Stability-complexity relationships within models of natural systems

BioSystems, 23 (1990) 359--370 Elsevier Scientific Publishers Ireland Ltd.

359

Stability-complexity relationships within models of natural systems R. Pilette,

R . S i g a l a'* a n d J . B l a m i r e

Department of Biology and aDepartment of Computer and Information Sciences, Brooklyn College of CUNY, Brooklyn, N Y 11210 (U.S.A.)

(Received June 12th, 1989) Drawing on the qualitative loop analysis models prepared by Lane for a Delaware Bay plankton community, we evaluated 12 systems that ranged from 14 to 18 entities (population, guild or nutrient). Our approach was to study models of extended trophic biotic communities and examine the stability-complexityissue not only as it exists between systems (the traditional approach) but also with respect to the entities and relationships within a given system. We found no statistically significant inverse relationship for stability and complexity between systems. Within a system, a significant inverse relationship at the entity level was observed embedded in an increasing stability positively related to increasing subsystem size. Also, within a system and from system to system, several entities were seen to vary their roles with respect to stability. These results extend the stability-complexity issue to models of relatively large biotic communities and raise issues concerning the roles, with respect to stability, played by entities within communities. Keywords: Stability-complexityrelationships; Loop analysis; Delaware Bay plankton community; Entity role analysis.

Introduction T h e s t a b i l i t y o f large, c o m p l e x c o m p u t e r g e n e r a t e d s y s t e m s a p p e a r s to b e i n v e r s e l y r e l a t e d to t h e i r c o m p l e x i t y ( G a r d n e r a n d A s h b y , 1970; M a y , 1972, 1973). H o w e v e r , in this t y p e o f a n a lysis, t h e c o n n e c t i o n s b e t w e e n e n t i t i e s w i t h i n t h e s y s t e m w e r e c r e a t e d r a n d o m l y a n d n o a t t e m p t was m a d e to e n s u r e t h a t t h e s y s t e m g e n e r a t e d h a d a n y " r e a l i t y " . T h e s e r e s u l t s s t o o d as a " c a u t i o n " a g a i n s t a g e n e r a l l y h e l d b e l i e f o f ecologists at t h e time that a positive relationship between stability and complexity existed based on formal principles ( M a y , 1973). I n t r o d u c i n g r e q u i r e m e n t s o f feasib i l i t y ( R o b e r t s , 1974) o r p l a u s i b i l i t y ( D e A n g e l i s , 1975) in s e l e c t i n g s y s t e m s to s t u d y , led to r e s u l t s showing a positive correlation between stability and complexity. L a w l o r (1978), n o t i n g t h e v e r y low p r o b a b i l i t y of constructing a "reasonable" ecosystem using a Correspondence to: R. Pilette. *Present address: Dipartimento di Matematica, Universita di Catania, 95125 Catania, Italy.

random-number generator, argued that stabilityc o m p l e x i t y issues w o u l d b e b e t t e r a p p r o a c h e d b y s t u d y i n g s y s t e m s k n o w n to be " b i o l o g i c a l l y acc e p t a b l e " . C o n t i n u i n g this r e a s o n i n g , w h e n c o m p l e x i t y was d e f i n e d in t e r m s o f o r g a n i z a t i o n (blocks, etc.), he f o u n d t h a t " o b s e r v e d s y s t e m s " w e r e m o r e c o m p l e x a n d stable t h a n " a n a l o g o u s r a n d o m i z e d o n e s " for a g i v e n n u m b e r o f entities ( L a w l o r , 1980). H o w e v e r , w h e n c o m p l e x i t y was v i e w e d in t e r m s o f c o n n e c t a n c e , an i n v e r s e rel a t i o n s h i p b e t w e e n s t a b i l i t y a n d c o m p l e x i t y was seen. Such analyses represent a structural approach to t h e s t a b i l i t y - c o m p l e x i t y issue in that e n t i t i e s are first i d e n t i f i e d (or g e n e r a t e d b y a c o m p u t e r p r o g r a m ) a n d t h e n p l a c e d w i t h i n an i n t e r a c t i o n matrix. Alternatively, a functional (ecosystem) a p p r o a c h to this q u e s t i o n has b e e n t a k e n b y V a n V o r i s et al. (1980). T h e s e w o r k e r s e x a m i n e d experimental microcosms using power spectral analysis a n d f o u n d a p o s i t i v e c o r r e l a t i o n b e t w e e n stability and complexity. I n all o f these s t u d i e s , r e s u l t s were o b t a i n e d b y

0303-2647/90/$03.50 q3 1990 Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland

360

comparing one system to another and no attempt was made to evaluate stability-complexity relationships among the various entities within the individual systems. Even in the work of Van Voris et al. (1980) systems were compared to one another and the issue of internally stabilizing entities within a system was not addressed. Recently Giavelli et al. (1988) have used loop analysis to investigate the stability of small (maximum of four entities), computer generated "natural communities". T h e i r results support the argument that there is an inverse relationship between complexity and stability. Natural communities complexity analysis

÷

-

÷

>{

0(+) +

and stability

Our approach has been to study extended (up to 18 entities) models of trophic biotic communities and examine the stability-complexity issue not only as it exists between systems but also with respect to the entities and relationships within a given system. For this work, we chose to carry out our analysis on 12 annual estuarine plankton samples characterized by Lane (1986). These data follow the relationships among a group of populations, guilds and nutrients identified in samples taken from the Delaware Bay plankton community for the period July 1974 to May 1975. Lane (1986) constructed loop digraphs for each of these samples. One such digraph is shown in Fig. 1. T h e basis for each digraph constructed by Lane was that, under perturbation analysis, a series of sign changes relating to entity numbers would be generated for each entity that matched that actually observed in the sampling regime. For example, the numbers of the various entities making up sample 1 were found to change significantly for several of these entities for sample 2 taken approximately 1 m o n t h later in Delaware Bay. Lane constructed a community interaction matrix (digraph) for sample 1 which, when subjected to a particular perturbation, would indicate the subsequent gain, loss or insignificant change in numbers experienced by each entity as observed in sample 2. Lane completed this exercise for the annual cycle. Each digraph was stable

Fig. 1. Loop digraph for Delaware Bay System 2 (August 1, 1974, one of 12 for the annual cycle). See Table 2 for entity names. The --* represents the parameter input which generates the change in abundance value noted for each entity. The ( + ) for entity A5 represents the actual data-directed change which is in conflict with the model-directed change. (Adapted from Fig. 4, Lane, 1986.)

according to the rules of loop analysis. However, this was not a requirement of the modeling procedure itself. For the Delaware Bay community analyzed by Lane, a series of stable digraphs were most appropriate for modeling the actual observed sampling regimes. Complex communities may be constructed in such a manner as to allow for multiple stable states (May, 1977). Sutherland (1981) has argued, on the basis of empirical observations, for the concept of a single community exhibiting a series of stable states. In addition, Austin and Cook (1974) modeled simple systems showing that they may possess multiple stable states and, when perturbed, these systems can move from one equilibrium point to another in the "vector field vicinity". T h e stability of a system was calculated using the mathematical treatment known as loop analysis (Levins, 1974), the treatment of which has been presented often in the literature, (Levins, 1974, 1975; Puccia and Levins, 1985; Lane, 1986; Pilette et al., 1987; Giavelli et al., 1988). A distinction of loop analysis from other methods, which also use the determination of eigenvalues of

361 the c o m m u n i t y interaction matrix for stability determinations, is that all types of biological interactions are considered. T h i s is a distinction between "full loop analysis" calculations and that used with food webs (e.g. Yodzis, 1988). All these approaches are based on near equilibrium assumptions. Ecological systems, especially, are seen to possess both near and far from equilibrium properties. T h e justification for using the loop analysis model derives f r o m L a n e ' s success in " d e s c r i b i n g " the complex field observations through use of this model. In addition, loop analysis allows discrimination between stabilizing and destabilizing aspects of a system - - a situation in keeping with ecological observations and theory (Sutherland, 1981). It should be r e m e m b e r e d that the digraphs used in loop analysis reflect the modeler's best estimate of the relationships between the various entities and is a model of " a p p a r e n t interactions" (Taylor, 1985 and personal communication). L a n e ' s (1986) modeling is exceptional in that she was able to make a series of correct descriptions of changes in relative abundance for the n u m e r o u s entities involved based on her interaction matrices and thus give confidence to her digraphs. In the following discussion, by definition, an entity is a population, guild or nutrient; a system is one of the 12 samples; the c o m m u n i t y is the sample set for the annual cycle; interaction strength is set at + 1 or - 1 and self-interaction effects at - 1 . T h i s qualitative (or equivalent interaction strength) assumption m a y seem too simplistic, however, using this approach L a n e correctly described 173 of 187 (93%) of the changes in relative abundance in the observed samples. I n addition, using the Delaware Bay data, we are investigating the issue of qualitative versus quantitative interactions. Preliminary observations indicate that r a n d o m assignment of quantitative values within the matrix rarely resuits in changes of stability evaluations at the system level. Assignment of a " d o m i n a n t " entity does appear to effect the overall system stability, but is dependent on which entity is assigned dominance. Results at the subsystem level are being evaluated.

T o aid in these lengthy calculations, and also to extend the utility of this method, we have written a c o m p u t e r p r o g r a m , now i m p l e m e n t e d on the John von N e u m a n n Center C Y B E R 205 supercomputer, that carries out a loop analysis calculation for each connected subsystem within a given sample. T h i s m e t h o d (Pilette et al., 1987) of examining a system as a collection of subsystems is consistent with the reasoning of Margalef and Gutierrez (1983) that biotic systems are composed of sets of t e m p o r a r y associations among species, i.e. there is a natural discreteness within large systems. Such p h e n o m e n a as switching and resting are encompassed within the notion of discreteness.

Stability and complexity At the entity (population, guild or nutrient) level, 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. An actual subsystem is one where each entity is connected to at least one other entity of the subsystem. Each of these subsystems is feasible in that it represents connections modeled in a digraph and treated as temporarily discrete. Complexity was defined as the n u m b e r of subsystems in which an entity participated. T h u s , for sample C if we let ss(C,e,k) be the n u m b e r of subsystems of size k in which entity e participates and st(C,e,k) be the n u m b e r of stable subsystems of size k in which entity e participates, then complexity and stability are I7

c(C,e) = ~ ss(C,e,k)

(1.1)

k=l

s(C,e) = lO0 (k~= St(C,e,k))/c(C,e )

(1.2)

respectively, where N = n u m b e r of entities in sample C. At the system level, stability was defined as the percentage of all the connected subsystems of that system which were stable. Complexity was defined as the ratio of actual to possible subsystems for

362 t h a t p a r t i c u l a r system. T h u s if we let ss(C) be the n u m b e r o f c o n n e c t e d s u b s y s t e m s o f C a n d st(C) be t h e n u m b e r o f stable c o n n e c t e d s u b s y s t e m s o f C, t h e n s t a b i l i t y a n d c o m p l e x i t y are

s( C) = [st( C) / ss( C) ] lOO

(1.3)

c(C) = ss(C)/(2 N - 1)

(1.4)

respectively. I n a d d i t i o n to t h e classical s t a b i l i t y - c o m p l e x i t y issue, t h e s e statistics p e r m i t a d e t a i l e d analysis o f s t a b i l i t y - c o m p l e x i t y r e l a t i o n s h i p s at the. e n t i t y level. F o r e x a m p l e , t h e s t a b i l i t y value for a p a r t icular e n t i t y w i t h i n t h e s a m p l e can be f o l l o w e d t h r o u g h i n c r e a s i n g s u b s y s t e m sizes. I n a d d i t i o n , a c o m p a r i s o n can be m a d e o f an e n t i t y ' s s t a b i l i t y r a n k f r o m s a m p l e to sample. S u c h d a t a give i n f o r m a t i o n on t h e role a p a r t i c u l a r e n t i t y p l a y s w i t h i n t h e c o m m u n i t y . A l s o , overall t r e n d s o f t h e v a r i o u s entities w i t h i n s a m p l e s can be c o m p a r e d and evaluated. P r e v i o u s l y it has b e e n t h o u g h t t h a t l o o p analysis w o u l d b e r e s t r i c t e d to small s y s t e m s ( G i a v e l l i et al., 1988) b e c a u s e o f the difficulty o f determining the important interactions between entities. H o w e v e r , L a n e ' s careful c h a r a c t e r i z a tions o f t h e D e l a w a r e Bay p l a n k t o n s y s t e m allow us to a p p l y this a p p r o a c h to m o d e l s o f large, naturally occurring, communities.

Results

TABLE 1 Relationship between stability and complexity at the sample (system) level. Spearman Rank test (df = 10) r = 0.014.

I 2 3 4 5 6 7 8 9 10 11 12

Sample date

C (o~,)a

S (~,)b

Jul 19, 1974 Aug 1, 1974 Adg 21, 1974 Sep 17, 1974 Oct 15, 1974 Oct 30, 1974 Nov 14, 1974 Dec 13, 1974 Jan 16, 1975 Mar 9, 1975 May 9, 1975 May 28, 1975

4.6 10.5 7.2 5.1 3.2 3.8 5.1 9.3 3.7 10.8 6.2 5.1

74.1 66.8 57.7 64.8 53.3 56.0 83.7 44.8 80.3 68.1 99.2 47.4

a C= percentage which actually system. bS~percentage subsystems of stable.

of possible subsystems existed for that particular of all the connected that system which were

the c o r r e l a t i o n b e t w e e n t h e s e s t a b i l i t y a n d c o m p l e x i t y values. T h e r - v a l u e for t h e S p e a r m a n R a n k test was 0.014. U n l i k e p r e v i o u s s t u d i e s (see Introduction), which supported either a negative relationship or a positive relationship, our results ( b a s e d o n an analysis o f m o d e l s b a s e d u p o n field data) w o u l d s u g g e s t t h a t t h e r e is no statistically significant r e l a t i o n s h i p b e t w e e n s t a b i l i t y a n d c o m p l e x i t y at t h e s y s t e m level.

W e i n v e s t i g a t e d the f o l l o w i n g q u e s t i o n s : (1) W h a t was the relationship between stability and complexity at the sample (system) level? E a c h o f t h e 12 s a m p l e s were e v a l u a t e d for s t a b i l i t y a n d c o m p l e x i t y u s i n g t h e s y s t e m level criteria d i s c u s s e d above. A s can be seen in T a b l e 1, t h e r e is a c o n s i d e r a b l e r a n g e in values f o u n d for b o t h the s t a b i l i t y a n d c o m p l e x i t y m e a s u r e s . T h e value for s t a b i l i t y ranges f r o m 44.8°o to 99.2°0, differing b y m o r e t h a n a factor o f 2, while the values for c o m p l e x i t y r a n g e f r o m 3.2°,o to 10.8°o, differing b y m o r e t h a n a factor o f 3. T h e S p e a r m a n R a n k test was u s e d to e s t i m a t e

(2) W h a t was the relationship between stability and complexity at the entity level? U s i n g t h e e n t i t y level criteria as defined a b o v e , we d e t e r m i n e d t h e p e r c e n t a g e o f stable s u b s y s t e m s a n d t h e n u m b e r o f total s u b s y s t e m s for each s a m p l e ( T a b l e 2). I t can be seen that t h e m e a n s t a b i l i t y f r o m s a m p l e to s a m p l e ( T a b l e 1) varies c o n s i d e r a b l y m o r e t h a n the r a n g e o f values w i t h i n a g i v e n s a m p l e , a n d in no case d i d values w i t h i n a s a m p l e differ b y m o r e t h a n a factor o f 2 (usually m u c h less). O n t h e o t h e r h a n d the r a n g e o f c o m p l e x i t y values w i t h i n each s y s t e m shows c o n s i d e r a b l e

S

PC M MD D

0

Si N1 N2 A1 A2 A3 A4 A5 Z1 Z2 Z3 Z4

Silicate Nitrogen/phosphorus Organic n u t r i e n t s D iat om s Dinoflagellates L u x u r y - c o n s u m i n g diatoms Small flagellates Miscellaneous algal groups Co pep od adults 1 I m m a t u r e copepods Co pep od adults 2 Cladocerans Oikopleura sp. Polychaete larvae + cirripeds Mo llu s c larvae Med u s ae Decapods Sagitta spp.

729 1431 641 1344 1280 1008 . 956 675 676 644 505 479 -675 323 963 -. 69.4 79.6 73.2 73.4 88.1 84.6 -82.1 82.7 76.5 .

82 74.4 82.5 75.8 73.4 67.7

1677 3009 1401 3200 2800 1752 . . 2008 1685 2472 1869 . . 1005 1602 1602 1237 1401 . . . 62.6 65.9 66.4 69.4 . 82.1 73.8 71.7 73.3 71.4 .

55.5

78.7 69.2 71.4 67.5 66.9

S(°o)

C

C"

S (% ) b

2 A ug 1, 1974

1 July 19, 1974

. -529 509 491 575 .

267 820 509 683 1016 532 . 411 795 1056 980

C

.

.

-69.4 62.7 60.7 60.2

61.6 47.4 53.4 54.9

71.5 59.5 62.7 67.3 60.3 71.1

S(%)

3 A ug 21, 1974

.

. 2460 1813 2560 2398 1231 1231 1281 1206 1200 2410

1193 3070 1302 1996 2602 2384

C

64.9 67.5 60.3 69.1 64.8 64.9 71.9 68.1 69.7 64.8

88.2 64.8 66.5 60 66.2 69.6

S (%)

4 Sept 17, 1974

4811 7767 3393 6604 6784 4992 5180 5180 3752 6790 7080 2591 2591 4754 4692 3541 5925 3541

C

58.6 52.5 58.4 52 51.3 48.1 48.3 48.3 60.9 50.9 51.4 60.9 60.9 52.6 57.1 68.5 56.8 68.5

S(%)

5 Oct 15, 1974

2161 4293 3543 3784 4720 2752 2864 2364 2521 3360 4356 1183 1433 2653 2361 2180 -2180

C

64.7 55.9 58.5 57.3 56.5 49.8 55.9 56.6 56.3 45.9 55.1 62.3 56 79.3 62.2 58.7 -53

S(%)

6 Oct 30, 1974

Percentage of stable s ub s ys t e ms and the total n u m b e r of s ubs ys t e ms for each sample Delaware Bay P l a nkt on C o m m u n i t y . (See Lane, 1986 for en tity characterization and terminology).

TABLE 2

Silicate Nitrogen/phorphorus Organic nu tr ien ts D iato m s Dinoflagellates L u x u r y - c o n s u m i n g diatoms Small flagellates Miscellaneous algal groups C ope po d adults 1 I m m a t u r e copepods C ope po d adults 2 Cladocerans Oikopleura sp. Polychaete larvae + cirripeds Mollusc larvae M ed us ae Decapods Sagitta spp.

85 83.5 87.6 83.6 83.1 83.6 77.3 77.3 85 85 87.6 95.9 95.9 85 85 87.6 -87.6 -

. . 1042

-

977 1431 977

1600 1952 1952 1431 1840 2082

2392

55 47.4 54.8 -. 66.4

. 191 380 191 . --

233 560 380 513 567 464 281 191 234 234 380 .

45.5 44.9 53.7 46.5 39.2 44.6 39.9 39.8 47.4 45.2 33.2

1425 2925 1197 2400

3301 6633 3021 6384 6040 3384 4424 4424 3194 3194 4836 2213 2213 3193 3193 2419 -2419 --

90.6 74.7 88.5

89.7 80.4 80.5 79.5 79.5 78.4 80.4 80.6 89.7 89.7 74.2 .

. 1696 1588 -1615 1078

2214 1743 1743 1696 2016 2154

3228

1750 3484 1615 2956

C

. 68.5 69.5 -68.8 68.8

78.8 68.2 68.8 67.3 68.2 68.7 68.2 68.2 67.8 63 68.7

S(%)

10 M a r 9, 1975

998 989 -846 621

1536 1890 845 1976 1688 1085 946 769 1240 1310 846

C

99.4 100

1651 1315

1634 1625

1531 2875 2790 2142 3248 2178 1396 1862 1090 1796 2628 932

100 99.6 100 99.3 99.4 100 99.6 100 99.3 99.1 100 -99.6 100

c

71.9 54.6

56.6 57.4

59.7 44.4 46.7 43.5 46.6 41.7 46.7 46.7 57,2 49.2 45.3 46.8

s(oo)

12 M ay 28 1975

s(%)

11 M a y 9, 1975

C = total n u m b e r of subsystems par ti c i pa t e d in by each entity for that p a r t i c u l a r system. b S = percentage of all connected s u b s y s t e m s which were stable pa rt i c i pa t e d in by each entity for that particular system.

S

PC M MD D

0

Si N1 N2 A1 A2 A3 A4 A5 Z1 Z2 Z3 Z4

S (% )

C

S(%)

c

C

S(°0)

9 Jan l 6, 1975

8 D e c 13, 1974

7 N o v 14, 1974

365

variation. While differing by a factor of 3 between maximum and minimum values in most cases, there are several instances of a larger factor difference (see systems 1, 3, and 6). We also found that the less connected an entity was to the overall system, the greater the percentage of stable relationships of which it was a part. These results would be accounted for, in part, by entity self-regulating effects since, on the average, 61% of the entities of any given system were selfregulating while 75% of the less connected entities are self-regulating. This inverse relationship between stability and complexity was significant, at the P < 0.05 level, for 11 of the 12 Delaware Bay plankton samples. Table 3 gives the results of the Spearman Rank test as applied to the data in Table 2. T h e statistical significance holds under a variety of stability and complexity conditions, even that of the May 9, 1975 sample where the stability approached 100%. T h e s e results support the proposition of an inverse relationship between stability and complexity, but at the entity rather than the system level. (This inverse relationship held in simulations of systems level instability so that the stability of the systems as a whole does not seem to account for the entity level inverse relationship between stability and complexity).

(3) Was there a relationship between the subsystem size and the role with respect to stability played by an entity? Stability increases with increasing subsystem size, a trend that holds for the entities involved as well (Table 4). F o r example, as seen in the August 1, 1974 sample (system 2), the total subsystems of a particular size in which an entity participated (i.e. ss(Cz,e,k), 1 <~k <~ 15) and the percentage of these that were stable (i.e. 100 st(C2,e,k)/ss(C2,e,k)) was used to calculate the degree of stability for each entity for each subsystem size (Table 4b). T h e range among entities in average stability varies from 55.6% to 82.2%. T h e average stability of all entities of a given subsystem size increase consistently as the subsystem size increases (from size 3 onwards for this example). T h e s e results need to be viewed in the context of the system-level stability that was found for each of the 12 Delaware Bay samples. (Not surprisingly, in simulations of system-level instability, stability decreases with increasing subsystem size). This stability was not required in Lane's models but is a phenomena observed as a result of the most useful modeling procedure for "describing" the actual field observations.

TABLE 3 Significance test for stability-complexity relationships (entity level). Delaware Bay Plankton Community. Spearman rank test.

1 2 3 4 5 6 7 8 9 10 11 12

Sample date

Subsystems (% stable)

df

r-value

Significance (P)

Jul 19, 1974 Aug 1, 1974 Aug 21, 1974 Sep 17, 1974 Oct 15, 1974 Oct 30, 1974 Nov 14, 1974 Dec 13, 1974 Jan 16, 1975 Mar 9, 1975 May 9, 1975 May 28, 1975

1494 3456 1183 3333 8344 5018 6695 3055 604 3553 2027 3341

13 13 12 14 16 15 15 13 12 13 13 14

-0.691 -0.675 -0.674 -0.541 -0.725 -0.380 -0.797 -0.787 -0.726 -0.523 -0.568 --0.649

<0.005 <0.006 <0.009 <0.031 <0.001 <0.132 <0.001 <0.001 <0.004 <0.046 <0.028 <0.007

(74) (67) (58) (65) (53) (56) (84) (45) (80) (68) (99) (47)

66.7 52.6 60 58.8 70 65 94.4 72.2 71.4 61.1 76.5 66.7

51.6 42.5 51.9 58.6 59.4 51.6 82.9 55.9 84.2 65.7 85.7 58.1

3 69.1 56.3 58.8 57.1 61 55.9 87.3 61.3 82.8 69.7 94.1 42.1

4 66 59.4 55.4 57.8 51.9 53.4 79.2 49.7 81 68.2 97.8 40

5 67.4 59.6 49.3 58.4 54.8 49.8 76.4 44.7 77.2 68.2 98.7 35.5

6 68.3 60.6 47.6 56.5 48.5 49.1 76.9 40.8 75.3 67.3 99.6 35.8

7 70.3 63 48.8 58.3 47.3 49.9 79.3 39.5 76 66 100 38.2

8

Si N1 N2 A1 A2 A3 A5 Z1 Z2 Z3 0 PC M MD D Average

50 75 100 14.3 100 33.3 50 50 50 50 0 50 50 100 100 52.6

2

28.6 62.5 100 24 60 11.1 25 37.5 41.7 71.4 100 16.7 16.7 100 100 42.5

3

Su b s ys te m size

66.7 63.3 71.4 54.5 58.6 26.9 53.3 47.6 54.1 65 66.7 52.6 57.9 77.8 71.4 56.3

4

69.2 61.9 72.7 59.7 57.6 35 43.5 52.9 58.5 62.7 83.3 63.8 63.8 75 72.7 59.4

5

71.8 61.3 69.8 60 57.1 39.3 43 57.3 58 61.9 67.6 65.7 61.8 71.2 69.8 59.6

6

71 62.5 65.5 61.5 58.1 43.3 48 57.2 58.8 61.4 72.5 67 63.8 69.3 65.5 60.6

7

(b) System n u m b e r 2 (Aug 1, 1974) (see T a b l e 2 for entity key).

1 2 3 4 5 6 7 8 9 10 11 12

2

Su bs y s tem size

72.8 64.8 65 63.8 61 46.9 53.7 59.9 60.9 64.2 75.5 68.8 65.9 68.2 65 63

8

75 67.1 57.5 60.9 45.8 52 81.6 40 78.2 65.5 100 41.6

9

77.5 68.5 67.2 67.8 65.9 53.9 60.6 63.3 65 67.9 78.5 72.1 70 69.1 67.2 67.1

9

78.8 72.9 70.9 64.6 46.4 54.5 83.6 43.4 82.1 67.1 100 46.3

10

82.9 73.6 71.6 73.2 71.9 63.2 68.7 69.7 70.7 73.4 83.2 77.1 75.3 72.6 71.6 72.9

10

84 79.3 83.3 70.9 48.5 57.8 85.8 49.1 88.7 71.1 I00 53.1

11

88.1 79.5 77.4 79.3 78.8 72.4 76.9 76.7 77.6 78.9 88.6 82.7 81.2 77.5 77.4 79.3

11

91 86.5 93.5 79.4 52.9 61.8 88.4 59.7 95.8 76.8 I00 62.9

12

93.2 86.4 84.3 86.4 86.4 82.5 85.5 84.5 85.6 84.5 93.6 89.2 88.2 84.2 84.3 86.5

12

96.8 93.6 100 88 59.7 66.3 91.6 74.5 100 83.9 I00 76

13

Role analysis data s ho w ing percent stability for each s ubs ys t e m size. D e l a w a re Bay P l a nkt on C o m m u n i t y . (a) All systems

TABLE 4

97.4 93.6 92.1 93.6 93.6 92.1 93.5 92.1 93.5 92.1 97.3 94.7 94.7 91.9 92.1 93.6

13

100 100 100 95.1 69.3 72.8 95.2 90 100 91.7 100 90.6

14

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

14

100 100 100

100 82.3 81.1 98.2 100

100 100

15

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

15

100

100 94.5 90.1 100

16

78.7 69.2 71.4 67.5 67 55.6 62.7 65.9 66.5 69.4 82.2 73.8 71.8 73.3 71.4 66.8

Average ( %)

100 100 100

17

100

18

o~

367

An entity can play both stabilizing or destabilizing roles within the system or subsystem. For example, in system 2, mollusc larvae (M) for subsystem size 6 participates in 102 subsystems of which 61.8% are stable (Table 4). But that still leaves over 38% of the subsystems of size 6 in which it participates that are unstable. T h e role that an entity plays with regard to stability often changes with subsystem size. Some, like Oikopleura sp. (Fig. 2a), consistently play a stabilizing role. Apart from the lowest subsystem size, the stability for this entity is always above the average for that particular subsystem size. Others, such as the " l u x u r y " consuming diatoms (Fig. 2b), are relatively destabilizing in that their stability is always below the average for any given subsystem size. Several entities have stabilities which follow the average quite closely (e.g. diatoms, from subsystem size 4 onwards, Fig. 2c). While a few entities, such as medusae (Fig. 2d), vary in their role depending on the subsystem size. We are currently evaluating the consistency of these observations for the remaining 11 Delaware Bay systems.

(4) Did the stability rank for each entity vary from sample to sample? T a b l e 5 gives the ranks, with respect to stability, from sample to sample for each entity for the overall Delaware Bay model. Based on the mean value, the medusae (Fig. 3a), Sagitta spp., Oikopleura sp., cladocerans, mollusc larvae, polychaete larvae and cirripeds, silicate and organic nutrients play stabilizing roles. T h e various species of algae (including dinoflagellates, Fig. 3b), immature copepods and nitrogen/phosphorous are destabilizing, while the remaining entities play moderate roles (e.g. copepod adults type 1, Fig. 3c) with respect to stability. In addition, when the coefficient of variation is examined (Table 5), Sagitta spp., Oikopleura sp., and cladocerans can be seen to vary their role with respect to stability from sample to sample, while dinoflagellates, small flagellates and nitrogen/phosphorus vary little in their relative stability ranking from sample to sample.

l e60 so

40

./

,,

,,

,

8, loo ~

80

g.

6o 40 20

2

4

6

8

10

12

14

2

4

6

8

10

12

14

Subsystem Size Fig. 2. Stability related to subsystem size. O - - ~ , average stability for that subsystem size; O--O, entity. (a) Oikopleura sp.; (b) "luxury" consuming diatoms; (c) diatoms; (d) medusae.

368

TABLE 5 Coefficient of variation and rank from sample to sample for each entity. Delaware Bay Plankton C o m m u n i t y (see Ta bl e 2 for entity key). 1

2

3

4

5

6

7

8

9

Jul 19, 1974

Aug 1, 1974

Aug 21, 1974

Sep 17, 1974

Oct 15, 1974

Oct 30, 1974

N ov 14, 1974

Dec 13, 1974

Jan 16, 1975

6 10 4 9 11.5 15 -14 7 13 11.5

2 9 6.5 10 11 15 -14 13 12 8

1 11 5.5 4 9 2 -7 14 13 12

--

--

Si N1 N2 AI A2 A3 A4 A5 ZI Z2 Z3 Z4 O PC M MD D

2 -5 3 8

S

- -

- -

- -

10 M a r 9, 1975

11 M a y 9, 1975

12 May 28, 1975

Si N1 N2 AI A2 A3 A4 A5 Z1 Z2 Z3 Z4

1 10.5 4 14 10.5 6.5 10.5 10.5 13 15 6.5 --

4 9 4 13.5 11.5 4 9 4 13.5 15 4 --

2 14 10 15 12 16 10 10 4 7 13 8

O

- -

- -

1

PC M M

D S

D

1 3 5 4 6.5

8 2 - -

- -

9 4

5 3

- -

4 4

-3 5.5 8 10

- -

11.5 4

1 6

1

13 8 16 9 4 --

10.5 7 15 5 13 10.5 2 6 3 13 --

6

2

9

8

3

11 7 12 14 18 16.5 16.5 4 15 13 4 4 10 8 1.5 9 1.5

12.5 6 7 9 16 12.5 8 t0 17 14 3 11 1 4 5 . 15

14 4.5 12.5 15 12.5 16.5 16.5 9 9 4.5 1.5 1.5 9 9 4.5 . 4.5

10 4 7 14 11 12 13 5.5 9 15 -2 5.5 3 -.

8.5 7 10.5 10.5 12 8.5 6 3 3 14 --1 13 5

.

Average

Rank

Range

3.75 11.04 5.88 10.88 11.42 11.01 11.94 10.83 8.58 11.92 10.04 5.08 4.57 5.14 5.63 4.25 7.88 5.14

1 15 8 13 16 14 18 12 10 17 11 4 3 5.5 7 2 9 5.5

1--9 8.5--14 4--10 4--16 9--15 2--18 8.5--16.5 4--16.5 3--14 3--17 4 - - 15 1 - - 13 1 - - 11 1--10 2 - - 13 1.5--8 1- - 13 1--15

75.50 17.40 32.70 33.50 17.90 50.10 26.20 37.60 47.40 34.60 41.30 90.70 94.60 66.00 54.80 45.40 50.20 91.00

ities. Choice not we used computerized

lysis to investigate

--

Coefficient of Variation (%) n

Discussion

In this work

1

large, natural,

loop ana-

biotic commun-

arbitrary.

12 12 12 12 12 12 8 12 12 12 12 6 7 11 12 8 8 7

of the data to use in this study Lane's

ization of large plankton in that

(1986)

careful

communities

she was able to make

was

characterwas notable

a series of correct

369

0

2 4 6 8 ib 1 2' 4' 6' 8' i0 '

28 4'' 6' '

~3 1

(a)

Dates

(c)

Fig. 3. Stability rank from sample to sample. (a) Medusae; (b) dinoflagellates; (c) copepod adults type 1. descriptions of changes in relative abundance for the n u m e r o u s entities involved. T h i s enabled us to get beyond the assumed limits of loop analysis with respect to system size (Giavelli et al., 1988). In previous work, at the system level, the original inverse relationship between stability and complexity ( G a r d n e r and Ashby, 1970; May, 1972, 1973) was often lost when requirements of reasonableness were imposed (Roberts, 1974; DeAngelis, 1975). However, in these cases, the systems analyzed consisted of c o m p u t e r generated, de novo, models, examined purely at the macro level. Working with large matrices, based on models derived from field data studies, our near z e r o correlation between stability and complexity at the system level, falls somewhere between the negative and positive relationships found by others. T h i s indicates that, when dealing with natural communities, the theoretical sets of relationships which might be expected to give either a positive or negative correlation between stability and complexity, m a y indeed by tempered by other factors which blur such sharp distinctions. It is therefore important to isolate and investigate these other factors and look at levels of interaction below that of the entire system when taken together. At the entity level we found an inverse relationship between stability and complexity to hold for these reasonable and large systems. In addition, for stable systems, stability increases with increasing subsystem size both for the entities involved and for the subsystems themselves. T h u s larger systems are composed of a combination of smaller subsystems plus additional link(s) (see Fig. 4). T h e inverse relationship at the entity level has to be viewed in the context of subsystem interactions.

(b)

Fig. 4. Larger systems composed of smaller subsystems plus additional links. (a) Unstable; (b) unstable; (c) stable. Margalef and Gutierrez (1983) argued that discreteness (both temporal and spatial) is a stabilizing p h e n o m e n o n in an attempt to account for the discrepancy between the theoretical work and the observed stability of large natural systems. O u r results indicate that the p h e n o m e n o n of discreteness may have broader implications and may very well allow for the existence of areas of stability and instability within an otherwise stable system. T h i s precludes the need for accounting for the previously perceived contradiction. A large, complex, stable system can have e m b e d d e d within it, in near decomposable fashion (Simon, 1962), a flux of subsystems, m a n y of which are unstable. T h e relationship between the contribution to stability that an entity makes and its necessity to the system is also being investigated. Preliminary results indicate that it is possible for an entity to be necessary to a system (in that its removal results in the loss of other entities as well) and yet to be a relatively destabilizing presence itself. An important reason for carrying out investigations such as this one is to elucidate underlying mechanisms in ecological systems over time. T h e questions asked under (3) and (4) above regarding the role played by an entity within a system based on substystem size and the role played by an entity from system to system should eventually be viewed over the annual cycle of the community. For example, the concept of near decomposable discreteness could be one way of accounting for the observed stability of natural systems during

370

seasonal changes where entities change roles, enter, and leave the system in dramatic ways. We are therefore currently analyzing a complete set of " w i t h i n - s y s t e m " changes, of which T a b l e 4 is an example, and are exploring seasonal trends over an annual cycle such as that shown in T a b l e 5. It is also possible that in addition to the intrinsic interest in elucidating relationships both within a system and between systems separately, it may very well be important to see if " w i t h i n - s y s t e m " changes affect or are affected by "betweensystem" changes. T h e method we have described above allows us to undertake this "role-analysis" as well as the more traditional stability/ complexity analysis.

Acknowledgments T h i s research was supported in part by grants from N S F award no. D M B 8612828 and T h e City University of N e w York P S C - C U N Y Research Award Program n u m b e r s 6-67192 and 6-67293. We would like to thank Dr. P.A. Lane for consultations and providing the digraphs used in this and other studies, Dr. R. Levins for his helpful discussions, and Drs. D. DeAngelis, R. May, S.N. Salthe and P. T a y l o r for useful comments on an earlier version of this manuscript.

References Austin, M.P. and Cook, B.G., 1974, Ecosystem stability: A result from an abstract simulation. J. Theor. Biol. 45, 435--458. DeAngelis, D.L., 1975, Stability and connectance in food web models. Ecology 56, 238--243. Gardner, M.R. and Ashby, W.J., 1970, Cormectance of large dynamical (cybernetic) systems: critical values for stability. Nature 228, 784.

Giavelli, G., Rossi, O. and Siri, E., 1988, Stability of natural communities: loop analysis and computer simulation approach. Ecol. Model. 40, 131--143. Lane, P.A., 1986, Symmetry, change, perturbation and observing mode in natural communities. Ecology 67, 223-239. Lawlor, R.L., 1978, A comment on randomly constructed model ecosystems. Am. Nat. 112, 445--447. Lawlor, R. L., 1980, Structure and stability in natural and randomly constructed competitive communities. Am. Nat. 116, 394--408. Levins, R., 1974, Quantitative analysis of partially specified systems. Ann. N.Y. Acad. Sci. 231, 123--138. Levins, R., 1975, Evolution in communities near equilibrium, in: Ecology and Evolution of 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. May, R.M., 1972, Will a large complex system be stable? Nature 238, 413--414. May, R.M., 1973, Stability and Complexity in Model Ecosystems (Princeton University Press, Princeton, NJ). May, R. M., 1977, Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature 269, 471--477. Pilette, R., Sigal, R. and Blamire J., 1987, The potential for community level evaluation based on loop analysis. BioSystems 21, 25--32. Puccia, C.J. and Levins, R., 1985 Qualitative Modeling of Complex Systems (Harvard University Press, Cambridge, MA). Roberts, A., 1974, The stability of a feasible random ecosystem. Nature 251,607--608. Simon, H. A., 1962, The architecture of complexity. Proc. Am. Phil. Soc. 106, 467--482. Sutherland, J.P., 1981, The fouling community at Beaufort, North Carolina: a study in stability. Am. Nat. 118, 499-519. Taylor, P., 1985, Construction and turnover of multispecies communities: a critique of approaches to ecological complexity, Harvard University, Thesis. Van Voris, P., O'Neill, R.V., Emanuel, W.R. and Shugart, H.H., Jr., 1980, Functional complexity and ecosystem stability. Ecology 61, 1352--1360. Yodzis, P., 1988, The indeterminacy of ecological interactions as perceived through perturbation experiments. Ecology 69, 508--515.