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Stability patterns and trophic structure in a Delaware Bay plankton community
BioSystems, 26 (1991) 7 5 - 8 8
75
Elsevier Scientific Publishers Ireland Ltd.
Stability patterns and trophic structure in a Delaware Bay plankton community Ron Pilette, Ron Sigal a and John Blamire b Science Department, John Jay College of CUNY, 445 West 59th Street, New York, N Y 10019, aDepartment of Mathematics, Yale University, Box 2155 Yale Station, New Haven, CT 06520 and bDepartment of Biology, Brooklyn College of CUNY, Brooklyn, N Y 11210 (USA) (Received August 2nd, 1.991)
Using qualitative loop analysis we have extended our examination of a Delaware Bay plankton community to include an investigation of the roles :played by the various entities (population, guild or nutrient) in the community. In an entity removal exercise, we used stability relationships as a probe into community structure. Six types of stability change are possible as a result of entity removal from the system: stable to stable (s - s); stable to unstable (s - u); stable to disconnected (s - d); unstable to stable (u - s}; unstable to unstable (u - u); unstable to disconnected (u - d). Using these changes as an investigative tool, we found that in order to account for the stability-instability patterns, it was necessary to construct a refined trophic structure model. The observed connections between the entities in the larger model could be grouped into two different types of stability substructures: a simple pattern and a more complex branching pattern. These patterns map easily onto the refined trophic structure model. Using stability analysis it is also possible to model community structure in ways other than the traditional trophic approach. Patterns of system necessity and relative contribution to stability are observed. These patterns match the refined trophic structure model derived previously. The roles that the various entities play in the overall community were followed over an annual eycle. Entities were seen to change their roles as a function of time and status within a subgroup. These results show that stability determinations have the potential to be used as a valuable tool in community analysis.
Keywords: Community modeling; Population roles; Loop analysis; Stability patterns; Trophic structure; Delaware Bay plankton community.
1. Introduction
We have previously shown (Pilette et al., 1990) that the quali~ative loop analysis models of a Delaware Bay plankton community, prepared by Lane (1986), can be evaluated for relationships between stability and complexity. No statistically significant relationship was found for stability and complexity between systems (monthly community samples), but, within a system, a significant inverse relationship was found at the entity (population, guild or nutrient) level, and this was embedded in an increasing stability positively related to increasing subsystem size. Correspondence to: R. Pilette.
It was also noted that the stability role played by several of the entities within the system varied with respect to subsystem size and over the course of the annual cycle. This has prompted us to carry out a more detailed analysis of this aspect of our data. In the research reported here, we were concerned with the roles played by the entities of the Delaware Bay plankton community. Are there particular stability patterns for these entities and are these related to trophic structures? Are there patterns to the temporal sequences? Which entities share similar patterns? This work is based on 12 estuarine plankton samples collected over the period July 1974 to May 1975. These data were characterized by Lane (1986) and follow the relationships among
0303-2647/91/$03.50 @) 1991 Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland
76
a group of populations, guilds and nutrients using loop digraphs, where each digraph represents a stable system according to the rules of loop analysis (Levins, 1975; Puccia and Levins, 1985; Pilette et al., 1987). Stability from digraph to digraph (sample to sample) is similar to the concept of a moving equilibria as discussed by Bodini and Giavelli (1989). This stability is not a modeling requirement but represents the most appropriate characterization of the samples. Lane's modeling is exceptional as she was able to make a series of correct descriptions (173 of 187 or 93°70) of the observed changes in relative abundance for the numerous entities involved based on her interaction matrices, thus giving confidence in her digraphs. 2. Stability-instability calculations For this analysis an entity is defined as a population, guild or nutrient identified as playing a distinct role in the plankton community. In each of the 12 major systems, our computer program analyzed (according to loop analysis criteria) every subsystem in which each entity played a role. It also determined how the removal of the entity affected the stability (or instability) of that subsystem. (Note, an actual subsystem was defined as one where each entity of a group of entities is connected to at least one other entity of the group). Following the methods used previously (Pilette et al., 1990), stability was defined as the percentage of stable (according to loop analysis criteria) subsystems within which an entity participated as calculated over all subsystem sizes. Thus, for sample C with N entities, if we let ss(C,e,k) be the number of subsystems of size k in which entity e participates and st(C,e,k) be the number of stable subsystems of size k in which e participates, then stability is n
st(C,e,k) s(C,e) = 100
k=l
ss (C,e,k) k=l
These results show a detailed picture of the
stability relationships at the entity level. For each entity, the stability of every subsystem in which the entity participates is calculated before and after the removal of that entity. The computer program then gathers these data and gives statistics on the number of subsystems affected and the nature of the effect. From these data, complex but highly detailed patterns emerge as to the role played by each entity with regard to stability or instability, in both the overall system and the various subsystems. These patterns can also be followed over the annual cycle. We can enumerate six types of changes that occur when an entity is removed from a subsystem: (a) a previously stable subsystem can remain stable (stable - stable); (b) a previously stable subsystem can become unstable (stable unstable); (c) a previously stable subsystem can fragment and the entities become disconnected (stable disconnected); (d) a previously unstable subsystem can become stable (unstable - stable); (e) a previously unstable subsystem can remain unstable (unstable - unstable); (f) a previously unstable subsystem can become fragmented and the entities become disconnected (unstable - disconnected). The computer program tallies each time one of the above events occurs due to the removal of an entity from each and every subsystem in the community matrix. Percentages are then calculated and reported for each type of change. This use of entity removal as an analytical probe is similar to techniques used by Conrad (1983) in studying small artificial ecosystems. Cooper (1990) has characterized the observation of cell killing in immunological studies as both the assay to investigate other problems and the phenomena itself to be investigated. Similarly, in this study, we are interested in entity removal both as a simulation of switching, resting and local extinction observed in a plankton community and as a probe of the patterns found in the community itself. 3. Stability relationships and trophic structure Table 1 shows our results for stability relation-
77
TABLE 1 Annual mean percentage by category following entity removal. Entity
Nitrogen/phosphorus Organic nutrients Silicate Diatoms Dinoflagellates Luxury-consuming diatoms Small flagellates Miscellaneous algal groups Immature copepods Copepod adults 2 Copepod adults 1 Cladocerans Oikopleura sp. Mollusc larvae Polychaete larvae + Cirripeds Decapods Sagitta spp. Medusae Average
1a
2
3
4
5
6
SPI b
s-s
s--u
s-d
u-s
u-u
u-d
1~1 h2 Si A1 A2 A3
4.6 46.0 46.9 10.7 5.7 34.6
0.7 7.5 14.4 1.1 0.5 3.8
57.4 12.5 10.8 53.3 57.1 24.2
+0,0 0.0 0.0 2.4 3.9 11.5
2,3 22,9 23.1 8.4 4.0 16.1
35.0 11.2 4.8 24.1 28.7 9.8
39 20 20 34 36 25
A4 A5
33.4 28.4
+0.0 0.1
26.8 32.1
9.4 10.0
18.8 15.2
11.6 14.3
18 778 24 410
212 213 211 Z4 O 1H Pc
30.0 15.4 46.4 49.7 48.0 52.5 49.0
1.7 2.6 13.4 21.0 23.8 10.3 13.1
28.8 43.9 9.0 0.0 0.0 5.4 8.1
4.0 4.8 5.5 O.0 0.0 1.3 0.6
11.8 7.2 21.9 29.3 28.2 26.1 23.2
23.7 26.1 3.8 0.0 0.0 4.4 6.1
27 30 21 8 10 19 19
I) S MD
35.1 50.8 56.8
10.6 18.1 14.9
18.0 0.0 0.0
0.5 1.3 0.0
18.8 29.8 28.3
17.0 0.0 0.0
14 470 12 289 11 574
30.0
6.2
28.8
3.5
15.4
16.1
755 625 542 070 453 383
492 239 010 469 122 845 549
al, s -- s, stable - stable; 2, s - u, stable - unstable; 3, s - d, stable - disconnected; 4, u - s, unstable - stable; 5, u u, unstable - unstable; 6, u - d, unstable - disconnected, s - s, a previously stable system or subsystem upon removal of
the entity in question remains stable, etc. bSPI, subsystems in wl:.ich the entity participated.
ships between all 18 entities found by Lane within the Delaware Bay plankton community, and Fig. 1 shows the structural relationships between these entil;ies. Despite the obviously broad diversity of roles exposed by this type of analysis, there are some commonalties to be seen. For example, the Delaware Bay plankton community entities carl be placed in large composite groups according to straightforward trophic/ structural considerations. This is the classic simplification and can readily be seen in many large natural systems. Silicate (Si), nitrogen/ phosphorus (N1) and organic nutrients (N2) are resources; diatoms (A1), dinoflagellates (A2),
luxury consuming (probably auxotrophic, requiring B12) (Lane, 1986), diatoms (A3), small flagellates (A4) and miscellaneous algal groups (A5) are producers; copepod adults 1 (Z1), iramature copepods (Z2), copepod adults 2 (Z3), cladocerans (Z4), Oikopleura sp. (0) and mollusc larvae (M) could be termed primary consumers; polychaete larvae + cirripeds (PC) and decapods (D) could be termed omnivores; while medusae (MD) and Sagitta spp. (S) could be termed secondary consumers. If stability data are grouped according to these categories, the results can be seen in Table 2 and Fig. 2. This is the broadest possible grouping of this data and yet stability roles can still be
78
Fig. 2. Broad trophic structural relationships (for trophic groups, see Table 2).
71eqS
detected. For example, both the resources and the producers are central to the system since, in over half of the instances for each, removal of any one of these entities results in the subsystems fragmenting and becoming disconnected (columns 3 and 6, Table 2). Loss of a resource results in subsystem fragmentation 55.4% of the time and loss of a producer results in subsystem fragmentation 61% of the time. In contrast, loss of primary and secondary consumers results in subsystem fragmentation 34.3% and 13.5% of the time. However, this broad grouping of entities and their stability roles masks a more subtle and interesting series of entity subgroups. This can be seen by examining each stability change category and looking not only at the total change reported but also at the variance. In the resources the average value for s - s is 25.9%; however, by looking at Table 1 it can be seen that at one extreme N1 has only a 4.6% value in this category while Si has a 46.9% value. Of the 108 values in Table 1, 66 differ from their group
"",
Fig. 1. Loop digraph (adapted from Lane 1986, Fig. 5) of the Delaware Bay plankton community, based on most frequent links between entities over the annual cycle (for entity key see Table 1). Key: - - - - , a positive effect from one entity to another; - - O , a negative effect from one entity to another; solid lines, most persistent connections; dashed lines, connections present 3 0 - 5 0 % of the time. TABLE 2
Annual mean percentage by category following entity removal broad trophic groups. Entity
Resources (Si, N1, N2) Producers (A1, A2, A3, A4, A5) Consumers (primary) (Zl, Z2, Z3, Z4, O, M) Consumers (secondary) + Omnivores (PC, D, S, MD) Average
1
2
3
4
5
6
SPI
s-s
s--u
s-d
u-s
u-u
u-d
R
25.9
5.9
34.1
+0.0
12.8
21.3
80 922
P
19.9
1.1
41.7
6.7
11.3
19.3
139 094
C
36.0
8.8
20.6
3.4
17.5
13.7
117 357
O/S
47.5
13.9
7.2
0.6
24.5
6.3
57 882
30.0
6.2
28.8
3.5
15.4
16.1
79 TABLE 3 Annual mean percentage by category following entity removal-refined trophic groups based on minimization of variance. Entity
Resources 1 (Si, N2) Resources 2
1
2
3
4
5
6
SPI
s-s
s-u
s-d
u-s
u-u
u-d
1~:1
46.5
10.9
11.6
0.0
23.0
8.0
41 167
F',2
4.6
0.7
57.4
+0.0
2.3
35.0
39 755
P1
8.1
0.8
55.3
3.2
6.1
26.5
70 523
P2
32.1
1.4
27.7
10.4
16.5
11.9
68 571
C1
49.2
15.2
5.0
2.4
25.5
2.8
59 626
C2
22.4
2.2
36.7
4.4
9.4
24.9
57 731
O1
43.1
12.0
12.3
0.5
21.3
10.7
34 019
S1
53.7
16.5
0.0
0.6
29.1
0.0
23 863
30.0
6.2
28.8
3.5
15.4
16.1
(NI)
Producers 1 (A1, A2) Producers 2 (A3, A4, A5) Consumers (primary) 1
(Zl, z4, O, M) Consumers (primary) 2
(z2, z3) Omnivores (PC, D) Consumers (second
ary) (MD, S) Average
average by more than 5%, 31 by more than 10%, and 10 by more than 20%. Clearly this type of major grouping by ~raditional trophic role is not accurate enough. Table 3 and Fig. 3 show the results of splitting each of the major gTOUpS into two smaller ones based on similarity of stability roles. Role dif-
ferences now become much more pronounced. In looking at fragmentation (columns 3 and 6, Table 2), for example, in resource group R1 (Si and N2), loss of these entities results in subsystem fragmentation 19.6% of the time, whereas loss of resource group R2 (N1) now has a 92.4% rate of fragmentation upon its removal from a subsystem. Additional confirmation that this grouping presents a more accurate picture of the community comes from the observation that there is little Variance left within each of the 8 groups. Of the 108 values (see Table 1), only 10 have variances greater than 5% (with the largest being 8.6%) when placed in these groupings (see Table 3). 4. Community substructures
Fig. 3. Refined trophic structural relationships, based on minimization of variance within a given trophic group (for trophic groups see Table 3).
How do the observed connections between the entities as seen in Fig. 1 relate to the more ab-
80
stract model seen in Fig. 3? For example, R2 resource N1 (nitrogen/phosphorus) is clearly playing a different stability role than resource R1 (Si and N2, silicate and organic nutrients), and producers P1 are playing different stability roles than producers P2. Analysis of the structures into which these groups of entities fall provides an answer. Taking Lane's original observed structural digraph (Fig. 1) as a basis (Lane 1986), it is possible to discern five basic subgroup classes based on trophic structural relationships. We have termed these community stability substructures and these are shown in Fig. 4. Two classes of pattern emerge from within
(1)
(2)
(3)
(4)
these stability substructures; a simple pattern in which resources are linked to producers and then onto primary consumers (patterns 1, 2 and 3 in Fig. 4) and more complex, branching patterns involving secondary consumers and omnivores (patterns 4 and 5). These patterns relate directly to the more refined trophic community structure we derived above (Fig. 3). Stability substructure patterns 1, 2 and 3 map onto the R2 - P2 - C1 sequence of the refined trophic structure and stability substructure patterns 4 and 5 map onto the more complex R1 P1 - etc. sequence. Figure 4, which is based on abstraction from the observed connections, maps onto Fig. 3, which is based on the minimization of variance with respect to stability participation within these subgroups. The match is almost perfect (compare Fig. 4 and Table 3); there are only two anomalies, Si and M. Si is part of both types of community stability substructures as is N1, yet Si's stability pattern resembles that of N2 which is part of only one type of substructure. M, while part of the more complex substructure pattern, is actually providing a connecting link between the two substructure patterns that comprise the stability substructure 4 and 5 (Fig. 4). M's substructure role is, therefore, different from that of the C2 members (Z2 and Z3) and its stability pattern resembles that of the C1 members. This connection was camouflaged by the abstraction made when constructing C2 in Fig. 3.
5. Non-trophic structure
(5)
Fig. 4. Community stability substructures, based on similarity of subgroup patterns found in Fig. 1.
The analysis so far has concentrated on model building based on traditional trophic relationships as seen in plankton community structure. We have shown that the simple trophic model (Fig. 2) is incompatible with our stabilityinstability analysis and that a more refined model (Fig. 3) more accurately describes the trophic relationships when viewed in the context of relative stability. However, the stability data can also be used in other ways to build a more complete picture of a community and the roles and relationships of the various entities. We can ask, for example, "How do the tradi-
81
tional trophic patterns (both the simple model and our m o r e refined model) relate to the six types of stability change that comes about following the removal of an entity (e.g. s - s, see above)?" Three new types of relationships emerge for each erLtity and its contribution to the overall system.
5.1. Relative participation This is the degree to which an entity participates in stable and unstable subsystems. Stabilizing is defined as the proportion of categories that are of the form s - ?, i.e. columns 1, 2 and 3 of Table 1; destabilizing is defined as the proportior~ of categories that are of the form u - ?, i.e. columns 4, 5 and 6 of Table 1. No large differences were found between the various entities. Relatively stable participation values vary from 60.2% to 72.1%; relatively destabilizing participation values vary from 27.9% to 39.8% (Table 1).
5.2. System necessary This is the degree to which removal of the entity results in fragmentation of the subsystem. This is defined as the proportion of categories that are of the form ? - d, i.e. columns 3 and 6 of Table 4. The system necessary grouping can be further subdivided into system necessary components that are relatively stabilizing (s d, i.e. column 3) and system necessary components that are relatively destabilizing (u - d, i.e. column 6). There are four entities that show very high fragmentation rates when removed from the subsystem (Table 4): nitrogen/phosphorus (N1) 92.6%, dinoflagellates (A2) 85.8%, diatoms (A1) 77.4%, and copepod adults 2 (Z3) 70.0%. The next highest percentage is 52.5%, and this 17.5% break is the largest found for adjoining entities. Eight of the fourteen remaining entities show percentages below 16% in this category.
TABLE 4 Entities ranked by system necessary categories (3 + 6). Entity
Nitrogen/phosphorus Dinoflagellates Diatoms Copepod adults 2 Immature copepods Miscellaneous algal groups Small flagellates Decapods Luxury-consuming diatcms Organic nutrients Silicate Polychaete larvae + Cirripeds Copepod adults 1 Mollusc larvae Cladocerans Oikopleura sp. Sagitta spp. Medusae
N1 A2 A1 Z3 Z2 A5 A4 D A3 N2 Si PC Z1 M Z4 O S MD
3 s - d
6 u - d
3+6
Trophic group
57.4 57.1 53.3 43.9 28.8 32.1 26.8 18.0 24.2 12.5 10.8 8.1 9.0 5.4 0.0 0.0 0.0 0.0
35.0 28.7 24.1 26.1 23.7 14.3 11.6 17.0 9.8 11.2 4.8 6.1 3.8 4.4 0.0 0.0 0.0 0.0
92.4 85.8 77.4 70.0 52.5 46.4 38.4 35.0 34.0 23.7 15.6 14.2 12.8 9.8 0.0 0.0 0.0 0.0
R2 P1 P1 C2 C2 P2 P2 O1 P2 R1 R1 O1 C1 C1 C1 C1 S1 S1
82
The first four entities can therefore be considered relatively system necessary and possibly play an important role in holding the plankton community together. Below these four entities are a transitional group of six entities with rates between 23.7% and 52.5%. The eight entities with rates 16% or lower can be considered relatively non-system necessary in that their removal does not often fragment the system. It is important to note that these relationships essentially hold whether or not the entity is relatively stabilizing or destabilizing. There is one exception: immature copepods (Z2) have as significant a contribution to fragmenting destabilizing systems upon removal, as do N1, A1, A2 and Z3. Rankings of system necessary entities readily fall into groups matching the more refined trophic grouping arrived at above (Table 4, last column). Only Decapods (D) need be moved in order for the entities of each of the eight trophic
TABLE 5a
TABLE 5b Entities ranked by contribution to instability (category 4). Entity
Luxury-consuming diatoms Miscellaneous algal groups Small flagellates Copepod adults 1 Copepod adults 2 Immature copepods Dinoflagellates Diatoms Sagitta spp. Mollusc larvae Polychaete larvae + Cirripeds Decapods Nitrogen/phosphorus Organic nutrients Silicate Medusae Oikopleura sp. Cladocerans
4 u - s
Trophic group
A3
11.5
P2
A5
10.0
P2
A4 Z1 Z3 Z2 A2 A1 S M PC
9.4 5.5 4.8 4.0 3.9 2.4 1.3 1.3 0.6
P2 C1 C2 C2 P1 P1 S1 C1 O1
D N1 N2 Si MD 0 Z4
0.5 0.0 0.0 0.0 0.0 0.0 0.0
O1 R2 R1 R1 S1 C1 C1
Entities ranked by contribution to stability (category 2). Entity
Oikopleura sp. Cladocerans Sagitta spp. Medusae Silicate Copepod adults 1 Polychaete larvae + Cirripeds Decapods Mollusc larvae Organic nutrients Luxury-consuming diatoms Copepod adults 2 Immature copepods Diatoms Nitrogen/phosphorus Dinoflagellates Miscellaneous algal groups Small flagellates
2 s - u
Trophic group
0 Z4 S MD Si Z1 PC
23.8 21.0 18.1 14.9 14.4 13.4 13.1
C1 C1 $1 Sl R1 C1 O1
D M N2 A3
10.6 10.3 7.5 3.8
01 C1 R1 P2
Z3 Z2 A1 N1 A2 A5
2.6 1.7 1.1 0.7 0.5 0.1
C2 C2 P1 R2 P1 P2
A4
+ 0.0
P2
groups to be listed consecutively. No such matching is found when simple trophic grouping is used.
5.3. Relative contribution to stability of the subsystem This is defined as the proportion of events that are either s - u, i.e. column 2 of Table 5a, or u s, i.e. column 4 of Table 5b. High scores in the category s - u can be regarded as evidence for relatively high contributions to subsystem stability, whereas high scores in the category u s can be regarded as evidence for relatively high contributions to instability. This is not the same measure as that found in (a) because in (a) it is possible for an entity to have a high participation rate in stable or unstable subsystems where removal does not result in a change in subsystem stability or instability. There are three patterns found for entities
83
regarding relative contribution to stability (Table 5a). Three entities Oikopleura sp. (0) 23.8%, cladocerans (Z4) 21.0% and Sagitta spp. (S) 18.1% show the highest percentages where removal from a stable subsystem results in instability for the remaining subsystem; seven entities show percentages in the range 7 . 5 14.9%; eight entities show percentages in the range 0 . 0 - 3.8%. There are three patterns found for entities regarding relative contribution to instability (Table 5b). Three entities, luxury consuming diatoms (A3) 11.5%, miscellaneous algal groups (AS) 10.0% and small flagellates (A4) 9.4%, show the highest ]percentages where removal from an unstable subsystem results in consequent subsystem stability; 9 entities show percentages in the range 0.5-5.5%; and 6 entities are at or near 0.0%. A good match is found between relative contributions to stability and the more refined trophic grouping (Table 5a,b, last column for each). The most interesting matching is found in the relatively destabilizing category (Table 5b) where entities of six of the eight trophic groups are listed consecutively.
90
/\
,2,.,o
80 70 60 50 40 30 20 10 i
0
i
2
3
i 4
i 5
i 6
i
i
i
i
i
7
8
9
10
11
12
7
8
9
10
11
12
Month
go 80
A 3 s -> s
70 60 %
50 40 3o 20 10 0 1
2
3
4
5
6 Month
Fig. 5. Examples of high and low variance over the annual cycle: (a) low variance; (b) high variance.
tities. However, no patterns relating to either simple or refined trophic or stability relationships were found.
6. Temporal sequences
6.1. Variance over the annual cycle
Changes in each of the measures discussed above were also followed over an annual cycle. These data show that, in addition to short term dynamics (switching, resting, disturbance), there are also long term seasonal patterns, which range from consistent stability relationships from month to month, to patterns showing extreme variation from one month to the next. In Lane's original analysis (Lane 1986), samples were taken at approximately 1 month intervals over a period of 1 year. It is therefore possible to follow Changes in the roles played by various of the entil;ies during a complete series of seasonal changes. Lane began her study in July 1974 (month 1) and concluded in May 1975 (month 12) and we have maintained this sequence for our ans~lysis. Viewed in this way, several different types of annual pattern can be seen for the various en-
In this analysis we used standard deviation rather than coefficient of variation because of the often low means seen in the various categories. In looking at the 55 single categories with means over 10%, six show a low variation (standard deviation one-third or less of the mean) and eight showed a high variation (standard deviation as high or higher than the mean). Figure 5 shows an example of each. For the six categories studied (i.e. s - s, s d, etc.), the average standard deviation of these annual values for A2 (7.0) and N1 (7.7) are very low with the remaining sixteen entities averaging between 11.1 and 16.5.
6.2. Parallel tracking This measure is based on the comparative change between two entities from one sample to
84
a
high
s->s
C
ooo Ioooooooo %
%
500 40.0 30.0 20.0 10.0 0.0
high
.~->d
•
50.0 40.0 30.0
2
~'
20.0
10.0 i
i
1
i
2
i
3
4
~
5
J
6
J
7
i
8
L
9
i
10
i
11
1
12
2
3
4
5
6
a
low
7
8
g
10
11
12
Month
Month
o.01 c
s->s
low
s->d
60.0 50.0
A1 & A 3 s -> s 70.0 60.0
A3
% 6o.o
%
A Z3
40.0
o
30,0
40.0 30.0 20,0 10.0
20.0
O
Z2
10.0 A1
0.0
--
= 1
0.0 2
I
3
4
5
6
7
8
g
10
11
2
3
4
5
6
7
8
10
9
11
12
Month
12
Month
0
high
S->u d
~
35.o
Si & Z l s -> u
u->s
A1 & A 2 u -> s
12.o
Zl
30.0
T
10.0
25.0
%
high
20.0
8.0 6.0
15.0
%
4.0
A1
/
A2 0--
10.0 2.0
5.0
0 0.0-
0.0 1
2
3
4
5
6
7
0
9
10
11
l
1
12
2
3
4
5
d low
.
9
10
11
8
9
10
11
12
low
u->s
0.0 o
Si&N2s->u
% %
8
o.o I z
s->u
4°Ql
7
Month
Month
b
6
20.0 15.0
15.0 10.0
Si
5.0
10,0
0.0
5.0
l
1
0.0 1
2
3
4
5
6
7
8
9
10
11
12
2
3
4
5
6
7
12
Month
Month
the next in the annual sequence. Parallelism is scored when both entities increase (or decrease) together in value for a particular category from one sample to the next, and non-parallelism is scored if the change is in the opposite direction. Overall, a much higher degree of parallelism was found than would be expected from pure
chance. The expected average is 5.5, whereas the actual average based on 146 sets of eight or more was 7.5. Figure 6 shows examples of high parallelism and low parallelism between entities. In addition, parallelism can be seen to extend over all six categories of removal. For example
85 e
high u->u
A1 & M, u -> u
f°
40.0
%
35.0 3O.O 25.0 20,0 15,0 10.0 5.0 0,0
M
. 1
; 2
;
;
3
4
; 5
;
.
6
.
7
f
i
8
Month
A1
g
10
11
12
high u->d
% 30,0 20,0 10.0 0.0
i 1
2
3
4
5
6
7
8
9
10
11
12
Month f
%
low u - > d
35°1,,,3u-,° ^ 20.0
1
/
15.0
O0
J 1
2
3
4
5
6
7
8
9
10
11
12
M onth
Fig. 6. E x a m p l e s of h i g h a n d low p a r a l l e l i s m in all c a t e g o r i e s o v e r t h e a n n u a l cycle. (a) s - s; (b) s - u; (c) s - d; (d) u -
s;(e) u--
u;(f) u-.
titles were also among those showing low variance and high parallelism and do not show any switching with regard to their roles over the annual cycle. These results are in contrast to that seen for entity Z2 (immature copepods), shown in Fig. 6c. In samples 2 - 6 and 11 and 12, Z2 shows high scores in the s - d category. However, in samples 1 and 7 - 1 0 the scores in this category are always lower than seen at any other time of the year. We have termed this type of change in behavior over the temporal cycle, role switching. Similar role switching is seen in other entities, for instance A4 (small flagellates). This entity, which only appears in samples 5 - 12, has values ranging from 49.9% to 50.0% for the combined categories s -- d and u - d, in samples 5 - 8 . These values for disconnectance fall to 0.0% for samples 9 - 1 2 . In a less seasonal manner, role switching is also seen for entity N2 (organic nutrients). Disconnectance values for s - d and u - d are always 0.0% except for samples 6, 9 and 12 when the values jump to 66.6%, 49.9%, and 83.3%, respectively. This role switching seems limited to individual entities and does not seem to extend to either simple or refined trophic groups. 7. Discussion
d.
high parallelism is seen for A1 + N1 (average 9) and low parallelism for A1 + A3 (average 4).
6.3. Role switching Complementing the patterns seen for total annual cycles, it is possible to observe interesting micro-patterns of behavior for a given entity at certain times of the year. Most entities exhibit little change in their stability-instability relationships. For instance, entities N1 (nitrogen/phosphorus ratio)and A2 (dinoflagellates) (Fig. 6c) remain highly system necessary for a full yearly cycle. These two en-
Previously, Pilette et al. (1990) have shown that complexity at the entity level is inversely related to the number of subsystems in which an entity participates, but that stability increases with increasing subsystem size both for the entities involved and for the subsystems themselves. It would appear therefore that large systems such as the plankton community in Delaware Bay are composed of various, ever changing combinations of smaller subsystems. Subsystems and their interactions thus become importi~nt modulators of overall system stability and resilience. Margalef and Gutierrez (1983) have also argued that discreteness (both temporal and spatial) is a stabilizing phenomenon. Large complex systems may be stable over significantly
86
long periods of time, but, embedded within these systems, in near decomposable fashion (Simon, 1962) there is a flux of subsystems, many of which may themselves be unstable. Thus, large biotic systems are composed of discrete but fluid sets of temporary associations among the various entities. Long scale analysis may assign certain traditional structural roles to entities within systems (i.e. diatoms - mollusc larvae); however, when viewed at shorter time scales, this link may be seen (for example) to hold 60% of the time. During the remaining 40%, the predator may establish other links (i.e. dinoflagellates - mollusc larvae). Such fluctuations in role performance may cause temporary changes in stability within the subsystem just vacated, but, for the system as a whole (which may be comprised of up to 8300 subsystems for these Delaware Bay plankton samples), this local change within one subsystem may be compensated for by changes in stability within other subsystems, hence leading to temporal equilibrium or moving equilibria (Bodini and Giavelli, 1989) in what appears to be a series of local non-equilibrium situations. In addition to the effects due to switching behavior, there are also effects due to resting and disruption to consider as well. In this study we have used the phenomena of switching, resting, and disruption as an investigative tool in an entity removal exercise. Initially, we searched for patterns related to trophic structure. Nitrogen and phosphorus are necessary resources; diatoms and dinoflagellates are producers; cladocerans and copepods are consumers; and the top carnivores in these plankton communities are decapods and Sagitta spp. When Lane (1986) was building a model to fit changes in abundance over an annual cycle, she grouped organisms in a structural manner such that species having identical links in her digraphs were placed in the same entity grouping. These groups are equivalent to guilds. In order to make her model accurate with regard to predicting the next system in her series, she found it was necessary to subdivide at least two groups: diatoms into diatoms and luxury consuming diatoms (the latter, probably aux-
otrophic requiring vitamin B12), and copepods into copepod adults 1 and copepod adults 2. Similarly, Andersen et al. (1987), while analyzing communities in enclosed water columns in a British Columbia inlet over a 40 day cycle, tried to model the basic parameters representing phytoplankton development. These authors found that the obvious simple model of nitrate - phytoplankton - herbivores - carnivores was not sufficient for them to predict phytoplankton development, and that this model required further subdivisions in order for it to have value. It appears, therefore, that simple models of trophic interaction or stability cannot always account for observed changes and that more detailed models are necessary. The results of our analysis of the Delaware Bay plankton community support these conclusions. Simple relationships between large groupings of these entities hide a more subtle series of roles played by various entities which are revealed when larger groups are split into smaller assemblages and when roles are examined over a complete annual cycle. The more complex stability-instability patterns are based on the minimization of variance within the various entity removal categories. The results indicate, for example, that there is more than one type of producer and more than one type of consumer. The observed stability-instability patterns match the more refined trophic structure (Table 3 and Fig. 3). The only anomaly of the 18 entities is Polychaete larvae + Cirripeds (PC), whose stability pattern more closely fits that of the Organic Nutrients (N2) and Silicate (Si) than that of Decapods (D), its relevant trophic group co-member (Table 1 and Fig. 1). In looking at the roles of relative contribution to stability (or instability) and system necessity, patterns also emerge for these non-trophic considerations. We can combine these roles with respect to contribution to subsystem stability and system necessity into a framework provided by Pilette (1989) as shown in Table 6. Core entities are considered relatively stabilizing and noncore entities relatively destabilizing. System necessary entities tend to result in fragmented
87 TABLE 6 Relative contribution to :~tability and system necessity a. Core b Not system necessary Somewhat system necessary System necessary
Outer core
Non-core
R1, O1
P2
S1,C1
R2,P1,C2
aSee text for discussion. bCore/outer core/non-core: ranked contribution to stability from most stabilizing to least stabilizing (core to non-core).
systems when removed, while those not system necessary tend to leave the system intact after removal. Again, as seen by Andersen et al. (1989), the more refined trophic grouping is required to tease out the various roles. Groups S1 and C1 are stabilizing, not: particularly destabilizing, and not system necessary. Groups R2, P1 and C2 are system necessary and make little contribution to either stabilization or destabilization. Groups R1 and O1 are moderately system necessary, stabilizing, and are not destabilizing. Group P2 is moderately system necessary, makes little contribution to stabilization, and makes the greatest contribution to destabilization. Thus, entities may be system necessary and yet relatively destabilizing. Pilette (1989, unpublished) has found this in an even stronger form: an entity may be required to hold a system or subsystem together and be highly destabilizing at the same time. We can make the useful observation that not only are a variety of patterns found for system necessity and relative contribution to stability, but, they fit rather closely those derived from trophic considerations and the more complex stability-instability patterns (Table 5a,b, last column). Examination of temporal sequences over a 12 month annual cycle shows that much is hidden
when role analysis is performed only on yearly averaged data. Resources such as N1 and producers such as A1 show a high degree of consistency over the annual cycle, contribute to subsystem stability, and are system necessary. This might have been anticipated from their trophic importance within the plankton community and from their biological role. The value of these entities can thus reasonably be gauged largely from yearly average scores in most categories. However, within primary consumers and omnivores, the yearly pattern for stability relationships shows various types of role switching. Both primary consumers C1 (low system necessary) and C2 (high system necessary) show one or more periods when roles, in this category, are reversed. This is often correlated with switches in relative contribution to stability and highlights what may turn out to be interesting biological phenomena taking place in the plankton community during these time periods. Omnivores are more opportunistic and these entities might indeed be expected to change trophic roles during an annual cycle. These entities, which are normally not system necessary, do show brief periods of role switching with regards to the plankton community stability. These results, however, do not map onto the refined trophic structure. As discussed above, the system necessary patterns and the relative contribution to stability patterns both match quite well with the complex stability patterns based on a minimization of variance. This complex stability pattern matches the refined trophic structure pattern. The refined trophic structure pattern is, thus, based on the minimization of variance within various stability categories. This refined trophic structure is composed of two substructures (Fig. 3). One of our most interesting results is that these abstracted trophic substructures match almost perfectly with the subtypes of the original community digraph for the annual cycle (Fig. 4). In a sense, we have used stability relationships as a probe into community structure. The entity removal exercise has allowed us to generate various categories regarding stability: group-
88
ings based on minimization of variance, system necessity and relative contribution to stability. Each of these categories of analysis maps poorly onto simplified trophic structure and maps very well on to more refined trophic structure. From our results, it would appear that ecologists can ill-afford to oversimplify community and trophic structure analysis.
Acknowledgements This research was supported in part by grants from NSF award number DMB 8612828 and the City University of New York PSC-CUNY Research Award Program numbers 6-67192, 6-67293 and 6-68186. We would like to thank Dr. S.N. Salthe for useful comments on an earlier version of this manuscript.
References Andersen, V., Nival, P. and Harris, R.P., 1987, Modelling of a plankton ecosystem in an enclosed water column. J. Mar. Biol. Assoc. U.K. 67, 407-430. Bodini, A. and Giavelli, G., 1989, The quantitative approach in investigatingthe role of species interactions on stability of natural communities. BioSystems 22, 289-299.
Conrad, M., 1983, Adaptability: The Significance of Variability From Molecule to Ecosystem (Plenum Press, New York). Cooper, E.I., 1990, Immune diversity thoughout the animal kingdom. BioScience 40, 720-722. Lane, P.A., 1986, Symmetry, change, perturbation and observing mode in natural communities. Ecology 67, 223 - 239. Levins, R., 1975, Evolution in communities near equilibrium, in: Ecology and Evolution in Communities, M.L. Cody and J.M. Diamond (eds.) (Harvard University Press, Cambridge) pp. 16-50. Margalef, R. and Gutierrez, E., 1983, How to introduce connectance in the frame of an expression for diversity. Am. Nat. 121,601-607. Pilette, R., 1989, System necessary populations that are destabilizing. Abstract: Annual Meeting of the International Society for Ecological Modeling. Pilette, R., Sigal, R. and Blamire, J., 1987, The potential for community level evaluations based on loop analysis. BioSystems 21, 25- 32. Pilette, R., Sigal, R. and Blamire, J., 1990, Stability complexity relationships within models of natural systems. BioSystems 23, 359-370. Puccia, C.J. and Levins, R., 1985, Qualitative Modeling of Complex Systems (Harvard University Press, Cambridge). Simon, H.A., 1962, The architecture of complexity. Proc. Am. Philos. Soc. 106, 467-482.
75
Elsevier Scientific Publishers Ireland Ltd.
Stability patterns and trophic structure in a Delaware Bay plankton community Ron Pilette, Ron Sigal a and John Blamire b Science Department, John Jay College of CUNY, 445 West 59th Street, New York, N Y 10019, aDepartment of Mathematics, Yale University, Box 2155 Yale Station, New Haven, CT 06520 and bDepartment of Biology, Brooklyn College of CUNY, Brooklyn, N Y 11210 (USA) (Received August 2nd, 1.991)
Using qualitative loop analysis we have extended our examination of a Delaware Bay plankton community to include an investigation of the roles :played by the various entities (population, guild or nutrient) in the community. In an entity removal exercise, we used stability relationships as a probe into community structure. Six types of stability change are possible as a result of entity removal from the system: stable to stable (s - s); stable to unstable (s - u); stable to disconnected (s - d); unstable to stable (u - s}; unstable to unstable (u - u); unstable to disconnected (u - d). Using these changes as an investigative tool, we found that in order to account for the stability-instability patterns, it was necessary to construct a refined trophic structure model. The observed connections between the entities in the larger model could be grouped into two different types of stability substructures: a simple pattern and a more complex branching pattern. These patterns map easily onto the refined trophic structure model. Using stability analysis it is also possible to model community structure in ways other than the traditional trophic approach. Patterns of system necessity and relative contribution to stability are observed. These patterns match the refined trophic structure model derived previously. The roles that the various entities play in the overall community were followed over an annual eycle. Entities were seen to change their roles as a function of time and status within a subgroup. These results show that stability determinations have the potential to be used as a valuable tool in community analysis.
Keywords: Community modeling; Population roles; Loop analysis; Stability patterns; Trophic structure; Delaware Bay plankton community.
1. Introduction
We have previously shown (Pilette et al., 1990) that the quali~ative loop analysis models of a Delaware Bay plankton community, prepared by Lane (1986), can be evaluated for relationships between stability and complexity. No statistically significant relationship was found for stability and complexity between systems (monthly community samples), but, within a system, a significant inverse relationship was found at the entity (population, guild or nutrient) level, and this was embedded in an increasing stability positively related to increasing subsystem size. Correspondence to: R. Pilette.
It was also noted that the stability role played by several of the entities within the system varied with respect to subsystem size and over the course of the annual cycle. This has prompted us to carry out a more detailed analysis of this aspect of our data. In the research reported here, we were concerned with the roles played by the entities of the Delaware Bay plankton community. Are there particular stability patterns for these entities and are these related to trophic structures? Are there patterns to the temporal sequences? Which entities share similar patterns? This work is based on 12 estuarine plankton samples collected over the period July 1974 to May 1975. These data were characterized by Lane (1986) and follow the relationships among
0303-2647/91/$03.50 @) 1991 Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland
76
a group of populations, guilds and nutrients using loop digraphs, where each digraph represents a stable system according to the rules of loop analysis (Levins, 1975; Puccia and Levins, 1985; Pilette et al., 1987). Stability from digraph to digraph (sample to sample) is similar to the concept of a moving equilibria as discussed by Bodini and Giavelli (1989). This stability is not a modeling requirement but represents the most appropriate characterization of the samples. Lane's modeling is exceptional as she was able to make a series of correct descriptions (173 of 187 or 93°70) of the observed changes in relative abundance for the numerous entities involved based on her interaction matrices, thus giving confidence in her digraphs. 2. Stability-instability calculations For this analysis an entity is defined as a population, guild or nutrient identified as playing a distinct role in the plankton community. In each of the 12 major systems, our computer program analyzed (according to loop analysis criteria) every subsystem in which each entity played a role. It also determined how the removal of the entity affected the stability (or instability) of that subsystem. (Note, an actual subsystem was defined as one where each entity of a group of entities is connected to at least one other entity of the group). Following the methods used previously (Pilette et al., 1990), stability was defined as the percentage of stable (according to loop analysis criteria) subsystems within which an entity participated as calculated over all subsystem sizes. Thus, for sample C with N entities, if we let ss(C,e,k) be the number of subsystems of size k in which entity e participates and st(C,e,k) be the number of stable subsystems of size k in which e participates, then stability is n
st(C,e,k) s(C,e) = 100
k=l
ss (C,e,k) k=l
These results show a detailed picture of the
stability relationships at the entity level. For each entity, the stability of every subsystem in which the entity participates is calculated before and after the removal of that entity. The computer program then gathers these data and gives statistics on the number of subsystems affected and the nature of the effect. From these data, complex but highly detailed patterns emerge as to the role played by each entity with regard to stability or instability, in both the overall system and the various subsystems. These patterns can also be followed over the annual cycle. We can enumerate six types of changes that occur when an entity is removed from a subsystem: (a) a previously stable subsystem can remain stable (stable - stable); (b) a previously stable subsystem can become unstable (stable unstable); (c) a previously stable subsystem can fragment and the entities become disconnected (stable disconnected); (d) a previously unstable subsystem can become stable (unstable - stable); (e) a previously unstable subsystem can remain unstable (unstable - unstable); (f) a previously unstable subsystem can become fragmented and the entities become disconnected (unstable - disconnected). The computer program tallies each time one of the above events occurs due to the removal of an entity from each and every subsystem in the community matrix. Percentages are then calculated and reported for each type of change. This use of entity removal as an analytical probe is similar to techniques used by Conrad (1983) in studying small artificial ecosystems. Cooper (1990) has characterized the observation of cell killing in immunological studies as both the assay to investigate other problems and the phenomena itself to be investigated. Similarly, in this study, we are interested in entity removal both as a simulation of switching, resting and local extinction observed in a plankton community and as a probe of the patterns found in the community itself. 3. Stability relationships and trophic structure Table 1 shows our results for stability relation-
77
TABLE 1 Annual mean percentage by category following entity removal. Entity
Nitrogen/phosphorus Organic nutrients Silicate Diatoms Dinoflagellates Luxury-consuming diatoms Small flagellates Miscellaneous algal groups Immature copepods Copepod adults 2 Copepod adults 1 Cladocerans Oikopleura sp. Mollusc larvae Polychaete larvae + Cirripeds Decapods Sagitta spp. Medusae Average
1a
2
3
4
5
6
SPI b
s-s
s--u
s-d
u-s
u-u
u-d
1~1 h2 Si A1 A2 A3
4.6 46.0 46.9 10.7 5.7 34.6
0.7 7.5 14.4 1.1 0.5 3.8
57.4 12.5 10.8 53.3 57.1 24.2
+0,0 0.0 0.0 2.4 3.9 11.5
2,3 22,9 23.1 8.4 4.0 16.1
35.0 11.2 4.8 24.1 28.7 9.8
39 20 20 34 36 25
A4 A5
33.4 28.4
+0.0 0.1
26.8 32.1
9.4 10.0
18.8 15.2
11.6 14.3
18 778 24 410
212 213 211 Z4 O 1H Pc
30.0 15.4 46.4 49.7 48.0 52.5 49.0
1.7 2.6 13.4 21.0 23.8 10.3 13.1
28.8 43.9 9.0 0.0 0.0 5.4 8.1
4.0 4.8 5.5 O.0 0.0 1.3 0.6
11.8 7.2 21.9 29.3 28.2 26.1 23.2
23.7 26.1 3.8 0.0 0.0 4.4 6.1
27 30 21 8 10 19 19
I) S MD
35.1 50.8 56.8
10.6 18.1 14.9
18.0 0.0 0.0
0.5 1.3 0.0
18.8 29.8 28.3
17.0 0.0 0.0
14 470 12 289 11 574
30.0
6.2
28.8
3.5
15.4
16.1
755 625 542 070 453 383
492 239 010 469 122 845 549
al, s -- s, stable - stable; 2, s - u, stable - unstable; 3, s - d, stable - disconnected; 4, u - s, unstable - stable; 5, u u, unstable - unstable; 6, u - d, unstable - disconnected, s - s, a previously stable system or subsystem upon removal of
the entity in question remains stable, etc. bSPI, subsystems in wl:.ich the entity participated.
ships between all 18 entities found by Lane within the Delaware Bay plankton community, and Fig. 1 shows the structural relationships between these entil;ies. Despite the obviously broad diversity of roles exposed by this type of analysis, there are some commonalties to be seen. For example, the Delaware Bay plankton community entities carl be placed in large composite groups according to straightforward trophic/ structural considerations. This is the classic simplification and can readily be seen in many large natural systems. Silicate (Si), nitrogen/ phosphorus (N1) and organic nutrients (N2) are resources; diatoms (A1), dinoflagellates (A2),
luxury consuming (probably auxotrophic, requiring B12) (Lane, 1986), diatoms (A3), small flagellates (A4) and miscellaneous algal groups (A5) are producers; copepod adults 1 (Z1), iramature copepods (Z2), copepod adults 2 (Z3), cladocerans (Z4), Oikopleura sp. (0) and mollusc larvae (M) could be termed primary consumers; polychaete larvae + cirripeds (PC) and decapods (D) could be termed omnivores; while medusae (MD) and Sagitta spp. (S) could be termed secondary consumers. If stability data are grouped according to these categories, the results can be seen in Table 2 and Fig. 2. This is the broadest possible grouping of this data and yet stability roles can still be
78
Fig. 2. Broad trophic structural relationships (for trophic groups, see Table 2).
71eqS
detected. For example, both the resources and the producers are central to the system since, in over half of the instances for each, removal of any one of these entities results in the subsystems fragmenting and becoming disconnected (columns 3 and 6, Table 2). Loss of a resource results in subsystem fragmentation 55.4% of the time and loss of a producer results in subsystem fragmentation 61% of the time. In contrast, loss of primary and secondary consumers results in subsystem fragmentation 34.3% and 13.5% of the time. However, this broad grouping of entities and their stability roles masks a more subtle and interesting series of entity subgroups. This can be seen by examining each stability change category and looking not only at the total change reported but also at the variance. In the resources the average value for s - s is 25.9%; however, by looking at Table 1 it can be seen that at one extreme N1 has only a 4.6% value in this category while Si has a 46.9% value. Of the 108 values in Table 1, 66 differ from their group
"",
Fig. 1. Loop digraph (adapted from Lane 1986, Fig. 5) of the Delaware Bay plankton community, based on most frequent links between entities over the annual cycle (for entity key see Table 1). Key: - - - - , a positive effect from one entity to another; - - O , a negative effect from one entity to another; solid lines, most persistent connections; dashed lines, connections present 3 0 - 5 0 % of the time. TABLE 2
Annual mean percentage by category following entity removal broad trophic groups. Entity
Resources (Si, N1, N2) Producers (A1, A2, A3, A4, A5) Consumers (primary) (Zl, Z2, Z3, Z4, O, M) Consumers (secondary) + Omnivores (PC, D, S, MD) Average
1
2
3
4
5
6
SPI
s-s
s--u
s-d
u-s
u-u
u-d
R
25.9
5.9
34.1
+0.0
12.8
21.3
80 922
P
19.9
1.1
41.7
6.7
11.3
19.3
139 094
C
36.0
8.8
20.6
3.4
17.5
13.7
117 357
O/S
47.5
13.9
7.2
0.6
24.5
6.3
57 882
30.0
6.2
28.8
3.5
15.4
16.1
79 TABLE 3 Annual mean percentage by category following entity removal-refined trophic groups based on minimization of variance. Entity
Resources 1 (Si, N2) Resources 2
1
2
3
4
5
6
SPI
s-s
s-u
s-d
u-s
u-u
u-d
1~:1
46.5
10.9
11.6
0.0
23.0
8.0
41 167
F',2
4.6
0.7
57.4
+0.0
2.3
35.0
39 755
P1
8.1
0.8
55.3
3.2
6.1
26.5
70 523
P2
32.1
1.4
27.7
10.4
16.5
11.9
68 571
C1
49.2
15.2
5.0
2.4
25.5
2.8
59 626
C2
22.4
2.2
36.7
4.4
9.4
24.9
57 731
O1
43.1
12.0
12.3
0.5
21.3
10.7
34 019
S1
53.7
16.5
0.0
0.6
29.1
0.0
23 863
30.0
6.2
28.8
3.5
15.4
16.1
(NI)
Producers 1 (A1, A2) Producers 2 (A3, A4, A5) Consumers (primary) 1
(Zl, z4, O, M) Consumers (primary) 2
(z2, z3) Omnivores (PC, D) Consumers (second
ary) (MD, S) Average
average by more than 5%, 31 by more than 10%, and 10 by more than 20%. Clearly this type of major grouping by ~raditional trophic role is not accurate enough. Table 3 and Fig. 3 show the results of splitting each of the major gTOUpS into two smaller ones based on similarity of stability roles. Role dif-
ferences now become much more pronounced. In looking at fragmentation (columns 3 and 6, Table 2), for example, in resource group R1 (Si and N2), loss of these entities results in subsystem fragmentation 19.6% of the time, whereas loss of resource group R2 (N1) now has a 92.4% rate of fragmentation upon its removal from a subsystem. Additional confirmation that this grouping presents a more accurate picture of the community comes from the observation that there is little Variance left within each of the 8 groups. Of the 108 values (see Table 1), only 10 have variances greater than 5% (with the largest being 8.6%) when placed in these groupings (see Table 3). 4. Community substructures
Fig. 3. Refined trophic structural relationships, based on minimization of variance within a given trophic group (for trophic groups see Table 3).
How do the observed connections between the entities as seen in Fig. 1 relate to the more ab-
80
stract model seen in Fig. 3? For example, R2 resource N1 (nitrogen/phosphorus) is clearly playing a different stability role than resource R1 (Si and N2, silicate and organic nutrients), and producers P1 are playing different stability roles than producers P2. Analysis of the structures into which these groups of entities fall provides an answer. Taking Lane's original observed structural digraph (Fig. 1) as a basis (Lane 1986), it is possible to discern five basic subgroup classes based on trophic structural relationships. We have termed these community stability substructures and these are shown in Fig. 4. Two classes of pattern emerge from within
(1)
(2)
(3)
(4)
these stability substructures; a simple pattern in which resources are linked to producers and then onto primary consumers (patterns 1, 2 and 3 in Fig. 4) and more complex, branching patterns involving secondary consumers and omnivores (patterns 4 and 5). These patterns relate directly to the more refined trophic community structure we derived above (Fig. 3). Stability substructure patterns 1, 2 and 3 map onto the R2 - P2 - C1 sequence of the refined trophic structure and stability substructure patterns 4 and 5 map onto the more complex R1 P1 - etc. sequence. Figure 4, which is based on abstraction from the observed connections, maps onto Fig. 3, which is based on the minimization of variance with respect to stability participation within these subgroups. The match is almost perfect (compare Fig. 4 and Table 3); there are only two anomalies, Si and M. Si is part of both types of community stability substructures as is N1, yet Si's stability pattern resembles that of N2 which is part of only one type of substructure. M, while part of the more complex substructure pattern, is actually providing a connecting link between the two substructure patterns that comprise the stability substructure 4 and 5 (Fig. 4). M's substructure role is, therefore, different from that of the C2 members (Z2 and Z3) and its stability pattern resembles that of the C1 members. This connection was camouflaged by the abstraction made when constructing C2 in Fig. 3.
5. Non-trophic structure
(5)
Fig. 4. Community stability substructures, based on similarity of subgroup patterns found in Fig. 1.
The analysis so far has concentrated on model building based on traditional trophic relationships as seen in plankton community structure. We have shown that the simple trophic model (Fig. 2) is incompatible with our stabilityinstability analysis and that a more refined model (Fig. 3) more accurately describes the trophic relationships when viewed in the context of relative stability. However, the stability data can also be used in other ways to build a more complete picture of a community and the roles and relationships of the various entities. We can ask, for example, "How do the tradi-
81
tional trophic patterns (both the simple model and our m o r e refined model) relate to the six types of stability change that comes about following the removal of an entity (e.g. s - s, see above)?" Three new types of relationships emerge for each erLtity and its contribution to the overall system.
5.1. Relative participation This is the degree to which an entity participates in stable and unstable subsystems. Stabilizing is defined as the proportion of categories that are of the form s - ?, i.e. columns 1, 2 and 3 of Table 1; destabilizing is defined as the proportior~ of categories that are of the form u - ?, i.e. columns 4, 5 and 6 of Table 1. No large differences were found between the various entities. Relatively stable participation values vary from 60.2% to 72.1%; relatively destabilizing participation values vary from 27.9% to 39.8% (Table 1).
5.2. System necessary This is the degree to which removal of the entity results in fragmentation of the subsystem. This is defined as the proportion of categories that are of the form ? - d, i.e. columns 3 and 6 of Table 4. The system necessary grouping can be further subdivided into system necessary components that are relatively stabilizing (s d, i.e. column 3) and system necessary components that are relatively destabilizing (u - d, i.e. column 6). There are four entities that show very high fragmentation rates when removed from the subsystem (Table 4): nitrogen/phosphorus (N1) 92.6%, dinoflagellates (A2) 85.8%, diatoms (A1) 77.4%, and copepod adults 2 (Z3) 70.0%. The next highest percentage is 52.5%, and this 17.5% break is the largest found for adjoining entities. Eight of the fourteen remaining entities show percentages below 16% in this category.
TABLE 4 Entities ranked by system necessary categories (3 + 6). Entity
Nitrogen/phosphorus Dinoflagellates Diatoms Copepod adults 2 Immature copepods Miscellaneous algal groups Small flagellates Decapods Luxury-consuming diatcms Organic nutrients Silicate Polychaete larvae + Cirripeds Copepod adults 1 Mollusc larvae Cladocerans Oikopleura sp. Sagitta spp. Medusae
N1 A2 A1 Z3 Z2 A5 A4 D A3 N2 Si PC Z1 M Z4 O S MD
3 s - d
6 u - d
3+6
Trophic group
57.4 57.1 53.3 43.9 28.8 32.1 26.8 18.0 24.2 12.5 10.8 8.1 9.0 5.4 0.0 0.0 0.0 0.0
35.0 28.7 24.1 26.1 23.7 14.3 11.6 17.0 9.8 11.2 4.8 6.1 3.8 4.4 0.0 0.0 0.0 0.0
92.4 85.8 77.4 70.0 52.5 46.4 38.4 35.0 34.0 23.7 15.6 14.2 12.8 9.8 0.0 0.0 0.0 0.0
R2 P1 P1 C2 C2 P2 P2 O1 P2 R1 R1 O1 C1 C1 C1 C1 S1 S1
82
The first four entities can therefore be considered relatively system necessary and possibly play an important role in holding the plankton community together. Below these four entities are a transitional group of six entities with rates between 23.7% and 52.5%. The eight entities with rates 16% or lower can be considered relatively non-system necessary in that their removal does not often fragment the system. It is important to note that these relationships essentially hold whether or not the entity is relatively stabilizing or destabilizing. There is one exception: immature copepods (Z2) have as significant a contribution to fragmenting destabilizing systems upon removal, as do N1, A1, A2 and Z3. Rankings of system necessary entities readily fall into groups matching the more refined trophic grouping arrived at above (Table 4, last column). Only Decapods (D) need be moved in order for the entities of each of the eight trophic
TABLE 5a
TABLE 5b Entities ranked by contribution to instability (category 4). Entity
Luxury-consuming diatoms Miscellaneous algal groups Small flagellates Copepod adults 1 Copepod adults 2 Immature copepods Dinoflagellates Diatoms Sagitta spp. Mollusc larvae Polychaete larvae + Cirripeds Decapods Nitrogen/phosphorus Organic nutrients Silicate Medusae Oikopleura sp. Cladocerans
4 u - s
Trophic group
A3
11.5
P2
A5
10.0
P2
A4 Z1 Z3 Z2 A2 A1 S M PC
9.4 5.5 4.8 4.0 3.9 2.4 1.3 1.3 0.6
P2 C1 C2 C2 P1 P1 S1 C1 O1
D N1 N2 Si MD 0 Z4
0.5 0.0 0.0 0.0 0.0 0.0 0.0
O1 R2 R1 R1 S1 C1 C1
Entities ranked by contribution to stability (category 2). Entity
Oikopleura sp. Cladocerans Sagitta spp. Medusae Silicate Copepod adults 1 Polychaete larvae + Cirripeds Decapods Mollusc larvae Organic nutrients Luxury-consuming diatoms Copepod adults 2 Immature copepods Diatoms Nitrogen/phosphorus Dinoflagellates Miscellaneous algal groups Small flagellates
2 s - u
Trophic group
0 Z4 S MD Si Z1 PC
23.8 21.0 18.1 14.9 14.4 13.4 13.1
C1 C1 $1 Sl R1 C1 O1
D M N2 A3
10.6 10.3 7.5 3.8
01 C1 R1 P2
Z3 Z2 A1 N1 A2 A5
2.6 1.7 1.1 0.7 0.5 0.1
C2 C2 P1 R2 P1 P2
A4
+ 0.0
P2
groups to be listed consecutively. No such matching is found when simple trophic grouping is used.
5.3. Relative contribution to stability of the subsystem This is defined as the proportion of events that are either s - u, i.e. column 2 of Table 5a, or u s, i.e. column 4 of Table 5b. High scores in the category s - u can be regarded as evidence for relatively high contributions to subsystem stability, whereas high scores in the category u s can be regarded as evidence for relatively high contributions to instability. This is not the same measure as that found in (a) because in (a) it is possible for an entity to have a high participation rate in stable or unstable subsystems where removal does not result in a change in subsystem stability or instability. There are three patterns found for entities
83
regarding relative contribution to stability (Table 5a). Three entities Oikopleura sp. (0) 23.8%, cladocerans (Z4) 21.0% and Sagitta spp. (S) 18.1% show the highest percentages where removal from a stable subsystem results in instability for the remaining subsystem; seven entities show percentages in the range 7 . 5 14.9%; eight entities show percentages in the range 0 . 0 - 3.8%. There are three patterns found for entities regarding relative contribution to instability (Table 5b). Three entities, luxury consuming diatoms (A3) 11.5%, miscellaneous algal groups (AS) 10.0% and small flagellates (A4) 9.4%, show the highest ]percentages where removal from an unstable subsystem results in consequent subsystem stability; 9 entities show percentages in the range 0.5-5.5%; and 6 entities are at or near 0.0%. A good match is found between relative contributions to stability and the more refined trophic grouping (Table 5a,b, last column for each). The most interesting matching is found in the relatively destabilizing category (Table 5b) where entities of six of the eight trophic groups are listed consecutively.
90
/\
,2,.,o
80 70 60 50 40 30 20 10 i
0
i
2
3
i 4
i 5
i 6
i
i
i
i
i
7
8
9
10
11
12
7
8
9
10
11
12
Month
go 80
A 3 s -> s
70 60 %
50 40 3o 20 10 0 1
2
3
4
5
6 Month
Fig. 5. Examples of high and low variance over the annual cycle: (a) low variance; (b) high variance.
tities. However, no patterns relating to either simple or refined trophic or stability relationships were found.
6. Temporal sequences
6.1. Variance over the annual cycle
Changes in each of the measures discussed above were also followed over an annual cycle. These data show that, in addition to short term dynamics (switching, resting, disturbance), there are also long term seasonal patterns, which range from consistent stability relationships from month to month, to patterns showing extreme variation from one month to the next. In Lane's original analysis (Lane 1986), samples were taken at approximately 1 month intervals over a period of 1 year. It is therefore possible to follow Changes in the roles played by various of the entil;ies during a complete series of seasonal changes. Lane began her study in July 1974 (month 1) and concluded in May 1975 (month 12) and we have maintained this sequence for our ans~lysis. Viewed in this way, several different types of annual pattern can be seen for the various en-
In this analysis we used standard deviation rather than coefficient of variation because of the often low means seen in the various categories. In looking at the 55 single categories with means over 10%, six show a low variation (standard deviation one-third or less of the mean) and eight showed a high variation (standard deviation as high or higher than the mean). Figure 5 shows an example of each. For the six categories studied (i.e. s - s, s d, etc.), the average standard deviation of these annual values for A2 (7.0) and N1 (7.7) are very low with the remaining sixteen entities averaging between 11.1 and 16.5.
6.2. Parallel tracking This measure is based on the comparative change between two entities from one sample to
84
a
high
s->s
C
ooo Ioooooooo %
%
500 40.0 30.0 20.0 10.0 0.0
high
.~->d
•
50.0 40.0 30.0
2
~'
20.0
10.0 i
i
1
i
2
i
3
4
~
5
J
6
J
7
i
8
L
9
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10
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11
1
12
2
3
4
5
6
a
low
7
8
g
10
11
12
Month
Month
o.01 c
s->s
low
s->d
60.0 50.0
A1 & A 3 s -> s 70.0 60.0
A3
% 6o.o
%
A Z3
40.0
o
30,0
40.0 30.0 20,0 10.0
20.0
O
Z2
10.0 A1
0.0
--
= 1
0.0 2
I
3
4
5
6
7
8
g
10
11
2
3
4
5
6
7
8
10
9
11
12
Month
12
Month
0
high
S->u d
~
35.o
Si & Z l s -> u
u->s
A1 & A 2 u -> s
12.o
Zl
30.0
T
10.0
25.0
%
high
20.0
8.0 6.0
15.0
%
4.0
A1
/
A2 0--
10.0 2.0
5.0
0 0.0-
0.0 1
2
3
4
5
6
7
0
9
10
11
l
1
12
2
3
4
5
d low
.
9
10
11
8
9
10
11
12
low
u->s
0.0 o
Si&N2s->u
% %
8
o.o I z
s->u
4°Ql
7
Month
Month
b
6
20.0 15.0
15.0 10.0
Si
5.0
10,0
0.0
5.0
l
1
0.0 1
2
3
4
5
6
7
8
9
10
11
12
2
3
4
5
6
7
12
Month
Month
the next in the annual sequence. Parallelism is scored when both entities increase (or decrease) together in value for a particular category from one sample to the next, and non-parallelism is scored if the change is in the opposite direction. Overall, a much higher degree of parallelism was found than would be expected from pure
chance. The expected average is 5.5, whereas the actual average based on 146 sets of eight or more was 7.5. Figure 6 shows examples of high parallelism and low parallelism between entities. In addition, parallelism can be seen to extend over all six categories of removal. For example
85 e
high u->u
A1 & M, u -> u
f°
40.0
%
35.0 3O.O 25.0 20,0 15,0 10.0 5.0 0,0
M
. 1
; 2
;
;
3
4
; 5
;
.
6
.
7
f
i
8
Month
A1
g
10
11
12
high u->d
% 30,0 20,0 10.0 0.0
i 1
2
3
4
5
6
7
8
9
10
11
12
Month f
%
low u - > d
35°1,,,3u-,° ^ 20.0
1
/
15.0
O0
J 1
2
3
4
5
6
7
8
9
10
11
12
M onth
Fig. 6. E x a m p l e s of h i g h a n d low p a r a l l e l i s m in all c a t e g o r i e s o v e r t h e a n n u a l cycle. (a) s - s; (b) s - u; (c) s - d; (d) u -
s;(e) u--
u;(f) u-.
titles were also among those showing low variance and high parallelism and do not show any switching with regard to their roles over the annual cycle. These results are in contrast to that seen for entity Z2 (immature copepods), shown in Fig. 6c. In samples 2 - 6 and 11 and 12, Z2 shows high scores in the s - d category. However, in samples 1 and 7 - 1 0 the scores in this category are always lower than seen at any other time of the year. We have termed this type of change in behavior over the temporal cycle, role switching. Similar role switching is seen in other entities, for instance A4 (small flagellates). This entity, which only appears in samples 5 - 12, has values ranging from 49.9% to 50.0% for the combined categories s -- d and u - d, in samples 5 - 8 . These values for disconnectance fall to 0.0% for samples 9 - 1 2 . In a less seasonal manner, role switching is also seen for entity N2 (organic nutrients). Disconnectance values for s - d and u - d are always 0.0% except for samples 6, 9 and 12 when the values jump to 66.6%, 49.9%, and 83.3%, respectively. This role switching seems limited to individual entities and does not seem to extend to either simple or refined trophic groups. 7. Discussion
d.
high parallelism is seen for A1 + N1 (average 9) and low parallelism for A1 + A3 (average 4).
6.3. Role switching Complementing the patterns seen for total annual cycles, it is possible to observe interesting micro-patterns of behavior for a given entity at certain times of the year. Most entities exhibit little change in their stability-instability relationships. For instance, entities N1 (nitrogen/phosphorus ratio)and A2 (dinoflagellates) (Fig. 6c) remain highly system necessary for a full yearly cycle. These two en-
Previously, Pilette et al. (1990) have shown that complexity at the entity level is inversely related to the number of subsystems in which an entity participates, but that stability increases with increasing subsystem size both for the entities involved and for the subsystems themselves. It would appear therefore that large systems such as the plankton community in Delaware Bay are composed of various, ever changing combinations of smaller subsystems. Subsystems and their interactions thus become importi~nt modulators of overall system stability and resilience. Margalef and Gutierrez (1983) have also argued that discreteness (both temporal and spatial) is a stabilizing phenomenon. Large complex systems may be stable over significantly
86
long periods of time, but, embedded within these systems, in near decomposable fashion (Simon, 1962) there is a flux of subsystems, many of which may themselves be unstable. Thus, large biotic systems are composed of discrete but fluid sets of temporary associations among the various entities. Long scale analysis may assign certain traditional structural roles to entities within systems (i.e. diatoms - mollusc larvae); however, when viewed at shorter time scales, this link may be seen (for example) to hold 60% of the time. During the remaining 40%, the predator may establish other links (i.e. dinoflagellates - mollusc larvae). Such fluctuations in role performance may cause temporary changes in stability within the subsystem just vacated, but, for the system as a whole (which may be comprised of up to 8300 subsystems for these Delaware Bay plankton samples), this local change within one subsystem may be compensated for by changes in stability within other subsystems, hence leading to temporal equilibrium or moving equilibria (Bodini and Giavelli, 1989) in what appears to be a series of local non-equilibrium situations. In addition to the effects due to switching behavior, there are also effects due to resting and disruption to consider as well. In this study we have used the phenomena of switching, resting, and disruption as an investigative tool in an entity removal exercise. Initially, we searched for patterns related to trophic structure. Nitrogen and phosphorus are necessary resources; diatoms and dinoflagellates are producers; cladocerans and copepods are consumers; and the top carnivores in these plankton communities are decapods and Sagitta spp. When Lane (1986) was building a model to fit changes in abundance over an annual cycle, she grouped organisms in a structural manner such that species having identical links in her digraphs were placed in the same entity grouping. These groups are equivalent to guilds. In order to make her model accurate with regard to predicting the next system in her series, she found it was necessary to subdivide at least two groups: diatoms into diatoms and luxury consuming diatoms (the latter, probably aux-
otrophic requiring vitamin B12), and copepods into copepod adults 1 and copepod adults 2. Similarly, Andersen et al. (1987), while analyzing communities in enclosed water columns in a British Columbia inlet over a 40 day cycle, tried to model the basic parameters representing phytoplankton development. These authors found that the obvious simple model of nitrate - phytoplankton - herbivores - carnivores was not sufficient for them to predict phytoplankton development, and that this model required further subdivisions in order for it to have value. It appears, therefore, that simple models of trophic interaction or stability cannot always account for observed changes and that more detailed models are necessary. The results of our analysis of the Delaware Bay plankton community support these conclusions. Simple relationships between large groupings of these entities hide a more subtle series of roles played by various entities which are revealed when larger groups are split into smaller assemblages and when roles are examined over a complete annual cycle. The more complex stability-instability patterns are based on the minimization of variance within the various entity removal categories. The results indicate, for example, that there is more than one type of producer and more than one type of consumer. The observed stability-instability patterns match the more refined trophic structure (Table 3 and Fig. 3). The only anomaly of the 18 entities is Polychaete larvae + Cirripeds (PC), whose stability pattern more closely fits that of the Organic Nutrients (N2) and Silicate (Si) than that of Decapods (D), its relevant trophic group co-member (Table 1 and Fig. 1). In looking at the roles of relative contribution to stability (or instability) and system necessity, patterns also emerge for these non-trophic considerations. We can combine these roles with respect to contribution to subsystem stability and system necessity into a framework provided by Pilette (1989) as shown in Table 6. Core entities are considered relatively stabilizing and noncore entities relatively destabilizing. System necessary entities tend to result in fragmented
87 TABLE 6 Relative contribution to :~tability and system necessity a. Core b Not system necessary Somewhat system necessary System necessary
Outer core
Non-core
R1, O1
P2
S1,C1
R2,P1,C2
aSee text for discussion. bCore/outer core/non-core: ranked contribution to stability from most stabilizing to least stabilizing (core to non-core).
systems when removed, while those not system necessary tend to leave the system intact after removal. Again, as seen by Andersen et al. (1989), the more refined trophic grouping is required to tease out the various roles. Groups S1 and C1 are stabilizing, not: particularly destabilizing, and not system necessary. Groups R2, P1 and C2 are system necessary and make little contribution to either stabilization or destabilization. Groups R1 and O1 are moderately system necessary, stabilizing, and are not destabilizing. Group P2 is moderately system necessary, makes little contribution to stabilization, and makes the greatest contribution to destabilization. Thus, entities may be system necessary and yet relatively destabilizing. Pilette (1989, unpublished) has found this in an even stronger form: an entity may be required to hold a system or subsystem together and be highly destabilizing at the same time. We can make the useful observation that not only are a variety of patterns found for system necessity and relative contribution to stability, but, they fit rather closely those derived from trophic considerations and the more complex stability-instability patterns (Table 5a,b, last column). Examination of temporal sequences over a 12 month annual cycle shows that much is hidden
when role analysis is performed only on yearly averaged data. Resources such as N1 and producers such as A1 show a high degree of consistency over the annual cycle, contribute to subsystem stability, and are system necessary. This might have been anticipated from their trophic importance within the plankton community and from their biological role. The value of these entities can thus reasonably be gauged largely from yearly average scores in most categories. However, within primary consumers and omnivores, the yearly pattern for stability relationships shows various types of role switching. Both primary consumers C1 (low system necessary) and C2 (high system necessary) show one or more periods when roles, in this category, are reversed. This is often correlated with switches in relative contribution to stability and highlights what may turn out to be interesting biological phenomena taking place in the plankton community during these time periods. Omnivores are more opportunistic and these entities might indeed be expected to change trophic roles during an annual cycle. These entities, which are normally not system necessary, do show brief periods of role switching with regards to the plankton community stability. These results, however, do not map onto the refined trophic structure. As discussed above, the system necessary patterns and the relative contribution to stability patterns both match quite well with the complex stability patterns based on a minimization of variance. This complex stability pattern matches the refined trophic structure pattern. The refined trophic structure pattern is, thus, based on the minimization of variance within various stability categories. This refined trophic structure is composed of two substructures (Fig. 3). One of our most interesting results is that these abstracted trophic substructures match almost perfectly with the subtypes of the original community digraph for the annual cycle (Fig. 4). In a sense, we have used stability relationships as a probe into community structure. The entity removal exercise has allowed us to generate various categories regarding stability: group-
88
ings based on minimization of variance, system necessity and relative contribution to stability. Each of these categories of analysis maps poorly onto simplified trophic structure and maps very well on to more refined trophic structure. From our results, it would appear that ecologists can ill-afford to oversimplify community and trophic structure analysis.
Acknowledgements This research was supported in part by grants from NSF award number DMB 8612828 and the City University of New York PSC-CUNY Research Award Program numbers 6-67192, 6-67293 and 6-68186. We would like to thank Dr. S.N. Salthe for useful comments on an earlier version of this manuscript.
References Andersen, V., Nival, P. and Harris, R.P., 1987, Modelling of a plankton ecosystem in an enclosed water column. J. Mar. Biol. Assoc. U.K. 67, 407-430. Bodini, A. and Giavelli, G., 1989, The quantitative approach in investigatingthe role of species interactions on stability of natural communities. BioSystems 22, 289-299.
Conrad, M., 1983, Adaptability: The Significance of Variability From Molecule to Ecosystem (Plenum Press, New York). Cooper, E.I., 1990, Immune diversity thoughout the animal kingdom. BioScience 40, 720-722. Lane, P.A., 1986, Symmetry, change, perturbation and observing mode in natural communities. Ecology 67, 223 - 239. Levins, R., 1975, Evolution in communities near equilibrium, in: Ecology and Evolution in Communities, M.L. Cody and J.M. Diamond (eds.) (Harvard University Press, Cambridge) pp. 16-50. Margalef, R. and Gutierrez, E., 1983, How to introduce connectance in the frame of an expression for diversity. Am. Nat. 121,601-607. Pilette, R., 1989, System necessary populations that are destabilizing. Abstract: Annual Meeting of the International Society for Ecological Modeling. Pilette, R., Sigal, R. and Blamire, J., 1987, The potential for community level evaluations based on loop analysis. BioSystems 21, 25- 32. Pilette, R., Sigal, R. and Blamire, J., 1990, Stability complexity relationships within models of natural systems. BioSystems 23, 359-370. Puccia, C.J. and Levins, R., 1985, Qualitative Modeling of Complex Systems (Harvard University Press, Cambridge). Simon, H.A., 1962, The architecture of complexity. Proc. Am. Philos. Soc. 106, 467-482.