O~~Y2-X24O/X~S3.o(I
f II.00
Pergamon Press Ltd.
Society for Mathematical Biology
AN ISOCLINAL APPROACH TO THE
COMPARATIVE STATICS OF BIOLOGICAL
COMMUNITIES*
m GRANT G. THOMPSON,~ W. MICHAEL BOOTY,
WILLIAM J. LISS and CHARLES E. WARREN
Oak Creek Laboratory of Biology,
Department of Fisheries and Wildlife,
Oregon State University, Corvallis, OR 97331, U.S.A.
An understanding of the comparative statics of biological communities is important both
as a means of explaining the long-term effects of changes in environmental conditions,
and as a framework for viewing community time trajectories. A general formulation of
community dynamics is presented here which, given full information about a particular
community’s dynamic behavior, describes the impact of a change in environmental conditions on the community steady state. However, since such full information is often
lacking in studies of biological communities, various approaches to partial information
analysis of comparative statics are presented and compared, including a generalized
protocol for isocline analysis. The suggested isocline protocol is shown to be a useful tool
for both full and partial information analyses, as welJ as for both general and partial
equilibrium studies.
Introduction.
In the course of their scientific maturation, a number of
disciplines have found it useful to formulate theories of comparative statics,
describing the equilibrium response of a relevant class of systems to exogenous shifts in environmental conditions. Ecology, too, has a long tradition
in this area of inquiry. Writers as long ago as Mobius (1877) were concerned
with the long-term impact of a change in environmental conditions. The
problem was approached more rigorously by Lotka (1925) in his classic
work.
Recently, Levins (1974, 1975), Riebesell (1974), Levine (1976), Holt
(1977), Lawlor ( 1979), May et al. (1979), Travis and Post (1979), Hallam
(1980), Vandermeer (1980), Schaffer (1981), Boucher er al. (1982), and
MacNally (1983) have contributed to the ecological theory of comparative
statics. A large part of this theory has either been responsible for, or been
developed in concert with, the emerging interest in the role of ‘indirect
*This work was supported over several years through funding from the International Biological
Program, the Oregon State University Sea Grant College Program. the U.S. Environmental Protection
Agency, the U.S. Forest Service. the Electric Power Research Institute, and the Northwest and Alaska
Fisheries Center.
t Present address: Northwest and Alaska Fisheries Center, Resource Ecology and Fisheries Management Division, 7600 Sand Point Way NE, Seattle. W.4 98115-0070, U.S.A.
60
G.G. THOMPSONet al.
effects’ in structuring communities. This interest has been heightened by a
realization that the equilibrium response of one species to a change in
density of another species may turn out to be opposite in sign to the instantaneous, pair-wise response.
As has been recognized by theoreticians in a variety of fields [e.g. Samuelson (1947) in economics] , an understanding of comparative statics can also
find importance as a framework for explaining a system’s time trajectory.
Ecologists, too, have profited by viewing community trajectories in the
context of comparative statics (Liss and Warren, 1980). For example, as a
biological community is observed through time, its course through phase
space may appear incomprehensible when interpreted as the path of a system
in pursuit of a single steady state. However, if that same community is
viewed as pursuing (but perhaps never reaching) a steady state which periodically changes as the result of a changing environment, its trajectory may
become more meaningful.
Thus, there is ample reason to suspect that further development of the
theory of ecological comparative statics will prove rewarding. Historically,
the starting point for this theory has been the use of a set of time-differential
equations to model community dynamics. In general form, each equation in
the model appears as follows (Lotka, 1925; Levins, 1974,1975):
dx,ldt =&(x1+2,.
. .
,x,;Y~,Y~,
. -.
,YA
(1)
where x1, x2, . . . , x, comprise a set ‘X’ of n state variables (usually thought
of as the populations constituting a particular community), andy,, y,, . . . ,y,,,
comprise a set ‘Y’ of m entities (distinct from other populations in the
community) that impact the state variables.
The yi may be termed environmentd variables to distinguish them from
the state variables. The difference between state and environmental variables,
from a mathematical point of view, is that changes in state variables take
place within the model, whereas any changes in environmental variables
take place exogenously.
Naturally, the equilibrium solution(s) of equations (1) must play a key role
in any mathematically-formulated
theory of comparative statics. As a set
of n equations in n + m unknowns, equations (1) may be solved (perhaps
only implicitly) to yield the equilibrium value(s) of any state variable as a
function of the set of m environmental variables, as shown below:
xi* = &(Y,, * * * ,v?rA
where the asterisk denotes
equilibrium
value, and gt is a function
(2)
specific
tOXi*
Another important construct in the ecological theory of comparative
statics is the community matrix (Levins, 1968, 1974, 1975). This matrix
AN ISOCLINAL APPROACH TO COMPARATIVE STATICS
61
(henceforth referred to as the ‘A’ matrix) is the pt X n Jacobian matrix of
partial derivatives of equations (1) taken at equilibrium with respect to each
of the state variables. In other words, each element in the A matrix is
defined as follows:
Qfj =
ah*/ a+
(3)
Analogously, it is possible to defme a ‘B’ matrix as the n X m matrix of
partial derivatives of equations (1) taken at equilibrium with respect to each
of the environmental variables:
ayi.
b, = aff*l
(4)
Now, if A is non-singular, let the matrix ‘C’ be an y1X m matrix defined as
follows:
c=
Importantly,
1947):
-A-‘B.
the C matrix also has the following interpretation
Cij = agi/ayi.
(5)
(Samuelson,
(6)
The C matrix thus gives a general equilibrium description of a system’s
comparative
statics, since it describes the (instantaneous)
equilibrium
response to changes in environmental conditions. Unfortunately, calculation
of the Cmatrix by equation (5) is a highly information-intensive
undertaking,
in that it requires complete knowledge of the A and B matrices. Such an
analysis shall be referred to here as a ‘full information’ analysis.
Alternatively, it is often necessary to perform a comparative static analysis
of a system in which at least one element of A or B is unknown (a ‘partial
information’
analysis). The following material represents an attempt to
formulate a method which is applicable in both situations, i.e. a method
which: (1) represents a useful means of performing a partial information
analysis of comparative statics, and (2) can also be used to perform a full
information analysis.
A Linear Example.
Prior to formulating such a method, it shall prove helpful to develop a hypothetical example. To this end, suppose the following:
(1) An ecosystem consisting of three species (x1, x2 and xg) and a pair of
environmental variables ( y1 and y2).
(2) A data set describing the state of the system and its environment at
quarterly intervals over a 3-yr period (Table I).
(3) An understanding that the system can be usefully modeled in linear
form, i.e. in the form dX/dt = AX + BY.
62
G. G. THOMPSON et al.
TABLE I
Time-series Data for Hypothetical
Time
Xl
Ecosystem
x3
x2
Yl
Y2
24.0000
24.0000
24.0000
24.0000
6.0000
6.0000
6.0000
6.0000
24.0000
24.0000
24.0000
24.0000
24.0000
12.0000
12.0000
12.0000
12.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
(Yr)
0.0000
0.2500
0.5000
0.7500
1.oooo
1.2500
1.5000
1.7500
2.0000
2.2500
2.5000
2.7500
3.0000
10.0000
5.4384
4.1153
3.9474
4.1279
3.7748
3.0242
2.3618
1.8794
1.9304
2.3733
2.8374
3.2116
10.0000
10.1600
11.7007
13.2698
14.5039
11.2486
8.7376
7.0076
5.8776
9.0770
11.4296
13.0520
14.1287
10.0000
3.5420
1.5213
1.0967
1.1829
2.6914
2.6256
2.2152
1.8255
1.9105
2.3660
2.8347
3.2 106
Levels of state variables (xi, x2 and xs) and environmental
listed at quarter-yr intervals over a period of 3 yr.
variables ( yI and yJ are
Given this information, the following equation
dynamics on an annual time scale (Appendix):
describes the system’s
dX/dt =
(7)
The solution to equation (7) may be written in the form
X(t) = &‘[X(O) - CY] + CY.
Pre-multiplying
(8)
B by the negative inverse of A gives
-A-‘B
= C=
213
l/12
l/6
l/12
[ l/6
-l/6
1
.
(9)
Figures 1A and B show phase plane representations of the data contained
in Table I. Clearly, it is difficult to derive much useful information from
simple inspection of the system’s time trajectory. The lack of any apparent
pattern in these figures illustrates the need for the methodology
to be
developed; without an appropriate tool for understanding
comparative
statics, information about a system’s trajectory may be largely meaningless.
AN ISOCLINAL
APPROACH TO COMPARATIVE
STATICS
63
I
01
1
2
3
4
5
6
xl
Figure 1. Phase portraits of community trajectory. Data from Table I are
reproduced here as trajectories in phase space. Arrows are placed at yearly
intervals.
Partial Information Analysis: Two Methods. For the case in which only
a certain portion of the system’s dynamic behavior is given, defme the target
subsystem to be the set of state variables for which expressions in the form
of equation (1) are given, and the non-target subsystem to be the set of state
variables for which expressions in the form of equation (1) are not given.
The underlying dynamics of a community so partitioned appear in the linear
case as shown below:
(10)
X, represents the target subsystem, Xb represents the non-target
subsystem, and the A and B matrices have been partitioned accordingly
(note that the contents of the Aab, Abb and BYb submatrices are unknown
by definition).
In the case of the system described by equation (7), this partitioning
might appear as follows:
where
64
G. C. THOMPSONet al.
-1
I-1
-3
;
i
-1
x2
--
1 I-5
1
x1
--
I[
x3
1
1
+O
,--I
0
0
-
I0
Y.
(11)
-1
One extreme means of partial information analysis could be described as
the ‘independent’ approach, in which the X, subsystem is assumed to behave
independently of the X, subsystem or, equivalently, the X, subvector is set
equal to zero. This approach is implicitly adopted in many studies of community ecology, as well as in most single-species research. When X, in equation (10) is set equal to zero, the dynamics of the X, subsystem can be
described as shown below:
dX,Jdt = A,,X,
+ Bya Y.
(12)
In the case of the system described by equation (1 l), the X, subsystem’s
dynamics are given by the following:
(13)
The solution of equation (13) is shown below in the form of equation (8):
(1 +
x0(t)
t)ee2'
=
[
tem2j
-te-2'
(1 -t)e-2j
[x,(0)-[::::
$1
+[;:::
:]y*
(14)
Another method of partial information analysis, known as ‘abstraction
theory’, has been presented by Schaffer (198 1). In the abstracted approach,
the X, subsystem’s dynamics are modeled as though the Xb subsystem
were always in equilibrium. Obviously, this method requires knowledge
of the equilibrium value of X, for a given value of Y. Schaffer asserts that
it may be possible in some cases to obtain empirical estimates of X,X.However, for purposes of comparison with other methods, the following equations for the ‘abstracted’ A,, and B,, submatrices (denoted Ai, and B&)
are derived by calculating X,* from the full-information system of equation (10):
A”oa = Aaa -Aba(&)-lAab
3
05)
B;a = Bya -AAa(Abb)-lByb
*
(16)
Given these abstracted submatrices,
X, subsystem may be written as
the abstracted
dynamics
for the
AN ISOCLINAL
dXJdt
=
APPROACH TO COMPARATIVE
STATICS
AO,,X, + B$ Y.
In terms of the system described by equation
and B,, submatrices can be calculated as follows:
65
(17)
(1 l), the abstracted
A,
(19)
The solution
equation (8):
to the abstracted
dynamics is shown below in the form of
where
tit
=
[
se-2t
_
ze-2.4t
_
se-2t
+
3e-2.4t
ze-2t
_
ze-2.4’
_
le-2’
+
3e-24t
1.
(21)
The actual solution of the X, subsystem’s dynamics [equation (8)]
is distinct from the solutions generated by either the independent approach
[equation ( 14)] or the abstracted approach [equation (20)] . The three
solutions are compared in Fig. 2 for the case in which X(0) = (10, 10, 10)
and Y = (24, 12). As emphasized by Schaffer (1981), the important distinction is that while neither the independent nor the abstracted trajectory
matches the actual trajectory, the abstracted trajectory does converge on the
actual steady state, while the independent trajectory in general does not.
10
11
12
13
14
15
16
17
I.3
XI
Figure 2. Comparison
of actual and calculated trajectories of target subsystem. Actual, abstracted and independent trajectories of the target subsystem
are shown for X(0) = (IO, 10, 10) and Y = (24, 12). The large solid circle
marks the subsystem’s steady state under these environmental
conditions.
66
G.G.THOMPSONetul.
Another
Method: ‘Generalized Isocline AnaI.vsis. Isoclines constitute
another common tool in ecological treatments of comparative statics,
dating from the early works of Lotka (1925) and Volterra (1931). Most
often, when isocline analysis is used in ecology, only two-species systems
[i.e. X‘= (xi, xp)] are considered. The analysis usually proceeds as follows:
(1) the time derivative of xi is considered in isolation from that of xi, set
equal to zero, and solved for XT; (2) the process is repeated, switching
subscripts for xt and xi; (3) the resulting equations are plotted in the phase
plane (the coordinate axes of which correspond to x1 and x2) as ‘isoclines.’
Generalizing this concept, let the non-reduced isocEine &niZy of state
variable xi be defined as the equation resulting from setting the time derivative of state variable xi equal to zero. The non-reduced isocline family can
be thought of as defining a function (perhaps implicit) describing equilibrium values of xi in terms of the other variables in its time derivative.
In the two-species case, the concept of the non-reduced isocline family
may be sufficient for most applications. However, when the number of
species is allowed to increase, other classes of isocline families may become
important. The following definition provides a general protocol for dealing
with such isocline families:
Given: (1) a system X consisting of n state variables; (2) an environment
Y; (3) a non-target subsystem Xb consisting of k - 1 state variables
(1 < k < n); and (4) a state variable xi within a target subsystem X, (where
X, is the complement of X,); define the k-dimensional isocline family of
xi to be the reduction of xi’s non-reduced isocline family to an equation in
only X, and Y, where such reduction involves substitutions of the nonreduced isocline families of only those state variables in X,.
Two points are worthy of immediate note: first, for non-linear systems,
any reduction of xi’s non-reduced isocline family may only be implicit.
Second, at the extreme values of k (n and l), the kdimensional
isocline
family becomes identical to the non-reduced isocline family (k = n), or to
the general equilibrium solution described in equation (2) (k = 1).
To derive an example of a k-dimensional isocline family, consider again
the partitioned system described by equation (1 l), where n = 3 and k = 2.
From the above definition, xi’s twodimensional
isocline family is derived
by reducing xi’s non-reduced isocline family to an equation in only xg
and Y, where this reduction may be obtained, by substitutions involving the
non-reduced isocline families of x1 and x2, but not x3.
So, derivation of xi’s two-dimensional isocline family begins with xi’s
non-reduced isocline family, which may be extracted from equation (7)
as follows:
dx,/dt
= -x1 -x2
-x3
+ yi = 0.
(22)
ANISOCLINALAPPROACHTOCOMPARATIVESTATICS
x7=
7x2
-x3
67
(23)
+Yl*
By definition, reduction of equation (23) is limited in this case to a
single substitution,
namely the one involving x2’s non-reduced isocline
family, which is derived as follows:
dx,/dt
= x1 -3x2
-x3
= 0.
x; = (1/3)x, - (1/3)x,.
(24)
(25)
Substituting
equation (25) for x2 in equation (23) gives x1’s twodimensional isocline family for the partitioned system of equation (11):
XT = -0.5x3
+ 0.75JQ.
(26)
Derivation of x2’s twodimensional
isocline family is accomplished by
substituting x1’s non-reduced isocline family [equation (23)] into x2’s nonreduced isocline family [equation (25)] , resulting in the following:
x2 * =
-0.5x3
+ 0.25~~.
(27)
Sample two-dimensional isoclines for different values of y, are shown for
x1 and x2 in Fig. 3.
Basically, calculation of xi’s k-dimensional isocline family involves viewing
the system as though a specified k - 1 state variables behaved as environmental variables. Thus, the isoclinal approach views the dynamics of the
X, target subsystem in equation (11) as shown below:
Relationship of Isocline Analysis to Other Approaches: Partial Information
Analysis.
As illustrated by equation (28), the suggested isocline protocol
in effect creates an expanded
and the original Y vector:
Y vector (Ye), consisting of the X, subvector
ye =
I I*
xb
Y
The expanded
matrix (BQ:
(‘9)
Y vector in turn affe;ts X= through an expanded B,, subB&I = (A,,, B,,).
(30)
From the isoclinal point of view, the dynamics of the X, subsystem thus
appear as shown below:
68
G. G. THOMPSON et 02.
dX,,/dt = A,,X,, + B;, Ye.
(31)
Setting the LHSs of equations (12), (17) and (31) equal to zero and
solving for X,* results in three approaches to the steady-state performances
of the X, subsystem:
Independent:
X,* = -(A&‘B,,
Y.
(32)
Abstracted:
X,* = -(A&)-’
Bfa Y.
(33)
Isoclinal:
X,* = -(A&-l
B;, Ye.
(34)
While each of these approaches enables an examination of the equilibrium
changes in X, resulting from changes in Y, they differ in terms of the role
assigned to X,. In the independent approach, for example, the X, subcommunity is assumed to be unimportant in the Y-X, interaction. Of course,
in cases where this assumption is violated, misleading results will be obtained
if the independent approach is used.
Unlike the independent approach, abstraction theory gives correct equilibrium results for systems in which X, subsystems are important, because
A
20 Y, -24
15 Y, = 15
Xl
10 Y, * 12
5;
-
Y, -5
-
B
r
10
Figure 3. Two-dimensional
isoclines for target subsystem. Each phase plane
shows how the steady state of one state variable in the target subsystem varies
with changes in the non-target subsystem (xs) and the environment.
Because
of the particular way in which this system is organized, y2 drops out as a
determining factor on both phase planes.
AN ISOCLINAL
APPROACH TO COMPARATIVE
STATICS
69
incorporates the effects of X,* on X,. The abstracted approach to community statics may be compared with the isoclinal approach by examining
Figs 3 and 4. Figures 3A and B show some two-dimensional isoclines for
x 1 in (x 1, x&space and for x2 in (x2, x&space, respectively. Figure 4
illustrates the abstracted view of X,* obtained by substituting values of Ai,
and B;, gr‘ven in equations (18) and (19) into equation (33). The major
difference is that in Fig, 3, steady state is determined by y1 and x3 with y2
implicit, while in Fig. 4, steady state is determined by y1 and y2 with x3
implicit.
In contrast to either the independent or abstracted approaches, then, the
isoclinal approach leads to an analysis of X=‘s comparative statics in which
no prior constraints are placed on the possible values of Xb. The independent arrd abstracted approaches, in which Xb is held equal to specific
constants (X, = 0 and Xb = X$, respectively), can thus be seen to be special
cases of the isoclinal approach, in which X, is held equal to an arbitrary
constant (X, = X,).
it
A
y2
F&ue 4.
abstracted
varjlable in
ment. The
Y = (24,
Comparative statics of the target subsystem as determined by the
approach. Each graph shows how the steady state of one state
the target subsystem varies with changes in the system environsolid circle on each graph marks the steady state obtained when
12), and corresponds to the point marked by the same symbol
in Fig. 2.
IO
G. G. THOMPSON et al.
This characteristic of isocline analysis may often be useful, since many of
the questions commonly posed in ecological studies of comparative statics
focus on the long-term effects of species upon other species, not just of
abiotic environmental
factors upon species. Furthermore,
the isoclinal
approach is helpful in situations where estimates of Xg are unobtainable.
In such a case, it might be prudent to consider a variety of possible values
for Xg. This, in fact, is exactly what isocline analysis does: it answers the
question, “Given a value for Y, what would X,* look like if Xg were suchand-such?”
Relationship of Isocline Analysis to Other Approaches: Full Information
Analysis. Unlike the abstracted approach, the isoclinal approach to partial
information analysis does not necessarily yield a general equilibrium picture.
However, when used as a method of conducting a full information analysis
of comparative statics, the isoclinal approach is well-suited to address questions of general equilibria, and so compares favorably with abstraction
theory. Two approaches may be employed. The first is to conduct a onedimensional isocline analysis; i.e. X, can be defined so as to include the
entire system (X, is then an empty set). Of course, when X, is defined in this
manner, the three approaches (independent, abstracted and isoclinal) are all
equivalent to the basic full information solution given by equation (2).
The second method is more interesting. In examining the general equilibrium behavior of a given k-dimensional subsystem, this approach amounts
to conducting a series of k separate isocline analyses, each based on treating
a different k - 1 dimensions of the given subsystem as the non-target subsystem. When the results of these k analyses are considered jointly, they
yield a true general equilibrium description. As mentioned earlier, this is the
manner in which isocline analysis has traditionally been applied to twospecies systems: first xj’s two-dimensional isocline family is derived based
upon xi being treated as the non-target subsystem, then the subscripts are
switched and the process repeated; when plotted in the phase plane, the isoclines’ intersections represent the general equilibrium solution.
A general equilibrium analysis may be conducted along these lines for the
(Xl, x3) and (x2, x,) subsystems of equation (7) by superimposing x3’s twodimensional isoclines on the isoclines shown in Fig. 3. This involves the
following steps: first, x1 is treated as the non-target subsystem and x3’s twodimensional isocline family in (x1, x,)-space is calculated and superimposed
on the isoclines shown in Fig. 3A. Then x2 is treated as the non-target subsystem and x3’s twodimensional
isocline family in (x2, x3-space is calculated
and superimposed on the isoclines shown in Fig: 3B. Calculation of x3’s
twodimensidnal
isocline families follows steps analogous to those used in
AN ISOCLINAL
APPROACH TO COMPARATIVE
STATICS
deriving equations (26) and (27), and results in the following equations
the (x1, x3) and (x2, x3) phase planes, respectively:
71
for
XT = (1/4)x, - (3/16)v,,
(35)
x3 = (1/6)u, - U/@Y,.
(36)
Superimposing isoclines derived from equations (35) and (36) on the
isoclines shown in Fig. 3 results in the general equilibrium representation of
Fig. 5. Here, x1’s isoclines depict a view of the system in which .Y1 (Fig. 5A)
and x2 (Fig. 5B) behave as environmental variables. Thus, considering both
sets of isoclines’on either phase plane (i.e. either two-dimensional subsystem)
gives a general equilibrium description of that subsystem’s comparative
statics. When both phase planes are considered together, a general equilibrium description of the entire system’s comparative statics results. The solid
circle on each phase plane in Fig. 5 depicts the system steady state identified
by Y = (24,12), and corresponds to the points marked with the same symbol
in Figs 2 and 4.
A
x.
Figure 5. General equilibrium
representation
of comparative
statics as
determined
by the isoclinal approach. Each phase plane gives a general
equilibrium
description
of the comparative
statics of its respective twodimensional subsystem. Descending lines represent two-dimensional
isoclines
of the state variable indexing the vertical axis, and correspond to the isoclines depicted in Fig. 3: Ascending lines represent xs’s two-dimensional
isoolines. The solid circle on each phase plane marks the steady state obtaining when Y = (24, 12), and corresponds to the points marked by the same
symbol in Figs 2 and 4.
72
G.G.THOMP!SONerul.
Such an analysis can be extremely helpful in attempting to ascribe meaning
to a system trajectory of the type described by Table I and Fig. 1. For
example, Fig. 6 shows the trajectories of Fig. 1 superimposed on the relevant
set of isoclines from Fig. 5. Simple inspection of Fig. 6 reveals significant
information regarding the meaning of the system trajectory: as the system
begins its course through phase space, it is moving toward the steady state
marked by the solid circle [Y = (24, 12)]. However, after 1 yr has elapsed,
the environment changes [Y = (6, O)], and the system embarks on a path
toward the steady state marked by the solid square. Once again, though, the
environment changes at the end of the year [ Y = (24, O)] , and the system
veers off toward a third steady state, marked in Fig. 6 by the solid triangle.
This type of explanation cannot be derived from simple inspection of the
trajectory alone (Fig. 1).
Y,
1
V, -24
V,' 0
v, -24
v,=12
-6
v1 - 0
2
3
4
5
6
Figure 6. System trajectory viewed in the context of system comparative
statics. Appropriate isoclines from Fig. 5 are shown superimposed onto the
trajectories of Fig. 1. The solid circle on each phase plane marks the steady
state obtaining when Y = (24, 12), and corresponds to the points marked
by the same symbol in Figs 2,4 and 5. The solid square on each phase plane
marks the steady state obtained when Y = (6, 0). The solid triangle on each
phase plane marks the steady state obtaining when Y = (24,0).
73
AN ISOCLINAL APPROACH TO COMPARATIVE STATICS
Conclusion. Because it facilitates either full or partial information approaches, isocline analysis provides a flexible tool for the study of ecological
comparative statics. It also possesses the distinct advantage of being able to
address either general or partial equilibrium issues. As a technique which
incorporates other methods (e.g. the independent and abstracted approaches)
as special cases, isocline analysis can be adapted to the particular needs of
a wide variety of situations. Importantly, in situations where neither the
dynamic behavior nor the steady states of a non-target subsystem are obtainable, isocline analysis presents a straightforward,
heuristic, and highly
visualizable means of examining the effects of various possible values of the
non-target subsystem’s steady states.
The authors would like to thank the following individuals for reviewing
drafts of this article in various stages of completion: Drs Charles S. Ballantine
and Robert D. Stalley of the Oregon State University Mathematics Department, Dr Paul Cull of the Oregon State University Computer Science Department, Dr Bruce G. Marcot of the U.S. Forest Service, Dr Mostafa A. Shirazi
of the U.S. Environmental Protection Agency, and two anonymous reviewers.
The authors would also like to thank Mr Douglas S. Lee of the Michigan
State University Zoology Department for directing us to Schaffer’s paper.
APPENDIX
By virtue of the linearity assumption stated in the text, the hypothetical
dynamics can be modeled by a difference equation of the form
x(r + 1) = d&t)
+ &y(f),
community’s
(Al)
where t represents time measured in quarter-years, Ad represents the discrete counterpart to the A matrix, and Bd represents the discrete counterpart to the B matrix. Any of
a number of statistical techniques for simultaneous equation estimation may be used to
estimate equation (Al) from the data (Table I). For this example, the “Full Information
Maximum Likelihood” routine from the Time Series Processor (TSP) library was used to
obtain the following estimated system:
0.7407
X(t + 1) =
-0.1638
-0.1045
0.1342
0.4427
-;.;;;I
[ 0.1342
0.0748
.
X(r) + [Et
ZZii
iEJY0).
(AZ)
The eigenvalues of Ad are 0.6065,0.4724
and 0.3679. Taking the natural logarithm of
each eigenvalue and multiplying
by a factor of four gives the eigenvslues of the (continuous) A matrix re-calibrated to an annual time scale: -2, -3 and -4.
In order to complete the conversion of equation (AZ) to a continuous form, let the
matrix L be defined as the diagonal matrix of A’s eigenvalues. Next, let the matrix V be
74
G.G.THOMPSONetizZ,
defined as the matrix whose columns
operation can be performed:
consist of Ad’s eigenvectors.
Given the A matrix derived in equation (A3), the continuous
matrix in equation (A2) can be calculated as follows:
B =A(Ad
[ -1 1
-3 1
-1
I[
--I)-‘Bd
-5
-1
Then, the following
counterpart
to the Bd
=
-0.6464
-3.1877
-1.5623
(A4)
Thus, the continuous
follows:
form of this system’s dynamics
(annual
time scale) appears as
LITERATURE
Boucher, D. H., S. James and K. H. Keeler. 1982. “The Ecology of Mutualism.” A. Rev.
Ecol. System. 13, 3 1S-347.
HalIam, T. G. 1980. “Effects of Cooperation
on Competitive Systems.” J. theor. Biol.
82,415-423.
Holt, R. D. 1977. “Predation,
Apparent Competition,
and the Structure of Prey Communities.” Theor. Populat. Biol. 12, 197-229.
Lawlor, L. R. 1979. “Direct and Indirect Effects of n-Species Competition.”
Oecologia
43,355-364.
Levine, S. H. 1976. “Competitive
Interactions
in Ecosystems.”
Am. Nat. 110, 903910.
Levins, R. 1968. Evolution in Changing Environments:
Some Theoretical Explorations.
Princeton: Princeton University Press.
-.
1974. ‘The Qualitative Analysis of Partially Specified Systems.” Ann. N. Y. Acad.
Sci. 231, 123-138.
-.
1975. “Evolution in Communities Near Equilibrium.”
In Ecology and Evolution of
Communities (Eds M. L. Cody and J. M. Diamond). Cambridge: Harvard University
Press.
Liss, W. J. and C. E. Warren. 1980. “Ecology of Aquatic Systems.” In Fisheries Management (Eds R. T. Lackey and L. A. Nielsen). New York: John Wiley.
Lotka, A. J. 1925. Elements of Physical Biology. New York: Dover.
MacNally, R. C. 1983. “On Assessing the Significance of Interspecific
Competition to
Guild Structure.” Ecology 64, 1646-l 652.
AN ISOCLINAL APPROACH TO COMPARATIVE STATICS
75
May, R. M., J. R. Beddington, C. W. Clark, S. J. Holt and R. M. Laws. 1979. “Management of Multispecies Fisheries.” Science 205,267-277.
Mobius, K. 1877. Die Ausfer und die Austernwirtschaft.
Berlin: Wiegundt, Hempel &
Parey.
Riebesell, J. F. 1974. “Paradox of Enrichment
in Competitive
Systems.” Ecology 55,
183-187.
Samuelson, P. A. 1947. Foundations of Economic Analysis. Cambridge: Harvard University Press.
Schaffer, W. M. 1981. “Ecological Abstraction:
The Consequences of Reduced Dimensionality in Ecological Models.” Ecol. Monogr. 5 1, 383401.
Travis, C. C. and W. M. Post, 111. 1979. “Dynamics and Comparative Statics of Mutualistic Communities.“J.
theor. Biol. 78, 553-571.
Vandermeer, J. H. 1980. “Indirect Mutualism: Variations on a Theme by Stephen Levine.”
Am. Nut. 116,441448.
Volterra, V. 1931. “Variations and Fluctuations
of the Number of Individuals in Animal
Species Living Together.”
In Animal Ecology (Ed. R. N. Chapman). New York:
McGraw-Hill. Reprinted from J. Cons. perm. int. ExpZor. Mer, III, 1, 1928.
RECEIVED S-3-84
REVISED 7-31-85