J. theor. Biol. (2002) 215, 441–448 doi:10.1006/jtbi.2001.2523, available online at http://www.idealibrary.com on
Species Positions and Extinction Dynamics in Simple Food Webs Ferenc JordaŁ n*wz, IstvaŁ n Scheuringy and GaŁ bor Vidaw ! ! u. 2., Hungary Collegium Budapest, Institute for Advanced Study, Budapest H-1014, Szentharoms ag wDepartment of Genetics, Evolutionary Genetics Research Group, Eotv . os . University, Budapest H-1117, ! any ! P. s. 1/c, Hungary and y Department of Plant Taxonomy and Ecology, Research Group of Pazm ! any ! P. s. 1/c, Hungary Ecology and Theoretical Biology, Eotv . os . University, Budapest H-1117, Pazm n
(Received on 22 June 2001, Accepted in revised form on 18 December 2001)
Recent investigations on the structure of complex networks have provided interesting results for ecologists. Being inspired by these studies, we analyse a well-defined set of small model food webs. The extinction probability caused by internal Lotka–Volterra dynamics is compared to the position of species. Simulations have revealed that some global properties of these food webs (e.g. the homogeneity of connectedness) and the positions of species therein (e.g. interaction pattern) make them prone to modelled biotic extinction caused by population dynamical effects. We found that: (a) homogeneity in the connectedness structure increases the probability of extinction events; (b) in addition to the number of interactions, their orientations also influence the future of species in a web. Since species in characteristic network positions are prone to extinction, results could also be interpreted as describing the properties of preferred states of food webs during community assembly. Our results may contribute to understanding the intimate relationship between pattern and process in ecology. r 2002 Elsevier Science Ltd. All rights reserved.
Introduction Interesting ideas in ecology frequently come from distant fields of science. The conclusions of recent studies on the structure of complex networks (Watts & Strogatz, 1998; Baraba! si & Albert, 1999; Albert et al., 2000; Jeong et al., 2000; Amaral et al., 2000; Williams et al., submitted), giving information about how metabolic and neuronal networks seem to be organized, may also be useful for ecologists. An interesting result is that metabolic networks follow a power-law connectivity distribution
zAuthor to whom correspondence should be addressed. E-mail:
[email protected] 0022-5193/02/$35.00/0
indicating robust functional insensitivity against random errors (Jeong et al., 2000). The dynamical behaviour of a well-defined set of small sink webs (Cohen, 1978) has been studied (Jorda! n & Molna! r, 1999; Jorda! n et al., submitted), giving rise to the problem of how the network position of frequently extinct species can be characterized. Since the large-scale structure of biological networks depends highly on internal dynamical processes and constraints (Sugihara, 1984), we analyse how the internal extinction dynamics (based on Lotka–Volterra models) in these webs is related to the topology of trophic interactions. We think that these basic equations, even if their applicability is limited, are good as a starting point for further analyses. r 2002 Elsevier Science Ltd. All rights reserved.
442
F. JORDA´N ET AL.
Methods
The orientation of neighbours is characterized by D0 ¼ DinDout (where Din is the in-degree: the number of links directed to a particular node, and Dout is the out-degree: the number of links coming from a particular node). Low and high values of D0 characterize multiple exploited producers and generalist consumers, respectively, in the directed graph. The concept of ‘‘net status’’ (Harary, 1959, 1961) has been modified slightly in order to be ecologically more reasonable (see Jorda! n et al., 1999; Jorda! n, 2000, 2001). The modified index was constructed to give a quantitative measure of the importance of species (species of outstanding importance are called ‘‘keystone species’’, Paine, 1969; Power et al., 1996). This is the positional keystone index (K ) which considers not only neighbours in the web (like D) but their neighbours as well, thus providing more information about network structure. If bottom-up and top-down forces are considered to be of equal importance, then let K ðiÞ ¼ KbðiÞ þ KtðiÞ ; where KbðiÞ and KtðiÞ refer to the bottom-up
We compared the extinction dynamics of species constituting 25 model sink webs (i.e. food webs with a single top-predator, see Fig. 1). This set of webs contains all of the 25 topologically different model sink food webs possibly built up from five species (graph nodes) and five trophic interactions (links). Since the connectedness of these webs equals uniformly C ¼ 0.5, we are able to analyse the effect of link pattern on population dynamics, distinguished from the effects of changing connectedness. Food webs and individual species are characterized by the following structural and dynamical properties. The degree (D) of a graph node is the number of its neighbours (both prey and predators in the food web graph). Because all these webs contain five links, the sum of D values (sumD) equals 10 for every web. However, the D values of individual species differ; this difference is measured by the standard deviation of D within a web (stdD). 1
1
2
1
3
2
2 5
4
3
1 2
5
4
3
4
3
1
5
2
2
4
5
1
2
1
8
9 1
10
2
2
2
3
5
4
14
5
4
4
3
15 1
1
2
3
5
4
22
1 2
3
4
1 24 2
3 4
4 5
13 1
1
12
2
2 3
3 5
4 18 1
5
4 1
19
4
2
3
1 25
3
4
5 1
5
5
20 1
2
3 4
5 23
1
2
21
1
5
4
2
5
17 2
5
4
5
5
3
1
2
3
3
3
11
4
1
16
2
2
4
1
5
3
5
2
3
4
5
3
3
6 1
3 4
7
1
4 5
2
2
3 3 4 5
3
4
4
5
5
Fig. 1. Twenty-five model sink webs containing five species (nodes) and five trophic links. Both species and webs are numbered for further reference. Higher species consume lower ones (species #1 is always the top-predator; the direction of links is not shown for simplicity). For data on structural (D, D0 , K) and dynamical (E) characterization of species, consult http://falco.elte.hu/Bjordanf/data2.prn.
443
EXTINCTION DYNAMICS IN FOOD WEBS
and top-down effects. These indices count the number of disconnected species after the removal of species i. If species i is in an important network position it has a rich system of trophic interactions, then deletion will lead to more disconnections in both directions. The effects of species deletion on the disconnection of both bottom-up material flows and top-down flows of trophic regulatory effects are quantified by these indices of positional importance. The Kb value of the i-th species can be calculated as n X 1 1 ðcÞ ðiÞ ; ð1Þ þ K Kb ¼ dc dc b c¼1 where n is the number of its predators, dc is the number of prey eaten by its c-th predator, and KbðcÞ is the bottom-up keystone index of its c-th predator. Thus, the bottom-up index should be calculated first for top species. Kt is calculated similarly after turning the web upside down. It should be emphasized that K refers to keystone species only in a trophic sense (i.e. importance in maintaining trophic network flows). K characterizes food web pattern at an intermediate scale, between the local (measured by D and D0 ) and global (measured by stdD) properties of webs. Nevertheless, in our small webs K seems to be a rather global index. Both the sum and the standard deviation of K-values within webs (sumK and stdK) were also calculated for analyses. We are interested in whether these structural indices are possibly as informative and interesting as traditional food web statistics having been used extensively in the literature (e.g. trophic height, compartmentalization, the level of omnivory, etc., see Pimm, 1980). The behaviour of species is characterized by dynamical Lotka–Volterra models. Population dynamics is described by the following set of equations: ! n X bij xj xi ; i ¼ 1; y; n: ð2Þ x’ i ¼ ai þ j¼1
Here, xi is the population density of species i, ai is the per capita rate of reproduction (for growth, ai40, for decay, aio0). The parameter bij indicates the per capita effect of species
j on the per capita reproduction rate of species i. Here, ai is a uniformly distributed random variable in the range (0 1) if species i is a basal species, while ai is selected randomly from (1 0) if species i is intermediate or top species. Similarly, bij falls into (0 1) if i is the predator of j, and bji is selected from (1 0) if i eats j. If i ¼ j, bij is selected from (1 0) in case of producers but equals zero for higher species (May, 1973; Pimm, 1982). The dynamics of system (2) is simulated with 2000 random parameter sets selected in the way described above. We considered a species to be extinct if its density fell below a critical level (here, d ¼ 1015; results do not differ qualitatively for other values of d) after a given time (here, t ¼ 200 steps). The extinction value (E) of a species gives the number of extinctions out of 2000 simulations (we suggest that our approach to stability is more general than to study stable fix points). Results The distribution of the basic structural indices (D and K) of species in these model webs is shown in Fig. 2: the majority of species have two neighbours (D ¼ 2), while K is distributed evenly (especially if sink species are not considered). The most frequently extinct nodes can be characterized by an intermediate degree (D ¼ 2, see Fig. 3). The keystone index itself does not seem to be in correlation with the probability of extinction (Fig. 3; but seemingly extinctions are not frequent with very low K). However, the sum of extinction events within a web is clearly higher if the sum of keystone indices is higher (Fig. 3). The fact that K correlates with E only at the level of whole webs reflects how keystone index is calculated: it also quantifies indirect effects, which are of more global nature than locally understandable direct interactions. In order to study the effects of both D and K on extinction dynamics, they are plotted together (Fig. 4). If D ¼ 1, species go extinct with high probability if K ¼ 4; however, the probability of extinction is low at smaller K indices. If D ¼ 2, extinction can be very probable only with low K values. In case of a ‘‘dangerous’’ number of neighbours (i.e. if D ¼ 2), additional indirect effects (e.g. K ¼ 4) may
444
F. JORDA´N ET AL.
Fig. 2. The frequency distribution for the degree of nodes is a discrete, unimodal distribution (on the left). K values are evenly distributed (on the right; note that for all nodes of the last column K ¼ 4: 25 out of 47 nodes represent the sink species; thus, basal and intermediate species are distributed much more evenly).
Fig. 3. Relationships between structural and dynamical properties: top left, the number of extinction events concerning each species (E, out of 2000 simulations) is plotted against the number of their neighbours (D, ( ) show the average; the Mann–Whitney test on a distribution of 105 randomly generated samples shows significance at po0.0003, po0.0114, and po0.0025 for differences between [D ¼ 1, 2], [D ¼ 2, 3], and [D ¼ 3, 4] pairs of categories, respectively); top right, the number of extinction events (E) is plotted against species’ keystone indices (K); bottom, the sum of extinction events (sumE) within each web is plotted against the sum of their species’ keystone indices (sumK ).
decrease the probability of dynamical extinction. If D ¼ 3 or 4, extinction is less probable and is less sensitive to K. Thus, measuring both
exclusively direct (D) and direct plus indirect (K) effects gives an interesting extinction probability landscape.
445
EXTINCTION DYNAMICS IN FOOD WEBS
Fig. 4. The effects of the keystone index (K) on extinction dynamics (E), for nodes characterized by different D values (D ¼ 1, 2, 3, and 4). If a species has a single direct interaction (D ¼ 1), then it is more dangerous to have stronger indirect connections (large K). However, in case of having two neighbours (D ¼ 2), species can go extinct more frequently only if K is lower. For D ¼ 3 and 4, extinction frequency depends weakly on K is lower. For D=3 and 4, extinction frequency depends weakly on K.
Since both the number of neighbours and their orientation can be important for extinction dynamics, we analysed the relationship between D0 and E (Fig. 5). For example, if D ¼ 2, D0 may equal 2 (two prey of a sink species, e.g. Fig. 1, web #3, species #1), 0 (one prey and one consumer of an intermediate species, e.g. Fig. 1, web #19, species #3) and 2 (two consumers of a source species, e.g. Fig. 1, web #21, species #5). The extinction probability (E) is the highest for species with D0 ¼ 0. Thus, both multiple exploited producers (D0 ¼ 2) and generalist consumers (D41) are at lower risk of extinction. ‘‘Generalist’’ feeding and ‘‘multiple’’ food supply is dynamically more advantageous, even if the number of direct interactions is the same. Within a web, the standard deviation of degrees (stdD) and keystone indices (stdK)
measure how homogeneously the species are connected. High stdD means, for example, one strongly and ‘‘many’’ poorly interconnected species, while a low stdD indicates that all species have roughly equal connectedness. Similarly, stdK quantifies the homogeneity of direct and indirect connectedness. Homogeneous webs can be characterised by a slightly higher sum of extinctions (Fig. 6). This finding may imply that species in key positions can, in general, increase dynamical stability at the community level. Discussion Because the pattern of population dynamics is not uniform for the species of our model webs, positional effects are clearly demonstrated (i.e. quantified position within the network gives information on the dynamical properties of
446
F. JORDA´N ET AL.
Fig. 5. The number of species extinction events (E) is plotted against species’ D0 indices ( ) show the average; see explanation in text): not only the number but also the orientation of direct trophic links influence extinction dynamics. The Mann–Whitney test on a distribution of 105 randomly generated samples shows significant difference between [D0 ¼ 1, 0] at po0.00001. [D0 ¼ 0, 1] and [D0 ¼ 1, 2] show different average E at po 0.1. D0 ¼ 2 and 3 differ significantly at po0.003, while [D0 ¼ 2, 1] and [D0 ¼ 3, 4] do not differ significantly.
Fig. 6. The sum of extinction events (sumE) for each web is plotted against the standard deviation of degrees (stdD, ( )) and the standard deviation of keystone indices (stdK, ( )) of species. Living in a homogeneous web is more dangerous.
nodes). These positional properties of nodes may characterize either extinction risk (as far as randomly chosen interaction parameters are regarded as perturbations) or preferences in community assembly (as far as possible structures created by invaders are compared). If positional properties (D, D0 , K) influence extinction dynamics (E), species do not die out randomly, but rather in an internally directed way (alike ‘‘attacks’’ are defined as directed node
deletions in large networks, see Albert et al., 2000). For this set of small sink webs, it can be concluded that the presence of species in key positions can be dynamically either advantageous [if only direct interactions are considered (see stdD in Fig. 6)] or disadvantageous [if both direct and indirect interactions are taken into account (see Fig. 3)] against these ‘‘internal attacks’’ [in contrast with Albert et al. (2000), where non-random deletions are more
EXTINCTION DYNAMICS IN FOOD WEBS
dangerous in heterogeneous networks]. The description of extinction-prone network positions is suggested to identify unpreferred structural arrangements during web assembly (cf. Sugihara, 1984): the success of invasion may depend characteristically on the particular position of the invader within the community network (and this may hold also in case of speciation). The main mechanisms behind the simulated network dynamics include direct (e.g. predation, food supply) and indirect (e.g. exploitative competition, trophic cascade) trophic interactions among species (see a catalogue of indirect interactions in Menge, 1995). Given the topology of a trophic network, one may predict either the nature of a particular interaction or the fate of a particular species, but the dynamics of whole webs remain unclear until simulated. Only joint dynamical equations can give higher-level predictions (e.g. the fate of a whole web). Our simulation results show typical patterns in the dynamics of food webs and their species. For example, let us consider food web #12 in Fig. 1. This web can be considered typical as for the frequency of extinctions (1760 extinction events occurred out of 2000 simulations). The extinction frequency of the top-predator (#1, E ¼ 482) is close to the average. Its position is neither too risky nor very safe. One of its prey (#2, E ¼ 832) is prone to extinction, as are species with D0 ¼ 0 in general. They have a single source to consume and their fate depends strongly on their predators’ behaviour. Its other prey (#4, E ¼ 241) has a high extinction probability compared to other producers. It is caused by the following: if the top-predator’s density increases, it exploits species #4 both directly (by predation) and indirectly (by having a negative effect on species #2, which is good for species #3, which is bad again for species #4). Species #3 (E ¼ 87) is well saved by topology: it has a single predator and two preys. Finally, there is a producer in this ‘‘community’’ (#5, E ¼ 118) which goes to extinction less frequently than the other one (#4), because its position is better as for the number of predators. We can conclude that dynamically caused species extinctions are less frequent if the following conditions are satisfied: at the local
447
scale, (1) species have either a single or ‘‘many’’ (3 or 4) direct trophic links (see, for example, species #5 and #1 of web #1, Fig. 1), (2) the neighbours are clustered in either bottom-up or top-down direction (D0 a0, see species #2 in web #3); at an intermediate scale, (3) they have either a single neighbour and a low K-index or two neighbours and a high K-index (see species #5 in web #7 and species #5 in web #21), (4) the sum of K-indices within a web is low (see web #3), and, at the global scale, (5) whole webs have differently connected nodes (heterogeneity, stdD, see web #2). We have to emphasize that the simplest dynamical model was analysed: asymmetries in interaction strength (bij parameters) may lead to qualitative differences in our conclusions (Jorda! n et al., submitted). If trophic control in a particular community is considered of primary importance, i.e. food web studies have strong predictive power, then linking trophic structure to species’ dynamics could provide useful information for conservationists. The quest for keystone species (Power et al., 1996) is a big challenge for community ecologists. Quantitative approaches to identifying keystone species in ecosystems (e.g. based on food web position) could increase the predictive power of studies on the relative importance of species within communities. In addition, if keystones really make communities less prone to extinction, we can get closer to explaining their outstanding importance. Thus, the prediction of extinction dynamics by food web analysis could give a useful tool for conservationists (cf. Simberloff, 1998). If biological diversity can be expressed in terms of positional diversity of species in trophic networks, then the diverse population dynamics of species can also be approached quantitatively. We focused only on a particular but welldefined set of small food webs. Thus, possible effects of scale- and context-dependence were neglected. First, it should be clarified whether our results based on the analysis of small webs would also be applicable for large networks (see Bersier & Sugihara, 1997). Second, the problem of how the dynamics of community webs differs from that of sink webs is to be studied. The behaviour of persistent nodes following perturbations is to be analysed in detail (some
448
F. JORDA´N ET AL.
preliminary results on whole webs are given in Jorda! n et al., submitted). Further, the manner in which certain network properties affect basic organizational principles, similar to the ‘‘ageing’’ and ‘‘cost’’ of nodes in the study of Amaral et al. (2000), should be considered. Nevertheless, we think that this study can contribute to the understanding of the ‘‘pattern and process’’ problems in ecology. Future studies on the homogeneity of ecological networks may reveal important aspects of the organization of food webs, and, in general, the ecological principles of community assembly and biotic extinctions. We are grateful to Ja! nos Podani for his help in statistics and correcting English. F. J. also thanks . Szathma! ry for enabling a fruitful stay Professor Eors at Collegium Budapest, Mauro Santos and Elias Zintzaras for some methodical help. Two anonymous reviewers are kindly acknowledged for helpful comments on the manuscript. F. J. is supported by two grants of the Hungarian Scientific Research Fund, OTKA F 029800 and OTKA F 035092. I. Sch. is supported by the grant OTKA T 029789. Both F. J. and I. Sch. are recipients of the Bolyai Research Grant of the Hungarian Academy of Sciences. REFERENCES Albert, R., Jeong, H. & BarabaŁ si, A.-L. (2000). Error and attack tolerance of complex networks. Nature 406, 378–381. Amaral, L. A. N., Scala, A., BartheŁ leŁ my, E. & Stanley, H. E. (2000). Classes of small-world networks. Proc. Natl Acad. Sci. U.S.A. 97, 11 149–11 152. BarabaŁ si, A.-L. & Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512. Bersier, L. F., & Sugihara, G. (1997). Scaling regions for food web properties. Proc. Natl Acad. Sci. U.S.A. 94, 1247–1251.
Cohen, J. E. (1978). Food Webs and Niche Space. Princeton, NJ: Princeton University Press. Harary, F. (1959). Status and contrastatus. Sociometry 22, 23–43. Harary, F. (1961). Who eats whom? Gen. Syst. 6, 41–44. Jeong, H., Tombor, B., Albert, R., Oltvai, Z. N. & BarabaŁ si, A.-L. (2000). The large-scale organization of metabolic networks. Nature 407, 651–654. JordaŁ n, F. (2000). Is the role of trophic control larger in a stressed ecosystem? Commun. Ecol. 1, 139–146. JordaŁ n, F. (2001). Trophic fields. Community Ecol. 2, 181–185. JordaŁ n, F. & MolnaŁ r, I. (1999). Reliable flows and preferred patterns in food webs. Evol. Ecol. Res. 1, 591–609. JordaŁ n, F., TakaŁcs-saŁ nta, A. & MolnaŁ r, I. (1999). A reliability theoretical quest for keystones. Oikos 86, 453–462. JordaŁ n, F., Scheuring, I. & MolnaŁ r, I. Persistence, flow reliability in simple food webs. Submitted. May, R. M. (1973). Stability and Complexity in Model Ecosystems. Princeton, NJ: Princeton University Press. Menge, B. A. (1995). Indirect effects in marine rocky intertidal interaction webs: patterns and importance. Ecol. Monogr. 65, 21–74. Paine, R. T. (1969). A note on trophic complexity and community stability. Am. Nat. 103, 91–93. Pimm, S. L. (1980). Properties of food webs. Ecology 61, 219–225. Pimm, S. L. (1982). Food Webs. London: Chapman & Hall. Power, M. E., Tilman, D., Estes, J. A., Menge, B. A., Bond, W. J., Mills, L. S., Daily, G., Castilla, J. C., Lubchenco, J. & Paine, R. T. (1996). Challenges in the quest for keystones. BioScience 46, 609–620. Simberloff, D. (1998). Flagships, umbrellas, and keystones: is single-species management pass!e in the landscape era? Biol. Conserv. 83, 247–257. Sugihara, G. (1984). Graph theory, homology and food webs. Proc. Symp. Appl. Math. 30, 83–101. Watts, D. J. & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature 393, 440–442. Williams, R. J., Martinez, N. D., Berlow, E. L., ! Dunne, J. A. & Barab Asi, A.-L. Two degrees of separation in complex food webs. Proc. Natl. Acad. Sci. U.S.A. (In press).
