Physica A 391 (2012) 125–131

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Physica A

journal homepage: www.elsevier.com/locate/physa

The elimination of hierarchy in a completely cyclic competition systemYongming Li, Linrong Dong, Guangcan Yang ∗

College of Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325035, China

a r t i c l e i n f o

Article history:Received 21 October 2010Received in revised form 7 July 2011Available online 25 August 2011

Keywords:Cyclic dominanceRock–paper–scissors gameHierarchySingularizationStochastic simulation algorithm (SSA)

a b s t r a c t

Interactions among competing units are crucial to maintaining biodiversity, and non-hierarchical interactions can promote biodiversity in cyclic competing systems. In thepresent study, we explore the role of hierarchical interactions, existing ubiquitously inreality, in the co-evolution of a cyclic competing system. In systems composed of cycliccompeting species with hierarchy interactions in which one predator species has morethan one prey, we find that hierarchy disappears in a rather short evolving time. In theprocess of co-evolution, a hierarchical competing system tends to transit to a cyclic non-hierarchical competing systemdescribed by the rock–paper–scissors game. In otherwords,the cyclic competing interactions appear to eradicate hierarchy. This conclusion is analyzedby a mean-field approach and is tested by stochastic simulations.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Cyclic game theory has led to novel and interesting insights into biodiversity maintenance in a multispecies ecologicalsystem [1–3]. The classical cyclic game theory is mostly based on the representative rock–paper–scissors model in whichrock crushes scissors, scissors cut paper, and paper wraps rock [4–9]. On the level of the mean-field rate equations, thissystem is characterized by a reactive fixed point, corresponding to species coexistence where every species yields onethird of the total population when the interaction rates are symmetric [10]. When intrinsic stochastic fluctuations areconsidered, spatial systems and well-mixed systems show fairly different evolving behaviors. For example, in a two-dimensional large local spatial system without diffusion, cyclic competition results in the stable coexistence of all speciesand long-term maintenance [11–14]. However, population mobility is an important feature of real ecosystems, and itcan promote or jeopardize biodiversity depending on its magnitude [15]. When the mobility exceeds a critical value,biodiversity is jeopardized and finally lost. In contrast, when the mobility is below the value, competing species coexistand spatial–temporal patterns can emerge in the form of rotating spirals [16], whereas, in a well-mixed system, thecoexistence of competing species is, as a result of stochastic fluctuations, always non-persistent. In the course of timeevolution, the deviation of the stochastic trajectory from a deterministic orbit leads to one of the three absorbing stateswhere only one species survives and the other two become extinct. Interestingly, cyclic dominance of three species withasymmetric interaction rates always endswith zero–one survival behavior, where the ‘‘weakest’’ species seemingly survivesat a probability that is equal to one for a large populationwhile the other two are extinguished [10,17]. On the other hand, fora systemwith symmetric interaction rates, we found that it would be bound to reach an absorbing state after a long enoughtime by extensive numerical simulations using the stochastic simulation algorithm (SSA) [18,19]. Before they reach the finalstates, each species chases each another; the three species appear to keep this dynamic, cyclic chasingmode, which is shownin Fig. 1. Generally, finite-size fluctuations invalidate the neutral stability and induce the extinction of two species after acharacteristic time T , proportional to the system size N [1,20,21]. This is consistent with the case of asymmetric rates [10].

∗ Corresponding author. Tel.: +86 577 86689033; fax: +86 577 86689010.E-mail addresses: yanggc@wzu.edu.cn, yanggc77@gmail.com (G. Yang).

0378-4371/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2011.08.019

126 Y. Li et al. / Physica A 391 (2012) 125–131

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Fig. 1. The time evolution of a rock–paper–scissors system with symmetric interaction rates equal to 1, and system population 2000. (A) demonstratesthe dynamic, cyclic chasing mode of three species: after a long enough time the system has reached one of three absorbing states, shown in (B). (A) is cutfrom the header of (B) in order to display the competing relationship among species in detail.

In reality, hierarchy is quite common in nature and human society. For example, in a grassland ecosystem containing eagles,snakes, rats, and other living beings, eagles prey on snakes and rats, snakes feed on rats, and rats feed on other living beings.In this case, hierarchy is a prominent feature of the system. In the present study, we focus on the role of hierarchy in acyclic competing system. Through theoretical analysis and numerical simulation, we find that hierarchy disappears in quitea short co-evolution time in our model systems. In the process of co-evolution, a hierarchical competing system tends totransit to a cyclic, non-hierarchical competing state described by the rock–paper–scissors game. Generally, cyclic competinginteractions tend to eradicate hierarchy, at least in our simple model systems.

This paper is organized as follows. In Section 2, the hierarchy model is introduced with every pair of species interacting.In Section 3, we evolve the system numerically with a stochastic simulation method and then analyze the system withmean-field theory, and find all the possible fixed points and absorbed states. Discussions are presented in Section 4 andsome conclusions are drawn.

2. The hierarchy model

Since hierarchy is a widespread phenomenon existing in ecosystems, we are interested in its influence on the co-evolution of the cyclic systems. In the present study, we turn to a competing system with hierarchy. For a system withm species (m ≥ 3), the cyclic competition relationship is given as follows:

S1 + S2K1,2−−→ S1 + S1

S1 + S3K1,3−−→ S1 + S1

· · ·

S1 + Sm−1K1,m−1−−−→ S1 + S1

S2 + S3K2,3−−→ S2 + S2

· · ·

S2 + SmK2,m−−→ S2 + S2

· · ·

Sm−1 + SmKm−1,m−−−−→ Sm−1 + Sm−1

Sm + S1Km,1−−→ Sm + Sm.

(1)

In this cyclic formulation S1 outperforms S2, S3, . . . , Sm−1; S2 outperforms S3, . . . , Sm−1, Sm; . . . ; Sm−1 outperforms Smand in turn Sm outperforms S1. The interactions between any two species maintains the relation of the cyclic competitionuntil some species extinct in the co-evolution, while the asymmetric competition among all species enhances the hierarchywith the species number m. We show a system containing five species as an example in Fig. 2, in which arrows denote thecompetition between two species: the species pointed to by the arrow is a prey while the species doing the pointing is apredator. As can be seen from Fig. 2, both S1 and S2 have three types of prey, S3 has more than one prey, but S4 and S5 haveonly one prey. In such a system, various species are located at different levels of competition hierarchy.

Y. Li et al. / Physica A 391 (2012) 125–131 127

S1S2

S5

S3

S4

Fig. 2. The competition relationship existing in a cyclic system containing five kinds of species. This demonstrates the existence of hierarchy because ofthe asymmetric predator–prey relationship.

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A B

C D

Fig. 3. The co-evolution of some completely cyclic, hierarchical competing systems inwell-mixed formationwith the initial number of every species being1000, and the interaction rate between every pair of species being 1. (A), (B), (C), and (D) demonstrate the time evolution ofm species as being a completelycyclic, hierarchical system with m equal to 4, 5,6, and 7, respectively. The results of time evolution present some non-hierarchical rock–paper–scissorscompeting modes, and the survivors can only be S1 , S2 , and Sm .

3. Theoretical analysis and numerical simulations

In order to demonstrate the co-evolution of the model system, we performed extensive theoretical simulations for twocases of well-mixed and two-dimensional spatial formations. The interaction rates are all assigned to 1 for simplicity, withno loss of generality.

Well-mixed system

In a well-mixed system, the number of combinations of pairwise interactions is exactly C2m. In our simulation,

the interacting time and combination were chosen randomly. The stochastic simulation algorithm (SSA) was used todemonstrate the time evolution of the systems containing various species. We present four typical cases form = 4, 5, 6, 7,as shown in Fig. 3(A)–(D), where the initial number of each species is specified as 1000. When m = 4, we can see thatthe population of S3 decreases significantly at the initial stage, and then approaches zero gradually. The same dynamicalbehavior applies when m > 4. Interestingly, the surviving species are S1, S2, and Sm for this kind of hierarchical system. Toinvestigate the influence of the initial condition, we repeated the same procedure but with random initial population. It wasfound that the surviving species were still S1, S2, and Sm. The result is shown in Fig. 4.

128 Y. Li et al. / Physica A 391 (2012) 125–131

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Fig. 4. The co-evolution of some completely cyclic, hierarchical competing systems in well-mixed formation with random initial number of every species.The other conditions are the same as in Fig. 3.

Two-dimensional spatial system

In order to investigate the spatial effect of a hierarchical system, we introduce a two-dimensional lattice for simplicity.In the system, each lattice site is occupied by a species, and all the species are randomly arranged with the same probabilityat the beginning. Periodic boundary conditions are applied in the simulation. At each simulation step, a random individualis chosen to interact with one of its four nearest neighbors, which is randomly determined. As is shown in Fig. 5, here wedisplay the four cases corresponding to species numberm equal to 4 and 7. In the left plot of Fig. 5(A), there are four speciesrandomly distributed on the lattices at the beginning. After a rather short evolving time, the pattern of species distributionbecomes that shown on the right of Fig. 5(A). We can see that only three species survive; S3 becomes extinct. Fig. 4(B)corresponds to the case when m = 7. As with the case when m = 4, only S1, S2, and S7 survive and the other species fadeout with time. We tried various cases of differentm and obtained the same result. The conclusion is that only S1, S2, and Smsurvive finally. From a series of snapshots of the lattice evolving process, we can see that the pattern of species distributionvaries from time to time, then approaches the mosaic consisting of S1, S2, and Sm, whose cyclic competition can remain forquite a long time. Finally, the pattern reaches a static absorbed state containing only one species due to the fluctuation inevolution. To eliminate the influence of the initial condition, we initialized the lattice species pattern with different ratiosof random seeds to evolve the system. The same result was obtained for the different initial starting states.

In Figs. 3 and 4, we can see that the cyclic competing systems usually evolve into the rock–paper–scissors gamesformation after a transient process, regardless of the initial species population or lattice distribution. Actually, we cananalyze this evolving process by using the rate equations. We write the rate equations which describe the deterministictime evolution of the densities (or population) a1, a2, a3, . . . , am−1, am of S1, S2, S3, . . . , Sm−1, Sm. For reactions (1), the rateequations can be written as

·

a1 = a1(K1,2a2 + K1,3a3 + · · · + K1,m−1am−1 − K1,mam)

·

a2 = a2(−K1,2a1 + K2,3a3 + · · · + K2,m−1am−1 + K2,mam)

·

a3 = a3(−K1,3a1 − K2,3a2 + K3,4a4 + · · · + K3,m−1am−1 + K3,mam)

· · ·

·

am−2 = am−2(−K1,m−2a1 − K2,m−2a2 − · · · − Km−3,m−2am−3 + Km−2,m−1am−1 + Km−2,mam)

·

am−1 = am−1(−K1,m−1a1 − K2,m−1a2 − · · · − Km−3,m−1am−3 − Km−2,m−1am−2 + Km−1,mam)

·

am = am(K1,ma1 − K2,ma2 − · · · − Km−2,mam−2 − Km−1,mam−1)

a1 + a2 + · · · + am−1 + am = 1.

(2)

Y. Li et al. / Physica A 391 (2012) 125–131 129

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Fig. 5. The co-evolution of some completely cyclic, hierarchical competing systems in a two-dimensional 1000 × 1000 lattice. Fig. 4(A) and Fig. 4(B)demonstrate the co-evolution of m species completely cyclic, hierarchical system with m equal to 4 and 7, respectively. The left figures display the initialstates of multispecies and the right figures display the resulting states. The top-left corner figures embedded in each figure are local areas that havebeen enlarged to display the system mode in detail; they correspond to a 50 × 50 lattice. The results of time evolution present some non-hierarchicalrock–paper–scissors competing modes and the survivors can only be S1 , S2 , and Sm .

For simplicity, we consider the case whenm = 4:

·

a1 = a1(K1,2a2 + K1,3a3 − K1,4a4)·

a2 = a2(−K1,2a1 + K2,3a3 + K2,4a4)·

a3 = a3(−K1,3a1 − K2,3a2 + K3,4a4)·

a4 = a4(K1,4a1 − K2,4a24 − K3,4a3)

a1 + a2 + a3 + a4 = 1.

(3)

When the system arrives at a fixed point, a1·= a2·

= a3·= a4·

= 0, we have

a1(K1,2a2 + K1,3a3 − K1,4a4) = 0

a2(−K1,2a1 + K2,3a3 + K2,4a4) = 0

a3(−K1,3a1 − K2,3a2 + K3,4a4) = 0

a4(K1,4a1 − K2,4a2 − K3,4a3) = 0

a1 + a2 + a3 + a4 = 1.

(4)

Clearly, its solutions are (1, 0, 0, 0), (a∗

1, a∗

2, a∗

3, a∗

4), (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), where a∗

1 =K2,4

K1,2+K2,4+K1,4, a∗

2 =

K1,4K1,2+K2,4+K1,4

, a∗

3 = 0, a∗

4 =K1,2

K1,2+K2,4+K1,4. If we set the rate Ki,j = 1 (i, j = 1, 2, 3, 4), we have a∗

1 =13 , a

∗

2 =13 , a

∗

3 =

0, a∗

4 =13 , and the solution reads (1/3, 1/3, 0, 1/3), (1, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0). The first fixed point is

a cyclic competition system with no hierarchy; the others are absorbed states. It is obvious that reactive fixed point F isone of the solutions. The approach is easily generalized to the case of any m. We can expect that this state corresponds to adeterministic reactive fixed point F , located at (a∗

1, a∗

2, . . . , a∗

m−1, a∗m) =

13 ,

13 , 0, . . . , 0,

13

; here, a∗

1, a∗

2, a∗

3, . . . , a∗

m−1, a∗m

indicate the densities of S1, S2, S3, . . . , Sm−1, Sm. Certainly, there arem− 1 absorbed states corresponding to cases in whichonly one species survives.

130 Y. Li et al. / Physica A 391 (2012) 125–131

Table 1Every species’ net number of prey species in a system containingm kinds of species;m > 3.

Species name Number of prey Number of predators Net number of prey species

S1 m − 2 −1 m − 3S2 m − 2 −1 m − 3S3 m − 3 −2 m − 5· · · · · · · · · · · ·

Sm−1 1 −(m − 2) −(m − 3)Sm 1 −(m − 2) −(m − 3)

Interestingly, the survivors of both the well-mixed and spatial systems containing m kinds of species can only be S1, S2,and Sm, while S3, . . . , Sm−1 are eliminated in the process of evolution.

4. Discussions and conclusions

In the analysis in the last section, we can see that a hierarchical cyclic system tends to evolve into a non-hierarchicalsystem. This is an interesting phenomenon worth further investigation.

First, let us analyze the hierarchy relationship among species. Since the system is cyclic, we cannot determine thehierarchical relationship according to the food chain. For example, we cannot conclude that Sm is at a higher level than S1according to the fact that S1 is a prey of Sm. Thuswe propose amore reasonable rule to determine the hierarchical level basedon the prey and predators of one species. The hierarchy number of each species is defined as the net sum of its predatorsand prey. As a rule, a prey contributes the hierarchy number +1 while a predator contributes −1. Table 1 shows all species’prey number, predator number, and the hierarchy level in a system consisting of m species. For example, S1 and S2 sharethe same net number of prey species, and Sm−1 and Sm correspond to the same situation. From the table, we can see that S1and S2 belong to the highest competition level of a system while Sm−1 and Sm belong to the lowest competition level. Otherspecies belong to intermediate competition levels that become lower with the increase of their ordinal number in the cyclechain because of their different net number of prey species.

Furthermore, if S1 or Sm becomes extinct, we can derive from Eq. (1) that this will lead to the breakdown of cycliccompetition and, eventually, only one species survive. In sociological terms, we can say that the diversity is destroyed,which in turn leads to unilateralism, which eventually breaks the cyclic property. However, this is not the case in oursimulation. In other words, a completely cyclic competition usually tends to avoid unilateralism and maintains its cyclicproperty.

Based on the above analysis, we make the following two hypotheses for the time evolution of a hierarchical system.

(1) Completely cyclic competition in multispecies systems tends to eradicate hierarchy and avoid unilateralism. Anotherequivalent statement is that completely cyclic competition in multispecies systems always tends to eradicate hierarchyand maintain a cyclic property.

(2) The species belonging to a higher level of hierarchy have the advantage to survive in a completely cyclic competitionsystem.

The two hypotheses can be used to predict the time evolution of a completely cyclic, hierarchical systemwith symmetricinteraction rates. For a system containing m(m > 3) species, S1 and Sm are bound to be the winners in the course oftime evolution otherwise this would lead to unilateralism (singularization), which is inconsistent with hypothesis (1).According to inference (1), S1 and Sm naturally are the final winners. The third participant (winner) in the subsequentrock–paper–scissors game (the cyclic competition) should be determined by inference (2). As can be see from Table 1, S2has the highest level of competition hierarchy when S1 and Sm are excluded. Therefore, S2 is the third survivor amongS2, . . . , Sm−1. This conclusion is consistent with the numerical simulation shown in Figs. 3–5. Our hypotheses can wellexplain the result of a completely cyclic, hierarchical system. However, it is needed to investigate further whether theycan be used in other hierarchical systems.

In summary, in systems composed of cyclic competing species with hierarchy interactions, rate equation analysis andnumerical simulations show that the hierarchy disappears in a rather short evolving time. In the process of co-evolution,a hierarchical system tends to transit to a cyclic, non-hierarchical competing state described by the rock–paper–scissorsgame. In addition, it seems that the accumulative advantage in the cyclic competition due to fluctuation is conductive tothe species with higher hierarchy, becoming the three survivors of the resulting rock–paper–scissors system. However,the nature ecosystems we observed intuitively are seemingly not non-hierarchical cyclic rock–paper–scissors systems, norare the unilateralism (monotypic) systems that are the absorbing states of systems. The present results might be relatedwith the specialty of the systems in reactions (1) in which each pair of species interacts, which usually is not the case for areal ecosystem. Moreover, in the modeling of the key influencing elements of co-evolution, the interaction rates were notconsidered in determining the hierarchy of systems. Possibly, the actual hierarchy relationships among species are relevantto the interaction rates. Therefore, it is necessary to redefine the hierarchy of a cyclic competition system on a more realbasis. We intuitively infer that the existing cyclic ecosystems are non-hierarchical and the elimination of hierarchy is animportant feature of cyclic competition systems in a broad hierarchical sense.

Y. Li et al. / Physica A 391 (2012) 125–131 131

Acknowledgments

Thisworkwas partially supported by the State KeyDevelopment Programof Basic Research of China (No. 2007CB310405),Natural Science Foundation of China (Grant No. 10974146), and Zhejiang Provincial Natural Science Foundation (Grant No.Y6090222).

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