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Abrupt community transitions and cyclic evolutionary dynamics in complex food webs

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Abrupt community transitions and cyclic evolutionary dynamics in complex food webs
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  Abrupt community transitions and cyclic evolutionary dynamicsin complex food webs $ Daisuke Takahashi a, n , Åke Brännström b,c , Rupert Mazzucco c , Atsushi Yamauchi a ,Ulf Dieckmann c a Center for Ecological Research, Kyoto University, Hirano 2-509-3, Otsu 520-2113, Japan b Department of Mathematics and Mathematical Statistics, Umeå University, 901 87 Umeå, Sweden c Evolution and Ecology Program, International Institute for Applied Systems Analysis (IIASA), Schlossplatz 1, 2361 Laxenburg, Austria H I G H L I G H T S   We present the  fi rst individual-based model of community evolution in which linear functional responses suf  fi ce to enable the emergence of multipletrophic levels.   Evolving communities stochastically alternate between two states that are either dominated by producers or additionally feature diverse consumers.   We explain these cyclic transitions by an inexorable evolutionary drive towards particularly fragile community structures that allow extinctioncascades causing consumer collapse.   Our  fi ndings are shown to be robust to a wide range of model variations. a r t i c l e i n f o  Article history: Received 30 August 2012Received in revised form1 August 2013Accepted 5 August 2013Available online 12 August 2013 Keywords: Individual-based modelExtinction cascadeTrophic-level evolutionConsumer collapse a b s t r a c t Understanding the emergence and maintenance of biodiversity ranks among the most fundamentalchallenges in evolutionary ecology. While processes of community assembly have frequently beenanalyzed from an ecological perspective, their evolutionary dimensions have so far received lessattention. To elucidate the eco-evolutionary processes underlying the long-term build-up and potentialcollapse of community diversity, here we develop and examine an individual-based model describingcoevolutionary dynamics driven by trophic interactions and interference competition, of a pair of quantitative traits determining predator and prey niches. Our results demonstrate the (1) emergence of communities with multiple trophic levels, shown here for the  fi rst time for stochastic models with linearfunctional responses, and (2) intermittent and cyclic evolutionary transitions between two alternativecommunity states. In particular, our results indicate that the interplay of ecological and evolutionarydynamics often results in extinction cascades that remove the entire trophic level of consumers from acommunity. Finally, we show the (3) robustness of our results under variations of model assumptions,underscoring that processes of consumer collapse and subsequent rebound could be important elementsof understanding biodiversity dynamics in natural communities. &  2013 The Authors. Published by Elsevier Ltd. All rights reserved. 1. Introduction Biodiversity emerges over time through speciation and extinc-tion. Species evolve subject to ecological constraints, which stemfrom the interactions among them. A recent study of environ-mental change and species extinction suggests that the dynamicalchange of species interactions is an important proximate causeof species extinction (Cahill et al., 2012), thus highlighting theimportance of understanding the eco-evolutionary processes andmechanisms that maintain evolved biodiversity.The last few decades have seen impressive advances in ourtheoretical understanding of eco-evolutionary dynamics. In commu-nity evolution, the main focus is on understanding the dynamics andcomplexity of food webs (e.g., Verhoef and Morin, 2010), and muchresearch has been devoted to analyzing models that describe food-web formation and maintenance (Caldarelli et al.,1998; Drossel et al., 2001, 2004; Christensen et al., 2002; Yoshida, 2002, 2006; Rossberg et al., 2005, 2006, 2008; Stauffer et al., 2005; He and Yu, 2006; Ito and Ikegami, 2006; Bell, 2007; Rikvold, 2007, 2009; Rikvold and Sevim, 2007; Guill and Drossel, 2008; Guttenberg and Golden fi eld, 2008;P ę kalski et al., 2008; Ingram et al., 2009; Ito et al., 2009; Powell and Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/yjtbi  Journal of Theoretical Biology 0022-5193/$-see front matter  &  2013 The Authors. Published by Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jtbi.2013.08.003 $ This is an open-access article distributed under the terms of the CreativeCommons Attribution-NonCommercial-ShareAlike License, which permits non-commercial use, distribution, and reproduction in any medium, provided thesrcinal author and source are credited. n Corresponding author. Tel.: + 81 77 549 8020; fax: + 81 77 549 8201. E-mail address:  dtakahashi@ecology.kyoto-u.ac.jp (D. Takahashi). Journal of Theoretical Biology 337 (2013) 181 – 189  Boland, 2009; Murase et al., 2010; see also the recent review by Brännström et al., 2012). Such models are typically extended pre-dator – prey models with interactions depending on assigned traits, sothat food webs can ultimately emerge through evolution of thesetraits. A surprising  fi nding in many studies is that communitiessometimes exhibit a sudden transition from one evolutionary stateto another (Christensen et al., 2002; Ito and Ikegami, 2006; Rikvold 2007, 2009; Guill and Drossel, 2008; Rossberg et al., 2008; Murase et al., 2010).Using an individual-based model of evolutionary food-webemergence without adaptive foraging, Rikvold (2009) found asudden transition between two states: a community with multipletrophic levels and a community with only producer species.Although that study suggested that the emergence of intraspeci fi cpredation could initiate successive consumer extinction in thediverged community, it did not provide an explanation of themechanisms that would quickly remove almost all consumerspecies from a community. Ito and Ikegami (2006) also foundevolutionary transitions between highly diversi fi ed and poorlydiversi fi ed communities. Other authors observed  fl uctuatingdynamics of species richness without signi fi cant transitionaldynamics (Rossberg et al., 2008; Guill and Drossel, 2008). So far, however, no mechanistic explanation of the intermittent evolu-tionary dynamics observed in all those models has been provided.Most models of community evolution mentioned above focus onspeciation – extinction dynamics by regarding species as the unit of the modeled community and by considering mutation as beingequivalent to speciation (Drossel et al., 2001, 2004; Christensen et al., 2002; Yoshida, 2002, 2006; Rossberg et al., 2005, 2006, 2008; Stauffer et al., 2005; He and Yu, 2006; Bell, 2007; Rikvold, 2007; Rikvold and Sevim, 2007; Guill and Drossel, 2008; Guttenberg and Golden fi eld, 2008; P ę kalski et al., 2008; Ingram et al., 2009; Powell and Boland, 2009; Murase et al., 2010). However, this approach to modeling speciation, which forgoes a detailed account-ing of the mechanisms of mutation accumulation and trait diver-gence, precludes an understanding of species emergence as anadaptive process.Here, we investigate trophic interactions in a multi-dimensionalcontinuous niche space through an individual-based stochasticmodel with the aim of elucidating the evolutionary processes thatlead to the emergence and collapse of multi-layered communities. 2. Methods We consider an individual-based stochastic model in contin-uous time, in which birth and death events are realized withprobabilistic rates that depend on foraging success, predationpressure, and interference competition. Selection on foraging andvulnerability traits, which are inherited nearly faithfully by theasexually produced offspring, over time leads to the emergence of clusters of related individuals in trait space, which we identify asspecies. These species, together with the trophic interactionsamong them, de fi ne the food web, of which we analyze thestructure, stability, and certain network properties. The details of our model are described below.  2.1. Evolving traits Each individual is assumed to be haploid with nearly faithfulasexual reproduction. All individuals are thus considered toreproduce clonally and to produce mutated offspring with a smallprobability. Each individual has two sets of quantitative trophictraits: foraging traits and vulnerability traits. Both sets of traits arerepresented by two-dimensional vectors. Following previous workby Ito and Ikegami (2006) and Rossberg et al. (2006), the foraging trait vector of the  i  th individual,  f  i , represent its niche as aconsumer, while the vulnerability trait vector  v i  represents itsvulnerability to foraging, that is, the niche it provides as aresource. Like these authors, we do not assign speci fi c biologicalinterpretations (with reference to features such as color ortoxicity) to any axes or points in the trait space; instead, weconsider this space as an abstract representation of all relevantbiological traits.  2.2. Demographic dynamics We consider birth and death events, which increase anddecrease the total population abundance by 1, respectively. Eventsare realized sequentially one after the other, and average waitingtimes are exponentially distributed, following a Poisson process.We implement the resulting stochastic demographic dynamicsusing the Gillespie algorithm (Gillespie, 1976, 1977). Event rates depend on the intensities  F   and  I   of foraging and interferencecompetition, respectively. We assume that those interactionintensities between two individuals are given by their traits,in conjunction with a foraging kernel and an interference compe-tition kernel, which are both assumed to be Gaussian functions, F  ð  f  i ; v  j Þ ¼  1  ffiffiffiffiffiffi 2 π  p   s F exp    12 s 2F jj  f  i  v  j jj 2 ! I  ð  f  i ;  f   j Þ ¼  1  ffiffiffiffiffiffi 2 π  p   s I exp    12 s 2I jj  f  i   f   j jj 2 ! ;  ð 1 Þ with  s F  and  s I  being the standard deviations, or widths, of thosekernels. Interactions become more speci fi c for small widths, andless speci fi c for large widths. The foraging intensity is higher whena consumer ′ s foraging traits and a resource ′ s vulnerability traitsare more similar, corresponding to an overlap of the utilizableniche of the consumer and the providing niche of the resource.Moreover, the intensity of interference competition is maximalbetween individuals with the same foraging traits, as consumerscan be expected to interfere with one another most strongly whenutilizing the same resource.To prevent runaway selection, we furthermore assume a costfor vulnerability traits that increases quadratically with theirdistance from the srcin,  D ð v i Þ¼jj v i jj 2 . We assume the availabilityof an external resource, with vulnerability trait vector  v R   andabundance  N  R   For simplicity, we set the vulnerability trait vectorof the external resource equal to the srcin,  v R   ¼ð 0 ; 0 Þ .Based on the assumptions above, the instantaneous rates of birth events,  r  b i , and of death events,  r  d i , of the  i  th individual aregiven by r  b i  ¼ aC  F ∑  j F  ð  f  i ; v  j Þ þ  aC  F F  ð  f  i ; v R  Þ N  R  ; r  d i  ¼ C  F ∑  j F  ð  f   j ; v i Þ þ  C  I ∑  j I  ð  f  i ;  f   j Þ þ  C  D D ð v i Þ þ  d :  ð 2 Þ Here, the summations extend over all individuals in the commu-nity, and the coef  fi cients  C  F ,  C  I , and  C  D  scale the intensity of foraging, the intensity of interference competition, and the cost of the vulnerability traits, respectively. The remaining parameters a  and  d  quantify the trophic ef  fi ciency and the natural deathrate, respectively. As event rates are determined by summingover terms that do not depend on total population size, thecorresponding averaged deterministic dynamics are described bymultispecies Lotka – Volterra dynamics.  2.3. Evolutionary dynamics As we assume haploid individuals with asexual reproduction,mutation is the only source of phenotypic variation. We assume amutation rate proportional to the reproduction rate of each D. Takahashi et al. / Journal of Theoretical Biology 337 (2013) 181 – 189 182  individual (Stauffer et al., 2005; He and Yu, 2006; Bell, 2007; Rikvold and Sevim, 2007; Rikvold, 2007, 2009; Powell and Boland, 2009; Murase et al., 2010), with the ratio of those rates being given by amutation probability. Rossberg et al. (2006) argued, based on theiranalysis of empirical data, that the mutation rate of foraging traitstends to be much higher than that of vulnerability traits. Wetherefore consider different mutation probabilities for the foragingand vulnerability trait vectors,  μ f   and  μ v , respectively, with  μ f  4  μ v .We assume that the occurrences of mutations in foraging andvulnerability traits are independent of each other, so mutations thatalter both foraging and vulnerability trait vectors occur with prob-ability  μ f   μ v . A mutation alters an offspring ′ s trait vector from that of its parent by adding a random vector whose components are drawnindependently from a normal distribution with expectation 0 andvariance  s 2m .  2.4. Parameter values and initial conditions Table 1 lists the parameter values we use in our investigations.These are chosen in agreement with previous theoretical studies,in particular Loeuille and Loreau (2005) and Rossberg et al. (2008). To induce predator – prey diversi fi cation, the differentiationbetween branched prey species needs to be suf  fi ciently large(Doebeli and Dieckmann, 2000): as the distances among thevulnerability clusters of species are controlled by the width of the foraging kernel, we assume that the foraging kernel isconsiderably wider than the competition kernel.We start our evolutionary investigations with a small popula-tion of 100 individuals with foraging and vulnerability traits equalto those of the external resource. This choice of initial conditionsonly affects the initial transient dynamics and has no impact onthe long-term outcomes of the investigations.  2.5. Species determination Determining what constitutes a species is not trivial whenmutational steps are small and reproduction is asexual. However,in our model, distinct clusters tend to form in trait space, andthe strains in a cluster are mostly close relatives of each other.We can thus de fi ne a species as a cluster of strains in trait space,in accordance with the genotypic-cluster species concept intro-duced by Mallet (1995). To identify these clusters, we apply theQT-clustering algorithm (Heyer et al., 1999) to the distributionof strains. Due to the small mutation rate, mutation – selectionbalance can remove all the relatives of some strains, which resultsin isolated strains being detected as outliers. Those outlier strainsare treated as species consisting of a single trait type.  2.6. Trophic-level determination For every species  i 4 0, its real-valued fractional trophic level  t  i is calculated following Odum and Heald (1972) as the weightedaverage of the trophic level of its prey species plus 1, t  i  ¼ 1 þ ∑  j w ij t   j :  ð 3 Þ Here, the trophic level of the external resource, which can bethought of as the 0th species, is de fi ned as  t  0  ¼ 0. The weights  w ij are de fi ned by  w ij  ¼ F  ij = ∑ k F  ik  with  F  ij  ¼ ∑  x A S  i ∑  y A S   j F  ð  f   x ; v  y Þ = n i .Here,  S  i  and  S   j  are the sets of individuals that belong to species  i and  j , respectively, and  n i  is the abundance of species  i . The weight w ij  thus measures the fraction of the average energy input anindividual of species  i  receives from all individuals of species  j .Eq. (3) de fi ne a linear system in which the trophic levels  t  1 ; t  2 ; ::: appear as unknowns; this system is solved by elementary matrixalgebra.For  i 4 0, the trophic levels thus determined are always largerthan or equal to 1. Species in our model community tend to clusteraround integer trophic levels; we can thus naturally classifyspecies by their trophic level as producers (1 r t  i o 1 : 5), trophic-level-2 consumers (1 : 5 r t  i o 2 : 5), trophic-level-3 consumers(2 : 5 r t  i o 3 : 5), and so on. 3. Results The individual-based stochastic model described above allowsfor the emergence of diverse communities with several trophiclevels.After an initial transient phase, the abundance of individuals fl uctuates over time, but mostly takes values in two markedlydifferent ranges (Fig. 1), similar to the  fl ip- fl op dynamics reportedby Rikvold (2009). These ranges correspond to two characteristiccommunity states. We refer to these community states as the low-trophic-level (LTL) state and the high-trophic-level (HTL) state. AnLTL community mainly consists of highly abundant producers,while trophic-level-2 consumers are rare and ephemeral (Fig. 1a).In contrast, an HTL communitycomprises also higher-trophic-levelconsumers (Fig. 1b).Evolution is characterized by long periods of HTL and LTL statespunctuated by fast transitions. Below we offer a process-basedexplanation for the observed evolutionary dynamics, and alsodemonstrate that our results remain robust to changes in para-meter values and model assumptions.  Table 1 Model parameters. The abundance of external resource,  N  R  , the scale of thevulnerability costs,  C  D , and the intrinsic death rate  d  can be considered as scalingthe units of population abundance, trait-space distances, and time, respectively.Description Symbol ValueAbundance of external resource  N  R   4500Scale of the intensity of foraging  C  F  0 : 9Scale of the intensity of interference competition  C  I  0 : 1Scale of the vulnerability costs  C  D  20Trophic ef  fi ciency  a  0 : 2Intrinsic death rate  d  0 : 1Width of foraging kernel  s F  0 : 3Width of competition kernel  s I  0 : 1Vulnerability traits of external resource  v R   ð 0 ; 0 Þ Mutation probability of foraging traits  μ f   0 : 001Mutation probability of vulnerability traits  μ v  0 : 0001Width of mutation kernel  s m  0 : 03  0 1 2  3 T r  o ph i   c l   ev  el   Low-trophic-level state (LTL)High-trophic-level state (HTL) Fig. 1.  Examples of the two distinct community states observed in this study. Eachcirclerepresentsaspecies,withtheirareasbeingproportionaltothespecies ′ abundance,theircolors indicating the species' trophic level, andtheirhorizontal positions indicatingthe species'  fi rst vulnerability trait. The cross at trophic level 0 represents theexternal resource. Arrows indicate trophic links, with darker shades indicating strongerinteractions. D. Takahashi et al. / Journal of Theoretical Biology 337 (2013) 181 – 189  183  We now describe these  fi ndings in turn. All model parametersused for this investigation are speci fi ed in Table 1 (for theparameters used for the robustness checks, see Section 3.4).  3.1. Emergence of complex food webs with multiple trophic levels Over time, demographic changes and small mutational stepslead to the emergence of a large number of species organized inseveral trophic levels. Fig. 1 shows the typical structures of theemerging communities. In the HTL state, communities includeproducers and higher-trophic-level species, exhibiting three dis-tinct trophic levels (Fig. 1b).  3.2. Community-level evolutionary cycles Fig. 2 shows the total abundance of individuals in the commu-nity on a long time scale. This abundance tends to remain aroundeither of two levels for long periods, each corresponding to one of the characteristic community states shown in Fig. 1. As thepresence of trophic-level-2 consumers effectively regulates theabundance of the producers, the HTL producer community tendsto have lower total abundance than the LTL producer community.Occasional mutations from producers to trophic-level-2 consu-mers do occur in the LTL state, but they typically fail to establish.Transitions between these states are relatively fast (Fig. 2a), andwe consistently observe cyclic evolutionary dynamics (Fig. 2b).The distributions of durations of both LTL and HTL states bettermatch exponential distributions than power-law distributions(Fig. 2c, d), suggesting that transitions between the two statesare triggered by rare random events that occur with constantprobabilities per unit time.  3.3. Understanding the evolutionary cycles We now present a detailed analysis of the observed evolutionarycycles (Fig. 2b). Starting from the LTL state, Fig. 3 shows the key steps in a schematic diagram. In practice, the steps constituting thefast transitions may occur nearly simultaneously.IntheLTLstate,producersinitiallymainlydiversifyintheirforagingtraits, so as to avoid interference competition. At the same time, theyform relatively large clouds in terms of their vulnerability traits,because there is little selectionpressure on those. Initially, the numberof such clouds almost equals the number of producers during thepreceding HTL state. Gradually, however, the number of those cloudsdecreases through random extinctions. Also, the occasional andtemporary emergence of a trophic-level-2 consumer imposes strongforagingpressureonone of those clouds, andtherebyincreasesitsriskof random extinction. Because of those processes, only a few vulner-ability clouds survive the LTL period. While all vulnerability traitvectors evolve toward the cost minimum at the srcin, directionalselection ceases at some distance from the srcin, since this allowsproducers to avoid being foraged by other producers.The transition from the LTL state to the HTL state is initiated by theappearance of a mutant individual with foraging traits that allow it toforage on the extant producer species. This mutant tends to be theoffspring of a producer with a foraging trait vector that is alreadyrelatively far away from the vulnerability trait vector of the externalresource (i.e., the srcin). As only a few vulnerability clouds exist atthe end of the LTL period, the newly emerged consumer species can 0500000150000025000000500010000150002000005000100001500020000Time (generations)    A   b  u  n   d  a  n  c  e 05101520Consumer species richness    P  r  o   d  u  c  e  r  a   b  u  n   d  a  n  c  e Duration (generations)    P  r  o   b  a   b   i   l   i   t  y   d  e  n  s   i   t  y Duration (generations)    P  r  o   b  a   b   i   l   i   t  y   d  e  n  s   i   t  y HTLLTLHTLLTL  AverageAverage HTLLTL Fig. 2.  Cyclic evolutionary transitions between the two community states. (a) Continuous curves represent the total abundance of producers (green), trophic-level-2consumers (orange), and trophic-level-3 consumers (red). (b) Frequency distribution of community states: 99% of community states are observed in the shaded areas, and75% of community states are observed in the dark-shaded areas. (c, d) Probability distributions of community-state durations (c: low-trophic-level communities, LTL; d:high-trophic-level communities, HTL). Minor tics indicate the bins used for constructing the histogram, red and blue curves indicate the best- fi t power-law distributions andthe best- fi t exponential distributions, respectively. The frequency distributions shown in (b – d) are obtained by convolving a Gaussian distribution with 72,060 sampledcommunity states from 60 independent model runs. D. Takahashi et al. / Journal of Theoretical Biology 337 (2013) 181 – 189 184  typically forage on a large number of producer species, making it asort of generalist. Consumer control now regulates producer abun-dance, leading to increasing producer evenness (Fig. 4a). Theproportion of foraged producers very quickly increases from 0 to 1(Fig. 4b). Because of the foraging pressure, the abundances of theproducers quickly decrease, leading to the eventual (stochastic)extinction of a number of producers due to overexploitation, in whatcan be viewed as a top-down process.The extinction of some producers leads to mounting foragingpressure by the generalist consumer on the remaining producers,generating a strong selection pressure towards a diversi fi cation of their vulnerability traits. This promotes differentiation of the vulner-abilitytraitvectorswithintheproducercommunity.Theforagingtraitsof the trophic-level-2 consumer undergo a corresponding specializa-tion, resulting in the emergence of trophic-level-2 consumers eachspecializedononeproducerspecies.Becauseweassumethatthecostsassociated with vulnerability trait vectors increase with their distancefrom the srcin, the process of diversi fi cation ceases once the viablevulnerability trait space is mostly occupied by producers. This is theHTL community state. The HTL producers are diversi fi ed in theirforaging trait vectors (because of interference competition) as well asin their vulnerability trait vectors (because of foraging pressure). Thetrophic-level-2 consumers of the HTL state are diversi fi ed in theirforaging trait vectors, but not so much in their vulnerability traitvectors (for the same reason that LTL producers are not, i.e., because of the absence of predation). The high evenness among producerssuggests that producer abundances are strongly controlled by con-sumers (Fig. 4a,c). A generalist trophic-level-3 consumer foraging ontrophic-level-2 consumers can also emerge. More complex commu-nities rarely evolve in our model, except for extreme parametersettings ( a ¼ 0 : 9, Fig. S2), because the strongly decreasing abundanceof the higher-trophic-level species makes their persistence less likely.The random extinction of a trophic-level-2 consumer initiates thetransition from the HTL state to the LTL state. Since producers aremostly foraged on by specialists, the extinction of such a specialistconsumer removes the foraging pressure from the correspondingproducer. As a consequence, the abundance of this producer quicklyincreases, which, in turn, increases the level of interference competi-tion exerted by it. Strong interference competition effectivelydecreases the abundance of the other producers, and consequently,the abundance of the corresponding trophic-level-2 consumers,threatening their survival (and the survival of all higher-trophic-levelconsumers). This destabilization of the producer level manifests itself in terms of decreasing producer evenness, which slightly precedes thedecrease in consumer richness (Fig. 4c). As more and more higher-trophic-level species become extinct, the proportion of producers thatare free from foraging pressure increases (Fig. 4d), and so does thecompetitivepressureon theremainingpairsof producersandtrophic-level-2 consumers. Ultimately, only a few producer species survive,which means that the community has reverted to its initial state. Thisextinction of the higher-trophic-level species can be seen as a bottom-up extinction process, as it is driven by the competitive dynamics of producer species.  3.4. Robustness of the evolutionary cycles To explore the robustness of our results, we consider alter-native minima of the vulnerability costs, different dimensionalitiesof the trophic trait space, variation in four salient model para-meters, and nonlinear functional responses. 0 0 LTLHTL Random extinction of a TL2 consumer increases the abundance of the corre-sponding producersStrong competition from the con-sumer-free producers reduces theabundance of the other producers A generalist consumer emerges atrandom, foraging on the producersand itself Foraging induces producer differ-entiation, followed by consumer specializationReduced producer abundance triggers an extinction cascade,eventually removing all consumersDifferentiation and specializationcontinue until the available nichesare filled SlowSlow Fig. 3.  Mechanistic explanation of the cyclic evolutionary transitions between the low-trophic-level (LTL) state and the high-trophic-level (HTL) state. In each panel, the topand bottom layers represent the trait spaces of foraging traits and vulnerability traits, respectively. The foraging traits and vulnerability traits of a species are indicated by twocircles, one on the top layer and one on the bottom layer, connected by a gray line. The area and color of each such circle indicates a species' abundance and trophic level,respectively, as in Fig.1. For ease of readability, vertical line segments with crosses at their lower ends indicate the mean foraging traits of each species, describing where theconsodered species forages most effectively. Dark arrows between the panels indicate fast and potentially concurrent transitions, while light arrows indicate slow transitionstriggered by rare random events. D. Takahashi et al. / Journal of Theoretical Biology 337 (2013) 181 – 189  185
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