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Ecological Modelling 161 (2003) 117 /124 www.elsevier.com/locate/ecolmodel Persistence and flow reliability in simple food webs Ferenc Jordán a,b,c,*, István Scheuring c, István Molnár a a Evolutionary Genetics Research Group, Department of Genetics, Eötvös University, Budapest, H-1117, Pázmány P.s. 1/c, Hungary b Collegium Budapest (Institute for Advanced Study), Budapest, H-1014, Szentháromság u. 2., Hungary c Eötvös University, Department of Plant Taxonomy, Research Group of Ecology and Theoretical Biology, Budapest, H-1117, Pázmány P. s. 1/c, Hungary Received 4 January 2002; received in revised form 31 May 2002; accepted 26 August 2002 Abstract Twenty-five model food webs can be designed from five points (species) and five links (trophic interactions), if they contain a single top-predator (i.e. sink webs). According to a simple topological approach, we presented elsewhere a reliability theoretical analysis of this set of food web graphs. The question addressed here is how network flow reliability is related to the dynamical behavior and stability of these ‘model communities’. We simulated the behavior of these webs and calculated their persistence, according to four models: (1) with symmetrical interaction coefficients between species; (2) with asymmetrical interaction coefficients and lower death rates for predators; and (3) with absolutely and (4) relatively perturbed, formerly persistent parameter sets of the asymmetrical model. We used both Lotka /Volterra (LV) and Holling II-type equations (with switching effect). Thus, we had eight persistence values for each web. Persistence (a dynamical property) and flow reliability (a structural property) were analyzed. We found that (1) reliable flow pattern is associated with high persistence in the Holling models, while the LV models predict no consistent correlations; (2) in asymmetrical situations, persistence is always much higher (in both LV and Holling models); and (3) the predictions of Holling models (versus LV models) are much less sensitive to local perturbations. Based on these results, we conclude that (1) reliable network flows can contribute to persistence only if switching is possible; (2) asymmetrical interactions increase persistence, independently of the switching effect but it indicates persistence in Holling II model; (3) switching makes the relationship much more predictable between structure and dynamics. Thus, the network design is less useful predictor of persistence without switching effect but well can ensure dynamical stability under the conditions of the Holling II model. We have presented how various dynamical models predict different behavior of modelled communities characterized by the same structure and complexity. Complementing dynamical with structural analysis may further increase our understanding of persistence in food webs. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Food web; Interaction symmetry; Network design; Persistence; Reliable flows; Switching effect 1. Introduction * Corresponding author. Tel.: /36-1-381-21-87; fax: /36-1381-21-88 E-mail address: [email protected] (F. Jordán). It was analyzed elsewhere how food web structure affects the reliability of nutrient flows in simple, model sink webs containing five points 0304-3800/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 0 0 ( 0 2 ) 0 0 2 9 6 - X 118 F. Jordán et al. / Ecological Modelling 161 (2003) 117 /124 and five links (Jordán and Molnár, 1999). However, the relationship between flow reliability and some characteristics of dynamical stability remained unclear. Here, we attempt to study how a particular type of dynamical stability (persistence) is related to reliability, a topological property. The reliability of energy flows in trophic flow networks can be increased by simply creating new links: enhanced complexity (i.e. connectance) provides new transport pathways of food and energy (cf. MacArthur, 1955). But dynamical stability may decrease with increasing complexity (Gardner and Ashby, 1970; May, 1973; Pimm, 1984). Thus, it is worth studying how dynamical behavior relates to reliability in equally complex food webs (containing the same number of points and links, i.e. characterized by the same connectance). Doing so, we can escape the traditional debates concerning the relationship between complexity and stability (e.g. De Angelis, 1975; McNaughton, 1977; see also Siljak, 1975; and for a recent contribution: McCann et al., 1998). Our main question is how persistence relates to reliability, a structural network property (defined below). Here, a web is considered persistent if the densities of all five species remain above a given threshold after an ‘ecologically reasonable’ time. This measure of persistence is general and may be more adequate stability property (McCann et al., 1998), than the widely used local asymptotic stability of fixed points (e.g. May, 1973). Various persistence values are calculated and plotted against flow reliability. Results are discussed in the context of connecting pattern to process in ecology (cf. Winemiller and Polis, 1996). 2. Methods First we used simple Lotka /Volterra (LV) equations for calculating persistence. Population dynamics is described by the following set of equations: n X dxi ri aij xj xi ; i 1; . . . n: (1) dt j1 Here, xi is the population density of species i, ri is the per capita reproduction (ri /0) or decay (ri B/0) rate. The parameter aij indicates the per capita effect of species j on the per capita growth rate on species i. If species i is a producer then aii B/0, otherwise let aii /0 (Pimm, 1982). Of course, if there is no predatory interaction between species i and j, then aij /aji /0. Two LV models were derived, as follows. 2.1. The basic models 2.1.1. Symmetrical LV model Here, ri is an evenly distributed random variable over the interval (0 1) if species i is a basal species, while ri is selected randomly from (/1 0) if species i is intermediate or top species. Similarly, aij is over the interval (0 1) if i is the predator of j, and aji is selected from (/1 0) if i eats j (or if i /j for producers, see May, 1973; Pimm, 1982). This basic model can be used to describe situations where we have little information about species and species interactions. 2.1.2. Asymmetrical LV model The parameter ri is defined as in the symmetrical LV model, and aij is in (0 1) for the predator i . However, aji //aij /(10/5j ), where j is a random variable in (/1 1). This asymmetry in interaction parameters comes from the observation that the size of a predator is in most cases larger than the size of its prey (Warren and Lawton, 1987; Cohen et al., 1993; now we do not consider the problems of parasites in food web theory, see Marcogliese and Cone, 1997) and only a fraction of energy directs to reproduction. The ½aij /aji ½ ratio may differ widely between trophic interactions, but it is generally around 1/100 for vertebrate/invertebrate, and 1/10 for vertebrate/vertebrate interactions (Pimm, 1982; Jonsson and Ebenman, 1998). Also, as a consequence of larger body size, predator death rates are lower. We choose ri over the interval (/1/15 0) for the predators. Thus, the asymmetrical model holds for situations where the consumers are characteristically larger than the consumed species. F. Jordán et al. / Ecological Modelling 161 (2003) 117 /124 119 2.2. Simulations 2.4. Switching effect Eq. (1) was numerically integrated by the second-order predictor-corrector method for each web (according to each model) after selecting 2000 sets of random parameters (ri and aij ). Initial densities (xi (0)) were selected randomly from the interval (0 1) for each parameter set. After the initial transients was discarded, the persistence index was measured as the ratio of parameter sets allowing the density of all the five species to remain over a critical threshold (d /1015) after t (t/200) steps. Here, we are interested in whether each population of the ecosystem can survive (at this ‘ecological’ time-scale). The sensitivity of persistence to various t (t /100, 200, 500) and d (d/1010, 10 15 and 10 20) parameters was studied: not surprisingly, extinction was more frequent if either t (longer simulation run) or d (high threshold density) was higher, but the results remained qualitatively similar. Each model presented below were simulated with the parameters t/200 and d/1015. In order to analyze whether switching of predators affects the relationship between reliability and persistence, we have studied models describing situations according to the Holling II-type functional response (Holling, 1959; Chesson, 1983). The main advantage of the LV model is its structural simplicity and the low number of parameters needed to describe it. However, as a Taylor expansion of a general dynamical model, it can be a good approximation of real dynamics only in the neighborhood of some equilibrium. As a more sophisticated model, the Holling II-type function takes into account the possibility of switching (we now consider only positive switching, Abrams, 1987), as well as the searching and handling times of predation. Here, the LV model modifies to: 2.3. Sensitivity to perturbation We analyzed two further stability indices, in order to study how local perturbations of formerly persistent parameter sets of the LV model affect dynamical behavior. 2.3.1. Absolute perturbation Here, 1000 random parameter sets that yield persistent webs were chosen and parameters were perturbed 10 times according to the rule ri? /ri / j , aij? /aij /j, where j is a random variable in (/ 0.1 0.1). (To change the sign of ri or aij by perturbation was not allowed, of course.) By this method, we tested how persistence is maintained if parameters change within given limits. 2.3.2. Relative perturbation It is similar to the previous case but ri and aij were modified to ri (1/j ) and aij (1/j ) after perturbation. n X dxi ri hij (x)aij xj xi bii x2i ; ¯ dt j1; (2) j"i where v Xij hij (x) ¯ gj vik xk (3) k Li if species i is the predator of species j, or vji X hij (x) ¯ gi vki xk (4) k Li if species i is the prey of predator j. The parameter gj is the half-saturation density of the prey j , vij is a constant between zero and one indicating the preference of species i for consuming species j , thus Sk Li vik /1, where Li denotes the indexes of all alternative preys of predator i. If species i is a primary producer (ri /0) then bii B/0 to set a finite carrying capacity. For species being in the higher trophic level bii /0. The variables ri , aij are selected symmetrically and asymmetrically as in the LV model previously, while the new parameters gj and vij are chosen randomly from the (0 1) interval in every case. 120 F. Jordán et al. / Ecological Modelling 161 (2003) 117 /124 2.5. The reliability of network flows The above defined persistence measures were compared to flow reliability, the probability that the sink point remains connected to at least one source, despite probabilistic deletions of points (see also Barlow and Proschan, 1965; Aggarwal, 1993), with fixed and equal probability for all four points except the sink (Jordán and Molnár, 1999), according to the equation: 4 X 4 i (4i) R 1 Fi ; (5) pq i i1 where the p deletion probability equals 0.01 (and ‘survival’, q /1/p /0.99), 4 indicates the number of points possibly deleted (all except the sink), and Fi is the ‘fatal ratio’. This is defined as the ratio of point sets containing i points which results in disconnection between the sink and the sources, when deleted (i/1,. . .,4). For example, consider web #15 in Fig. 1: two points out of four can be deleted in six combinations (with joint links, of course). In four out of six combinations, the sink species become disconnected to producers. Thus, F2 for web #15 is 4/6. Evidently, F4 /1 for each web. Based on their R -values, these 25 webs can be ranked in four reliability classes, as shown at Fig. 1 (the difference in R between two webs belonging to different reliability classes is at least two orders of magnitude larger, than that between two webs belonging to the same class). The structure of food webs clearly influences flow reliability (for an early idea of similar nature, see MacArthur, 1955). 3. Results The persistence values of the 25 webs (i.e. the ratio of persistent webs), according to the above simulations, are plotted against reliability (averaged for R -classes, Fig. 2; open boxes and dashed lines correspond to LV, while asterisks and solid lines correspond to Holling II-type simulation results). Under the conditions of the symmetrical LV model (Fig. 2a), flow reliability correlates positively with dynamical stability. According to the symmetrical Holling model, there is no significant correlation between persistence and reliability. (Note that the Holling simulations generally give lower persistence: this is an artefact, depending only on model parameters g and v. These were selected from (0 1), since we have no assumptions on their symmetry.) Fig. 2b shows how the asymmetry in interaction parameters and higher death rates of predators increase persistence. The correlation between persistence and reliability is positive according to the Holling model but negative in the LV case. If persistent parameter sets are slightly perturbed, extinction may appear. Holling models behave in a more or less similar way (significant positive correlation) in the cases of both relative (Fig. 2c) and absolute (Fig. 2d) perturbations. However, these local perturbations influence to a smaller extent, but qualitatively differently the behavior of the LV systems (no significant correlation in Fig. 2c, while negative correlation in Fig. 2d). 4. Discussion Persistence, as a type of stability (cf. Pimm, 1984) was calculated by LV and Holling II-type equations for a well defined set of model food webs. This set consists of all possible sink webs designed from five points and five links. The flow reliability of these webs was known (Jordán and Molnár, 1999). The average persistence values for the different reliability classes show (Fig. 2) that more reliable energy flows can be associated with either higher (e.g. symmetrical LV and asymmetrical Holling models) or lower (e.g. asymmetrical LV model) persistence. In some models, there is no significant correlation (relatively perturbed, asymmetrical LV and symmetrical Holling model). We found that asymmetry in interaction coefficients and lower death rates for predators lead to much more persistent ‘model communities’ (Fig. 2b/d, and cf. Hughes and Roughgarden, 1998). Switching effect makes predictable the reliabilitypersistence relationship only under asymmetrical conditions (cf. Pelletier, 2000). Thus, analyzing switching and interaction symmetry in combina- F. Jordán et al. / Ecological Modelling 161 (2003) 117 /124 121 Fig. 1. All of the possible sink food webs with five species (nodes) and five trophic interactions (links) (N/5, L /5, connectance: C/ 0.5), ranked according to four classes of reliability (calculated by (5); slightly modified after Jordán and Molnár, 1999). Trophic flows from producers (down) reach the single sink species (always on the top) in the most reliable way in R1 class webs. tion is more informative than studying switching in itself (for the relationship between switching, stability, and growth rates , see Matsuda et al., 1986). Various dynamical behaviors under different model conditions appeared because the exact mathematical conditions for the persistence are determined by the complex algebraic combinations of interaction parameters (Hallam, 1980). Thus, even if food web complexity is similar, the relationship between structure and dynamics can be strikingly unpredictable. Mechanisms being sensitive to the details of modelling include, for example, the interactions between trophic levels (Hairston et al., 1960; Fretwell, 1977; Oksanen et al., 1981), or reticulate control pathways (Polis and Strong, 1996). Our results suggest that, if interaction coefficients are nearly symmetrical (consider insectparasitoid interactions), switching makes the relationship between structure and dynamics less predictable (Fig. 2a). In this case, good network design can be dynamically advantageous without switching possibility, too. Note that the conditions of the symmetrical model seem to apply much more to specialist predators. However, if switching is possible and accompanied by asymmetrical interactions, reliable flow structure well predicts persistence. These findings support that network design principles have dynamical consequences on 122 F. Jordán et al. / Ecological Modelling 161 (2003) 117 /124 Fig. 2. The simulation results (i.e. persistence: the ratio of persistent webs) for model food webs were averaged for different reliability classes (from R4, containing the least reliable webs, to R1, containing the most reliable ones). Open boxes and dashed lines mark the results for the LV simulations, while asterisks and solid lines mark the results of the Holling simulations. Linear fit comes from the Marquardt /Levenberg algorithm, weighted by standard deviation values. We also give the asymptotic standard error of the slope: a, symmetrical interaction coefficients (LV: r/0.1329/0.0096, Holling: n.s.); b, asymmetrical interaction coefficients and lower death rates for predators (LV: r //0.07079/0.0039, Holling: r/0.0279/0.0067); c, asymmetrical case with relative perturbation (LV: n.s., Holling: r/0.02389/0.011); d, asymmetrical case with absolute perturbation (LV: r//0.01439/0.01, Holling: r/0.0369/0.0086). network dynamics and evolution. For example, the fragility of marine wasp/waist systems can be understood as a consequence of unreliable network architecture (Vasconcellos et al., 1997; Cury et al., 2000). Further, even if our approach does not take into account recycling pathways (Odum, 1969; Ulanowicz, 1986, 1988; Vasconcellos et al., 1997), we still can investigate the effects of the length and redundancy of pathways (web-like versus chain-like architectures, Pahl-Wostl, 1997; strictly corresponding to reliability classes) on stability and performance (Büssenschütt and Pahl-Wostl, 2000). Fig. 2 suggests that the sensi- tivity of chain-like structures (R4 class of reliability) is higher under the Holling II conditions (cf. Pahl-Wostl, 1997). The most important tasks for future are: (1) studying larger webs in order to detect possible scale-dependence, (2) studying community webs (i.e. not sink webs) in order to check whether the ‘sink web property’ causes any bias in results, and (3) analyzing the dynamical behavior of each point in each web in order to understand the relationship between their exact network position and extinction risk (cf. the ‘functional niche’ concept, PahlWostl, 1997, and Jordán et al., 2002). F. Jordán et al. / Ecological Modelling 161 (2003) 117 /124 Food web dynamics is a consequence of the structure and interaction intensity, hence, if the two need to be related, an analysis of dynamics should be complemented with structural analyses (cf. Silvert, 1983). The notorious question of how to connect structure to dynamics can be answered only by dual investigations (Winemiller and Polis, 1996; see also Jordán, 2000). 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