNESITE, 1976).
1
E
SIT
tM,r~l CAA/\DA
Af0.1!;~I!;?::~/"?_---W:::r~SN'-.j _~
I
~
~
------------ ... , -_ .
.\\---
[I~
N ----....... ~- '" -, - r-~i'.r-
If \··~·····f\~~~l. ~.t·1 cj" 1~...~ <~Jt::::j)j::::;:::I:~.
!~t
I, l, \ /,,; Lai\'e Co. IL' /I ~... " (;' ~"\m . ."
" A -J--- ~! V\i\aa{4§s:: .': <' /' '.I. @ ( . ;,'i' '; ,. :%/r"'ONITE HN\80R /HliaeJt-!G f. , ,Ir-r- .... ;,~ ·
I \ I 'Y , • 'y> /01 ~. ~-~'-r-t- ,y.r:«,\~ MlNNESOTA.
j ! ) Jf · 6 ~~ '~~VEIlBAV /" WIS(ONSIN-7 M1Cti
91
I I: &~k /' /I ~ : ~~~)\ ..,./
i \ LO~li! ~ .1 .~. ~~~~ /(. • if D /g'd ..~O._.-! : CJ;:::J , <::.l~~~ 0- ,.,./ ,0 .J\"Q 0 D e;, ~ /CD :=: ~~. -~. ~"\ \:-, './ ~ u a f:::/},,-y§ cd ,. I :>A~ ~~\'~~ .~~ AA Y'). ..~. ~~, 0 Cr-S c1
•., ... ill I ~~~.,~\'~ ~ D~" \?' /" tj.J ~~ ro -P I I ~~%~~~ t:'~~" ,. / /~.' ~.~ ~!:~..t'i~~~0%~' ~ / <\~j,~r~\~LuL .h cD I .~\~\ '. '.,,, "h\' •. . T ' nQ) b.D I ~:\\·':.\J.,.r<.. " ., I.p :>. r I " "\" '. j '1- /
M (j)''O .~ ' ,":. ~'.I;.:) .I SUPEf\lDf\ j "> l' -_?' •cd .-.( h : )\~\
~, ~ t·g l ~,. ...' I MlNESITE (1976) study are2~+.~ (;) Ie.
~ ~ ~ l ..-e;rlton Co., rmEl____~ ~ 7~._~_~_~~_~__~_________ r.~.~~··~··~·- ~ 5ME~i~G STUDY GIO
Successional Models.
Modeling strategy is dictated by the reali ty to be described
and an understanding of that reality. Given that:
generali-'cy measures the applicabili ty of the model indifferent locations,
realism measures the degree to which the model correspondsto the biological concep~s it represents, and
preci.sion is the closeness of model predictions to observed data,
three possible modeling s~rat~gies exist according to Levins' (1966)
widely-quoted classification.. Each strategy sacrifices one of
the above qualities so that the other two are maximized. Because
the basic purpose of a successional model in our context is the
accurate prediction of changes in vegetation over a relatively
short period of time (25-100 years) in a particular region of
northeastern Minnesota, a modeling strategy that sacrifices
generality to realism and precision seems most appropriate. In
this case, the most important parameters of these successional
processes need to be identified and accurately measured. Yet
succession is a poorly understood phenomenon, and consequently
it is difficult to identify the parameters let alone isolate the
more important ones. Hence, modelers of succession have resorted
to using the second modeling strategy (sacrificing realism to
generality and precision) in hopes that the unrealistic assumptions
implied by the general mathematical equations they use will affect
one another so that usefulL'predictions from the model can be obtained.
"'-.
Proposed successional models have talten on a varie"'ty of
forms. As outlined by May (1973), general mathematical models
are of four basic typese In both deterministic models, where
the dependent variables are continuous, and stochastic models,
~here they are discrete, the independent variable may be either
continuous (as in differential equations) or discrete (as in
difference equations). In a successional model proposed by
Shugart,Crow, and Hett (1973), a continuous dependent variable--
the acreage of land dominated by a particular forest type--depends
on the continuous variable of time. The flow of land between forest
type compartments is described by a set of first-order linear
differential equations. Similarly, Bledsoe and Van Dyne (1971)
used such a compartment model to describe the abstract flow of
energy or biomass from species to succeeding species during
oldfield succession. In the stochastic models of Horn (1975)
and Waggoner and Stephens (1970 )., the probabilities that a certain
tree or plot dominated by a particular species is replaced by
another is described by a finite Markov process. The dependent
and independent variables are discrete. In the successional
model by Leak (1970), probabilities that each tree in a forest
will either produce one offspring that becomes established,
produce no offspring, or die over the discrete interval of one
year are given by species-specific birth and death rates. Finally,
a complex stochastic model that indirectly simulates forest
succession by modeling tree growth was proposed by Botkin, J'anak,
and Wallis (1972). The model lumps deterministic equati.ons for
radial growth and growth in height. competition, and environmental
effects. But tree birth and death are determined by species-
specific probabilities so that both the dependent variable, the
number of trees on a plot, and independent variable--time--are
discrete.
Succession on the Mine Site.
Because the MINESITE Area is relatively large and thus
heterogeneous, the most appropriate model would seem at the
outset to be a form like that of Shugart et e1. (1973) or a
Markov model scaled up from describing the dynamics within a
particular forest to describing the dyna~ics of many forest
types within a region. In either case, the very large number
of concrete vegetation types occupying particular unit areas
are grouped into a finite number of abstract cover types.
Succession is then modeled in some deterministic or stochastic
fashion, while the total area of a region, constant over time,
is partitioned among cover types. The basic structure of this
model is described diagrammatically by what Shugart et ala (1973)
term a "model topology." In such a diagram (Fig. 2), the boxes
represent particular cover types, whereas the arrows represent
.5
the flow of area from dominance by one cover type to that
another.
6
A logical approach on the MlNESITE B. would involve ling
succession as it occurs naturally in the absence of natural or
unnatural perturba~ions. Disturbances could then be incorporated
later. Two basic problems are inherent in this technique. First,
natural succession in northeastern Mim1esota has not, as ye-t, been
well documented. Two recent studies (Heinselman 1973, Ohmann and
Ream 1-971a, 1971b) examine natural succession in the Boundary
Waters Canoe Area (BWCA) north of the MINESITE Area. A model
topology on the probable dynamics of eleven statistically-defined
cover types, is displayed in Figure 2. The magnitude of each
arrow is not known. Stressing the importance of fire in natural
ecosystems, Heinselman questions the existence of a true climax
conrrnuni ty for the area. Rather he views- climax on a regional
scale as a mosaic of cover types with no appreciable n~t change
in the total area occupied by each over time in the absence of
fire exclusion by man.
A second problem of modeling succession exists because the
forests on the MlNESITE Area are not 'virgin' (i.e. undisturbed
by European man). Natural succession may not be predictable even
by the best models for this intensively managed area where
L....
clearcutting is the most fre'quently used silvicultural system.
'7
A model topology describing natural succession in the B .. W C A.l'Dr those communities defined by~ Ohmann and Iteam (1971b) •
. probable trans.
pas sible trans.
PIKE
III)
I
IIII,II
_~ l
\'-mITE
NORTHERNWHI~~
CEDAR
RED PI1\~
IIL- __ --I _
JACK PINE JACK PII'1~
(BLACK (FIR)" SPRUCE)
//
// ASPEN-
LICHEN'\. BIRCH,", ,
"I " BLACK, , SPRUCE, (JACK, PINE) *
JACK PI1\TE RED MAPLE
(OAK) ASPEN-BIRCH
'* Includes the budworm disturbed COffiITlUni ty
Indeed, the accuracy of predicting future trends in vegetation
will rely heavily on an abili ty to predict fut-ure management
policies.
Within the framework of the modeling objective and these
associated problems, the remainder of th~s report will deal
with the advantages, disadvantages, and possible alternatives
to the deterministic model of Shugart et ale (1973).
8
~rHE MODEL OF-- --- -- ---~-----
Although deterministic Is yield c predictions,
9
the accuracy of those pre tions with respect to the r~al world
can be approximated only after the model is verified with field
data. The successional model of Shugar~
example of one that was not tested.
Description and Problems.
ale (1973) is an
The basic form of this model is a set of in' :first order
linear differential equations each of which represent the net :flow
rate of area into or out of cover-state x. (i=l,2,o •• ,n).1
dXl/dt=al1xl+a12x2+o .•• +alnxn
dX2/dt=a21xl+a22x2+····+a2nxn
•
•• •dXn/dt=anlxl+an2x2+····+annxn
n
dX./dt= La. ~X.1 j .. J lJ J
(where i=j=l, •• ~,n)
(1)
The a .. 's are coefficients that represent the transfer' of arealJ
f' th .th t th .th t t Th t trom e 1 a e.J cover-s a e. ese cons an 8 measure
the rate of movement along the arrows bet~een compartments in
the model topology diagrams.
The model represented in compact matrix notation is
d/dt(X)=AX (2)
where X is an n x 1 vector of the x. •s and A is an nxn matrix1
of the transfer coefficients, a .. being that value found inlJ.th .th
L.
the 1 row and J column of the matrix .. The diagonal elements of
10
this matrix (a .. ) have negative values and represent an output11
thflow of area from the i state or these values are zero in the
case of a climax state This flow is partitioned as i~puts of
area among the remaining states 80 thai; the sum of all column
elements is zerOe
(3)(for all i, j=l'lIme,n)n
La..=0i,.=\ lJ
Applying equation :3 over all colums implies that area can neither
enter nor leave the system or the region being modeled, which is
reasonablee Such a system is said to be closed (Bledsoe and
Van Dyne, 1971) and has the general solution:
n Aotx.=}:C .. e J ,
1 j::1 1J
where the ~.'s are the eigenvalues of matrix A and the C.. 'sJ lJ
(4)
are constants that depend on the initial distribution of ~he
region's area among the cover-states.
Each defined forest type '01' the region in question is
further divided by Shugart et ~. into three size classes--
seedling, pole, and saw-timber. Growth in these vegetation
~es is then simulated over the region if each of the size
catagories represents a separate cover-state and thus one in
the set of differential equations. As growth proceeds over
time, area would flow from seedling, through pole, to the
saw-timber size cover-state for any particular forest type.
The task of obtaining a matrix of trm!sfer coefficients
simplified by assuming that one forest type replaces 'another
only after that earlier forest type has reached saw-timber size m
The ae .§s are ~hen found by first measuring average gro~th inlJ '
stands that make up a forest type and, second, by finding the
probabilities- that one forest type is succeeded by another
(possibly measured by an average number of stems of the latter
found in the understories below the ~orrner)", The first step
isolates the diagonal elements of the coefficient matrix,
simplifying equation 1 to:
11
dX./dt=a .. x. with the solution1 11 1
-a·" tx.=x. e 111 10
where x. is the initial acreage in the i th cover-state. 'The10
diagonal coefficient, a .. , then is the reciprocal value of the11
time i~ takes for 63 percent of the initial area to leave the
i th cover-state because for a .. =l/t,11
- (a .. ) ( l/a .. ) -1 37 ( " )x.t=x. e 11 11 =X. e =. x. iD
1- 10 10 10
Coefficients other than the diagonals are determined by mult-
(6)
iplying the probability (p .. ) that the jth cover-state replaces1J
the i th forest saw-timber size cover-state by the diagonal
element of the latter:
aij=Pijaii·
Some time constants (l/a .. ) and transition probabilities11
(p .. ) used by Shugart et ale (1973) are shown in Tables 1 and 2.1J - -
According to the form of the model just described, the
'rable 1
darea ocsuccee
rcentars)eomes OC
e (1
The time consby one coveE.Lccording j~o S
Forest seedl
Aspen--- ---------------J
20 65
Jack pine 30 30 701------------;--------
Fir~spruce 30 45 cl
Northern hardwoods 100 100 2501-------------1---- -~-- ------- - --- ----~ ------11
Red pine
White pine
Black spruce
WamapaCK
Whi te cedar -..
100
100
80
100
57
100 100 300
150 200 1}50
80 80 24-0
100 90 290
1.15 1000 1172
Table 2
The probabilities that the row cover~£tate8 are succeeded by thecoluw1 cover-states according to Shug~rt ale (1973).
Succeeding state Birch-
.~ Red White N-&hard- Sugar Fir- Black ash- Whi ~e
'/;~% '/; nine nine woods HernIoe}r maple spruce spruce hemlock cedarAspen lIf~5 .55 .4
---:
-- - -~---,--
Jack pine 1-.. a
~- - - -- - _._._-- - --- .~-_.-
~J- ----.- -- -----
f{eq pine 1 .. 0- ---- -~_._----- --------- -- -- - ~- --...----
White·7 .2 .1pine I
;
.- -----_.~-- - -- -- ---- --..-
Northern 1.0hardwoods l
--- ----- ------_._--- f
Tamarack .8 .2....-- -- - ------ ----- ---- _._---------- ----~-_...- ---._-- - I
Whi te--i
1 .. 0 i
cedar !:
Statebeing
d
loss of area from a cover-s follows a e
:1
curve (Figures 3 and 4). Such an assumption is cle unre
This point illustrated if one examines growth in any fore
tJ~e simulated by the model. Consider, for example, the white
pine forest type of Shugart ~,,(1973) using their coeff'icients
in Tables 1 and 2. By removing all inputs to this subsystem and
initially setting 100 percent of the area dominated by white
pine in the seedling size-class, the behavior of the system over
time is· easily identified (see the appendiX) and is'shovffi graph-
ically in Figure 5. Two characteristics of these curves are
worth mentioning. First, pole and saw-timber are immediately
produced. Second and more important, only 50 percent OI the
initial seedling stan~s mature and give way to succeeding
overstories in 400 years. However, only the most vigorous white
pines reach such an age (Fowells, 1965). Growth is 'similarly
underestimated for the other forest. types listed in Table ~.
This inconsistency might have caused the underestimation in
abundffi1ce of the later successional forest types in Hahn and
Leary's (1974) trial simulation using Shugart et al.'s (1973)--model. If the transfer coefficients are increased to o£fset
this amazing longevity exhibited by the forest stands, ~he loss
of area becomes too great too quick so that the resulting curves
are still unrealistic.
1LI
Pigure J
s as implet al model
PercentTa-marack 60seedli:ngare a left 4·0
80
20
o '''''---r--_---.--"ii--_,20 .40 60 80 100
Time (years)
100
~rhe 58
area
-0 4---......-~_-_--__-_20 40 60 80 iDa
Time (years)
80
20
The loss of jack pine seedlarea to the j pole CDver-state as implied by the Shugartet tl model (197J).
100
-PercentJack pine ·60seedlingarea left 40
Figure 5
The decay of area from a white pine cover-state subsystem of theShugart et ale model (1973).- -
100
90
80
70
50
40
30
20
10
100L..
200 JOO
Time (years)
XXX){tXX seedling.
Ji.,(j,aJif1JJ,. pole- timber
oo00סס0 saw -timber
500
How does a curve behave which describes, over time, the
16
decrease in t;he area of a cover~· that results from stand
growth and mortality? As I have just shovm, the assumption that
the curve is negatively exponen'tial (as in -the model of Shugart
et m (1973)), is unrealistic. Nevertheless, such a simpli-
fication is probably acceptable over a very large and hetero-
geneous region. Over such a region, areas occupied by each
forest type are composed of many different stands of varying
ages and composition, occur on varying soil types and topography,
and are subjected to varying climatic and other environmental
conditions. These many stands are analogous to a population
of phenotypically varying individuals. Mortality in such a
population is often described by a continuous constant percent
decrease in numbers over time, i.e. a negative exponential
function (Lotka, 1925). The exponential curve then is a sim-
plification of the complex cover-state behavior that attempts
to account for heterogeneity over the region.
Instead of accounting for heterogeneity outright, an
alternative approach to describing the cover-state replacement
process involves building heterogeneity into a simpler math-
ematical system that describes replacement in a very homogeneous
region. For reasons that will become evident as this discussion
proceeds, 'this most c the at hand
Cons homogeneous region on which grows one large,
even-aged se stand where each surv seedling reaches
pole-size a.fter arse The loss in area occupied by this
\
seedling cover-state would occur all at once (Fig. 6). 1£ this
sarne hypothe region occupied equally by five even-aged
seedling stands of ages 0, 5, 10, 15, and 20 years, the loss
£rom the cover-state would follow a curve shovm in Figure 7.
Finally', in such a region occupied by many even-aged seedling
stands uniformly distributed in age and coverage, loss from the
cover-state would ideally follow a straight line (Fig. 8), not
an exponential curvee If the entire life span of an average
stand can be divided into discrete time intervals representing
ages, the growth of many stands over this homogeneous area can
be simulated by allowing the area occupied by stands of a -particu-
lar 'age to move to the next age-class as that interval of t~me
passes. The time interval might be set as low as one year or
as high as ten years. The form of this model for the interval
t=1 uni t is
(a Markov process). (8)
The diagonal elements of A are all set at -1. Those elements
just below the diagonals (a O +1 ,a.) are set at 1 in most cases.1 J. 1
It:....
A ve1--Y bulky model will result when this method is used
18
3010 15 205
20
80
60
%1. 00 ..d r, '-'"V" ,~. n r"f4. t}OOt::JfJ.X?O:T(>o:::e>CC:OQO'~~xtl
FiglJre 6
Loss in ar~~ from a seedlingcover-state i~ a homogeneousregion is occupied by oneeven~~aged seedling stand",-1t-
Time (years)
20
40
60
f---,----,.----r-----r--'~.~
5· 10 15 20 25 30
%area
100J,,{, 8<,,!
region ~
itt=()
Figure 7
Loss in area from a seedlingcover-state if a homogeneousis equally occupied by 0, 510, 15, and 20 year old evenaged seedling stands.*
Time (years)
Figure 8
Loss in area from a seedlingcover-state if a homogeneousregion is equally occupiedby seedling stands of allage9 between a and 25 years.*
% area100
80
60
40
20
*In this homogeneous region,surviving- seec.lings becomepole size in 25 years.
5 10 15 20 25, 30>:>,
Time (years)
to simulate average stand growth for many forest types because
a large number of variables or storage compartments (each rep-~
resenting an age-interval of a particular forest type) w,ill
be needed. However, such a simulation is not beyond the capacity
of today's computers.
Two major unrealistic assumptions are inherent in this
second model. First, all stands aren't necessarily ev~n-aged.
However, the abundance of uneven-aged stands in the MINESITE
Area and possibly in a greater portion of northeastern Minnesota
is probably negligible due to disturbance and the intolerant
(of light) character of most indigenous species. Second, stands
that make up a forest type grow at different rates depending ont' ..
environmental ,factors" not at the same rate as assumed in the
model. The level of complexity needed in the model to account
for such heterogeneity will depend on the degree to which these
growth rates vary about average values.
At the outset at least, the model is fundamentally and
conceptually very simple. Further, only a moderate number of
complications will probably be needed in the model since the
region isn't as heterogeneous as one might initially think.
For example, Grigal and Ohmann (1973) state that (1) "the narrow
range of climatic conditions" and (2) "the broad range of plant
tolerances" provide an explanation for the greater importance
19
20
of disturbance over other environmental factors in determining
the composition of plant communities in the BWCA. Disturbance
would tend to reduce regional heterogeniety. More importantly,
timber management in the area dramatically simplifies the complex
process of cover type replacement by maintaining most forest stands
in an even-aged structure.
21
~Phough stand age is often an important parameter of stand
growth, growth is usually measured in terms of the average size
of the -trees composing the stand--specifically the average diameter
at breast height (dbh). The Copper-Nickel Study group has rec
ognized five size categories--seedling (0-1" dbh), sapling (1-5" dbh) ,
pole (5-9" dbh), small saw (9-15"dbh), and a large saw-timber
size class (15+" dbh). The problem with the alternate model then
becomes one of relating stand age to the average dbh of trees in
the stand"
The site index of a forest stand (the predicted average
height of dominant trees at 50 years of age) gives a relative
measure of an area's suitability for supporting the particular
tree sources composing the stand.. This value is obtained from
pUblished species-specific site index curves or from knowledge of
the site itself--its soil type, topography, or its effect on other
species growing in the area.. If average stand dbh could be used
in place of stand height in obtaining site index values, radial
growth in many stands over a region could be modeled by further
partitioning stands of a cover type into groups found on excellent,
good, fair, and poor sites. Unfortunately, the average diameter
of trees in a stand over time strongly depends on density.. Although
height growth may be great in stands on good sites, radial growth
22
will be very slow if the stand is too heavily stocked.
A definite relation between height and diameter may exist
for the more intolerffiit species such as jack pine, aspen, and
paper birch as natural thinning lessens the effects of crowding
on tree diameter. Still, crowding in th~ stands of more tolerant
trees will arrest radial growth. Properly managed stands, however,
are periodically thinned to maximize growth. The stanqs are thinned
to a pres.cribed density or basal area at a specific age. This
unnatural thinning may also allow diameter to be roughly related
to height.
Modeling Management in the MINESITE Area.
Obviously forest management doesn't always strictly follow
prescribed guidelines. Such guidelines are also subject to change.
It's not.the purpose of this report to determine the extent that
such guidelines are followed or change, however. Instead, an
example of how the model might be applied to adequately stocked
stands in managed forests using present guidelines is' provided.
A model topology corresponding to the forest type management
policies used in the Superior National Forest is shown in Figure 9.
(Over 70 percent of the MINESITE Area is located in the Superior
National Forest and is intensively managed by the U.S. Forest
Service). This figure shows the flow of land between forestt:-'-_
types when various stands reach rotation age. A few of these
2J
VlHlTESPRUCE . <IQ9:J
all sites~
all sites
LOVlI~ND
HA RD'!l OODS
ASPEN
ASPEN
S . I :.c::60
CONIFERSWAIlP
all sites
ASPEN
ASPEN
S "I .~70
s I.~80
LOWLANDBLACKSPRUCES.I."=)O
LOWLANDBLACrSPRUCE
S .1 •~50
PAPERBIRCHS .. I 60
PAPERBIRCH
s.I.<'60
I A reo~el topolo[y describinr how de d forest types ( 2~)should be manaEed the Supe or Na st (S , 1975)The cjrcled numbers are the stand rotation ages in years",S.I.=site index (in feet)
JACKPINE
UPLAND s .. I ..~40BLACK RED
SPRUCE PINE
S .. I .< 50 JACKPINE
S.I.~50
UPLANDBLACI{SPRUCE "mITE
S.1.>50JACK PINE less)PIl\TE
all sitess.I.c.6p
BLACYSPRUCE
S.J .~23
Fjgure 9 (continued)
fana~ementOs forest def tions adequate s ekedstands. (Adequate s e sts when the basal area (B Ap)of a stand is ater than 40 'square et per acre)
A. Upland forest types
1. A stand is d for when aspen:a) maRes up more than percent of the crown density, orb) is d with paper birch, white spruce, or both
on sites that have a birch site index les~than 60 and a spruce B@A$ less than 50 £t-/acreat the aspen rotation ageo
2. A stand is managed fora) makes up more thanb) is mixed with aspen,
a birch site indexB.A. less than 50
when paper birch:of the crOVffi densi ty, or
spruce, or both on sites havingthan 60 and a white spruceat the birch rotation age.
3- A stand is managed for whi~e if white spruce:a) makes up more than 50 perc the cro~vn densi ty
. in a fir-spruce-aspen-birch m~xture, orb) has a B.A. greater than 50 ft-jacre at the rotation
age of the dominant tree species.
4. A stand is managed for balsam fir ifta) white spruce makes up less than 50 percent of the
crom1 density in a fir-spruce-aspen-birch mixture, orb) the total B.A., in fir stands, of paper birch, 2
pine, and/or white spruce does not exceed 40 :ft /acreor the site is poor for those long-lived species, or
c) :fir is needed for wildlife habitats.
5. A stand is managed for jack Rine if jack pine:a) makes up more than 80 percent of the crown density, orb) is mixed with red pine, white spruce, or aspen
on poor sites for those latter species or when notadequately stocked with those species~
6. A stand is managed for red pine ifsa) in mature stands, red pine is adequately stocked, orb) in stands just older than 35 years, red pine has a
B.A. greater than 25 and makes up at least 40 percent of thecro~~ density, or
c) in stands younger than 35 years and adequately stocked,there are at least 100 well distributed red pine per acre.
7. A stand is managed for white pine according to the sameprinciples that apply for red pine except when blisterrust threatens the stand.
B. Lowland forest types
1. A stand is mana[!ed for black snruce if it makes up greaterthan 50 percent of the-crovm density.
2. A stand is managed for lowland har9woods if red maple, ash, el~f
and/or basswood make up more than 50 percent of crovm density.
3. A stand is managed as a conifer swamn if neither black sprucenor lowland hardwoods rna}r,:e up 50 percent of the crovvn densi
forest types are handled as an exam})le.
1. Aspen.
Assuming first, that' all adequately stocked aspen stands
in the MlNESITE Area can be classified according to site index,
and second, that average radial growth in these stands can be
related to site index, Table 3a can be constructed using the
yield tables of Kittredge and Gevorkiantz (1929). Th~ model
expressi~ns corresponding to these roughly estimated parameters
are shown in Figure 10a. The restrictions imposed on the 8ub-
scripts require that, except for 50 percent of the aspen on
poor sites, all stands are regenerated as aspen. That area
not regenerated as asperi is arbitrarily converted to jack pine.
"
2. Paper Birch.
The same procedure used in the above treatment of aspen
is used for paper birch using the yield tables of Cooley (1962).
Birch stands on areas with a site index below 60 are 75 percent
regenerated as birch and 25 percent as jack pine. The model
expressions incorporating the parameters listed in Table Jb
are shown in Figure lOb.
J. Jack Pine.
Table 3c is constructed using the yield data of Eyre and
Lebarron (1956). All jack pine stands are regenerated as jack
pine and, for simplicity, birch and aspen are converted only
~r3.blc 3a
The approx sgraduate a size c sma.nagement guic3cl andGev (1)
on sare d accordyi Id tables of Ki
c ed 8iS.N F.
dge and
8i tE~index
'" b.,h" size class (in,Qb~
0 1 1 .5 5-9]c_ 9-15 15+
80 5 55 60 --
70 5 30 60 -- --60 5 35 50 - --50 10 35 -- -- ._-
Table Jb
The approximate ages that growing paper birch stands on specifiedsites graduate from a size class or are harvested according toSeN.F. management guidelines and the yield tables of Cooley (1962)
d.b.h. size class (inches)
si teindex
I 0 ... 1. 1-5 5-9 9-15~
70 5 30 60 80 --50 10 Lj-o 60 -- --
Table 3c
The approximate ages that growing jack pine stands on specifiedsites graduate from a size class or are harvested according toS .. N.F. management guidelines and the yield tables of Eyre andLebax'ron (1956).
deb .. h. size class (inches)
si-teindex
] 0-1 1-5 15-9 0-1 c::,' 15+.... -'
60 5 f 25 50 70 --50 5 30 60 -- --
I
J.}o J 5 I 35 50 -- I --
r lOa
5 arssome multiple
j 3,- elil,12j 1
j 1 for j=2,3,e.e,121=12 for j
A,?O. (t)1
The model equations for aspen usingand the parameters lis dof five years)"
ac~ '-' ofthe j aspen
1) age c s on A80.{S .. I.2:80 Jat tirne=t+5
A60. (t)l
j-1 for j=2 3,@"",101=10 j=1 i,j=1;2'Glo",,10
4) A501 Ct+5) =
A50 .. (t+5) =J '
( • .5)(A507(t) )
A50.(t) i=j-1 for'j=2,3, ••• ,71
i,j=1 ,1 2, .. e.,7
For any 5 year interval t:
aspen
aspen
aspen
... aspen
1-
seedling acreage= A801 (t)+A701 (t)+A601 (t)+ ~A50. (t)I:" l
5 b ~
sapling acreage=~A80.(t)+LA70. (t)+ A60. (t)+~A50. (t)"'1. 1 Z 1 1.... 1
II 12-
pole acreage=,zA80. (t)+,ZA70. (t)fQ 1 '1 1
small saw acreage= A8012 (t)
Figure lOb
The model equations for paper birch using a time interval of 5years and the parameters listed in table Jb.
1) PB70j(t+5) = PB70. (t) i=j-l for j=2,J, .... ,16i,j=1,2,tle.~16
1 i=16 for j=l
2) PESO. (t+S) ~FBSO. (t) i=j-l for j=2,3, ••• ,12::::onlyJ 1 i,j=1,2, .• ",12PB501(t+5) = ( .. 2S) (FB5012 ( t) )
For any 5 year interval t:
paper birch
paper birch
paper birch
paper birch
:l
seedling acreage= PB701(t)+~PB50.(t), ,1
sapling acreage=~PB70.(t)+~PB50.(t)2. 1 :3 l'
I;L 12
pole acreage=~PB70.(t):t ~PB50. (t)7 l «j l
small saw 'acreage=~PB70.(t)&::- t~.l.
Figure 10c
~Phe model equand the pararne
for j using a time interval 5 years
i=j 1 for j ,3'0&i=11+ for j=l
,14
= JPS012(t) + ( S)(A507(t» + ( 25)(PB5012(t))
- JP50.(t) j ,3,eeo,121
2) ,JP.501 (t+.5)
JP50j(t+5)
3)"JP40 j (t+S) = JP40. (t)1
j-l f'or j=2,3'11 e,10o j
For any 5 year interval ts
jack pine
jack pine
jack pine
jack pine
seedling acre JP601(t)+ JP501(t)+ JP401(t)6 -,
sapling acre = JP60.(t)+~JP50.(t)+~JP40. (t)1 ~ l' 2 1
12 10
pole acreage JP60. (t)+ <:"JP.50. (t)+L"JP40. (t)1 1 1 ~ 1
14small saw acreage= <:JP60. (t)
II 1
.. I"
to med ium site ,j model e ssions
are shown in Figure lOc
,This procedure could be repeated for some of the other
forest types using, for example t the data of Meyer (1929)
white spruce and fir. Fox and Kruse (1939) for black spruce,
and Eyre and Zehngraff (191}8) for red pine e The poin-tto be
made is this: the model's flexibility enables the use~ to easily
express when forests are harvested and the extent that a forest
..type is regenerated as the same or some other type.
Certainly not all stands are harvested at rotation age.
Many stands are lost to succeeding understories because no
markets exist for the overstory trees. Assuming that most stands
are harvested at rotation age and properly. regenerated, the model
at least provides a framework on which to base less optimistic
views of future management. Leuschner (1972) used a variety
of management strategies in his model to obtain a rough 'estimate
of future volumes of Lake States aspen. Likewise, in order that
a good estimate of management's effects on MlNESITE Area vegetation
is obtained, model simulations should be run incorporating different
management strategies.
Modeling Natural Succession in the MINESITE Area.
The portion of the region not subject to intensive management
~or one reason or another is subject to natural succession. In
the absence of management, the v ity of model creases
30
with the time of simulation ad as the model assumptions
become increasingly violated-~-stands naturally become uneven-aged
and mixed. Initially, stands are assumed to be even-aged and
relatively puree The two steps in applying the model to this
portion of the region are analogous to those outlined by Shugart
et ale (1973) described earlier.
1. Stand Growth is Modeled.
If-it's deemed necessary and if the MINESlTE Area vegetation
inventory permits it, variability of stand growth about an average
value can be accounted for by partitioning stands of forest types
among different site index classes and density classes. Such
subdividions should obviously be kept to a minimum, however.
The model is very large even before further complications are
added"
Finding values for the average longevity of forest stands
is a major problem. Longevity appears to depend on site quality
and may depend on density as well. These values might be assumed
to be a bit larger than the corresponding" rotation ages but more
study is needed in this area.
2. Transfer between Forest Types is Modeled.
The pondiagonal elements are calculated as before--using
equation 7 ..""~
The p .. 8 S should be obtained from field data .. ForlJ
example p inselman (1959) used observed dominance of 0
31
species in understories to reflect the de e to which those
species would replace aspen overstories in Minnesota. His results
are listed in Table 4 ..
Fire, Drought, and Epidemics ..
It is suggested that the quency and effect on vegetation
of fire, drought, and epidemics be ascertained from th.e past
history of the area. These effects can then be expressed in
the model by discontinuous mem1S. As in the case of modeling
different management strategies with different simulations,
the effect ffi1d timing of these factors can also be added dif-
.ferently in a number of simulations to arrive at an overall
average effeQt of each factor.
Table 1.1.
'J~he natural conversionthe dominance theplots in Minneso
Spruce
NorthernHardwoods
:Mixed
ConiferSwamp
Oak
cover s re cted byin the unders s of selec d(Heinselman, 195J+)
Acres ofOvers tory
.Aspen~Birch
Converted (xl0J )570
290
200
150
60
60
Inferior 1Hardwoods
Pine
TotalComplete 2Conversion
PartialConversion3
No 4Conversion
Total
50
30
1410
1020
4370
6800
1red maple, elm, ironwood, black and pin cherry.
2defined as >1050 ~>eedlings- acre'
(1 to 3" (3 to 5"2. 650 dbh trees), or >350 dbh trees)
acre acre
3defined as 600 to1049 "
400 to649 " , or 200 to
349 "
4defined as <600 " (400 " , or <200 " •
CONCLUSION
The model of Shugart, Crow, and Hett (1973) is not reccownended
for use by the Copper-Nickel Study team for projecting yegetative
trends in the MIIlliSITE Area. It is neither immediately evident
nor has it been demonstrated that this model accurately simu~a-tes
natural succession over a large region. Also, by focusing on
MINESITE's (1976) smaller and intensively managed region, the
study group can collect a mass of data allowing them to at least
qualitatively project, over a relatively short period of time,
what will becom~ of the many forest stands. The exponential
increase or decrease in the cover-state area implied by the
model of Shugart et ale (1973) would only prove to obscure these
.'
qualitative projections.
Instead, a Markov model is recommended. By reducing the
system of linear differential equations to one of linear dif-
ference equations, the important factors affecting growth and
replacement of forest stands can be more easily incorporated
into a simulation.
The present and future state of the area's vegetation is
greatly dependent on present and future forest management.
It is believed that different management policies and their
manifestations can be better simulated by the Markov model.
In ~act, the more linportant the role that management plays in
shaIJing the ve on the h will be the e to leh
the Marltov mod as ons are met, and thus the more
the model will in obtaining projectionso
One should remain critical of the abili ty of the S"A~"l'J-_
~ al~ (1973) model -to simulate natural succession", The same
holds even more so for the Markov model even when its i:nherent
assumptions are valid. As equations are added ~o account for
the fact that not all stands are pure or grow at similar rates,
the simulations may become more realistic.I
Still,the success
of this model, like the many other successional models which
embody the second Levins (1966) modeling strategy, depends
largely on whether urrrealistic effects inherent from the un-
realistic.assump~ionscancel each other.
Bledsoe, L .J' It and G Msimulation of sech~G. ( de)Ecology, Ac c
Van Dynesucees
.Sys·t;emr::;SS, New
A cp
modelInon in
.35
Botkin, DuBm, J.F Janak,logical consequencesJ. Ecole 60:849-872
a cliSe 1972@ Some eco
model of fores-t gl"ovrth
Cooley, J 1962. Sitein northern Wisconsin.Exp" S-ta. t Stag Pap. 105.
Eyre, F.R. and P.J. Zehnzraff.Minnesota.. USDA Cir., 778
191+8 II
JrOp •
yield of paper birchS erv m, LaJre S ta-tes For €I
Red pine management in
__~_ and R.K. Lebarron~
in the Lake States. USDA19360 Management of jack pine
ch. Bul. 863. 66pu
Fowells H.A. 1965. Silvics of forest trees of the United States.USDA Handb. 271€1 762pe
Grigal, D.F. and L.F'. Ohmann. 1973. Upland plant communitiesof the BWCA. Naturalist 24:17-20.
Fox, G.D. and G.W. Kruse. 1939. A yield table for well-stockedstands of black spruce in northeastern Mirmesotae J. For.
,>' 37: 565-567.
Hahn, J.Te and R.A. Leary. 1974. Comment on a paper by H.H.Shugart, T.R. Crow, and J.M. Bette For. Sci. 20:212.
Heinselman, M.L. 1954. The extent of nat~ral conversion toother species in the Lake States aspen-birch type.J. For. 52:737-738.
1973. Fire in the virgin forests of theBoundary Waters Canoe Area, Minnesota. Quatern. Rese3:]29-382.
Horn, H.S. 1975. Markovian properties of forest succession.p 196-211. In M.L. Cody and J.M. Diamond (ed.). Ecologyand Evolution of Communities. Belknap Press of HarvardUniv. Press. Cambridge, Mass. 545p.
Kittredge, J.3. and S.R. Gevorkiantze 1929. Forest possibilitiesof aspen lands in the Lake States. Minn. Agr. Exp .. Sm.Tech. Bul. 60. 84p.
Leak, W.B. 1970. Successional change in northern hardwoodspredicted by birth and death simulation. Ecology 51:794-801.
Leu~:tchner, W.A. 1972. Projecting the aspen-resource in theLake States. USDA For .. Serv .. Res. Pap. NC-81.. J2p.
Levins, R 1966. Thebiology. Am. Sci
of model build1
in population
36
Latka, ADJ. 1925. Elements of PhysicalWilkins. BaIt. Md@ 46op.
ology, Williams and
For" Serv",
Ohmann, IJ .17 • R R Ream 197 Wilderness ecology: amethod sampling and summarizing plant communityclassifica USDA Serv Rese e Nc-49 14pe
1971b. Wilderness ecology: virginBoundary Waters Canoe Area. USDA
NC-63e SSp.
May, R.M. 1973e Stability and Complexity in :Model Ec.osystems.Princeton Press Princeton, NDJ. 235p.
Meyer, W.R. 1929. Yields of second-growth spruce and fir inthe Northeast. USDA Tech. Bul. 142. 52pe
MINESITE. 1976. The MINESITE Data Manual.. Minn.: DNR Div ..Minerals.
Shugart, H.H., T.R. Crow, and J.M. Hett. 1973. Forest succession models: a rationale and methodology for modelingforest succession over large regionse For. Sci. 19:20)-212.
1974. Reply to J.T.Hahn and R.AD Leary on forest succession models. For. Sci.
,,20: 213"
Superior National Forest. 1975. Silvicultural Guides Summary.USDA For. Serve 45p.
Waggoner, P .. E.. and G.R. Stephens. 1970.. Transition probabilitiesfor a forest. Nature 255:1160-1161.
3?
Shugart et
Solving a System of LinearDifferential Equ ons
• didn;t obtain an explicit solution for
their model Numerical approximation techniques were us~d instead
An explicit solu on for a small subset of the set of differential
equations can easily found, however, by applying techniques
found in any introductory texbook of linear algebra or differential
equations. Such is done below so that growth and decay of white
pine stm1ds over a region can be visualized as predicted by the
model in the absence of input acreages from other cover types.
The ~hree equations that describe the growth of white pine
stands are
,seedling
pole
saw
dX1/dt=(-1/100)X1
dX2/dt=(1/100)X1+(-1/150 )X2
dXJ/dt=(1/150)x2+~-1/200)XJ
or
. [-1/100 . a ad/dt(X) = 1/100 -1/150 0
a 1/150 -1/200
x = AX
This system is co~plicatec because the second and third equations
contain variables other than that variable in the differential.
1£ the matrix A is similar to a diagonal watrix, D, ~ade
un of its e:genvalues, the syste~ can be reduced to eliminate
these other variables. All ~·s ttat are solutio~2 G~ 'det(~I _~)=~t~, ~
(~here I is the 3 x 3 identityn
ix) are eigenvalues of A
)B
This follows from the defini on of the eigenvector X as (I AX :::: AXe
where X is a nonzero solution of this equation only if the above
determinant is zero Since
A+ 1/100 0 0
Det -1/100 A+l/150 0 == 0
0 -1/150 A+1/200,
only when (A+l/100){A+l/150)(A+l/200)==O,A1==-1/100, ~2==-1/150,
and ~==~'1/200 are the eigenvalues and
[
-1/100 0 0 ]
D == 0 -1/150 . 0
o 0 -t/200 li
.+he trick now is to find a matrix of colu~ned eigenvectors,
d P, such that dt(X) Such eigenvectors are Iound
by substi tuting each eigenvalue, ~~ , into (Ai. In-A)X == 0 and solving
for X. When this is done for A above, one eigenvector is found
to be associated with each eigenvalue.
[1/4] ::A; [-1~4] , XA3{ 1JI
[1/4 0
1];:).,'" - 3~~ Jso :;: = -3~4 -1/4
1
The inverse of F, denoted 1'-1, as obtaineci using the so-called
matrix inversion alsorith~.
rlh (' 0 1 0 0 1 0 0 I l~ 0 0lineZlr I
I[J/Lf _1/i! C· 0 1 0 1"0\'1 0 1 0 1-16 12 0
transformations I
1 1 1 0 () 1 0 0 :1 : 4/3 ' -LJ, 1
[ Ji0 0]so 1 -1f 12 0::::
" ., _l~ 1.-:.,/_)
Tin2..11y, sett=_:;'I~......,T_ 1,':i th OJ'
dt P-l-;;
:::: ).:::: DY , the Iollowing
e:;,:pressions 2.re ot·tc:..':nec'1:
r:l" ~ /-: -!- - ( 1 11 f"\ 0 \ ... rL· J 1 \..~ lJ - - /- '~I : J 1 ';:.:.- tn solutio:!
\':i th solution
\':i th solution
,T ==C e (;-1/100) tJ 1 1
"I -c (-1/150)tY2-'2e
_0 (-1/200) tYJ-''"'3e . •
-1-'\t =T _,r - -or X ==?Y, an explicit solution for the subset
}:2 == ( - J /L:_ )y 1- ( l/h )y 2
or
or
or
Xl:::: ( 1/L~ )C1 8 ( -1/100 ) t
]{ :::: ( _3/1.~ )C e ( -1/100) t2 1
... ( ~ /L!_) C2 8 (-1/150) t,
"'r ::::(' e(-1/100)t + C e(-1/1:C')tJ~3 '"'1 2
..L r~ ,. (-J/200)t• v~·_
-'
,..l~r::-('< ,O!~.J..'- .J..lr.\"" __ 10.- '_' •• \,.., ,",-, __ ,-,..
( 1 /1 ,.... r\ ) .J.."'or _ ( 1 r'. r ":::c. ,- " - - - l-..... 1-\--~-1-
:' A 11 ' ,. , ...:-~",.....=/ _-:,--r '\.:- \- __ '. \... -;- " -- .- ,.., ..
I '-
.., 11" ,-I v
.... _"." " ~ ( -1,1 ~ Cr ) ','.. "": - •• ' ... I
, 1/1[" '-:... I .. /""r'..;.• r: r ' 'I ~ " - '-.' - _, . / L ... .',.., r " _ .. - - / t: 1-,-- ,- \ .;~,
•