Ghosts in high dimensional non-linear dynamical systems ...complex.upf.es/~josep/ghostshd.pdf · erties of the ghosts are explored for these networks. Hence, delayed transitions in

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Chaos, Solitons and Fractals xxx (2007) xxx–xxx

www.elsevier.com/locate/chaos

Ghosts in high dimensional non-linear dynamical systems:The example of the hypercycle

Josep Sardanyes *

Complex Systems Laboratory (ICREA-UPF), Barcelona Biomedical Research Park (PRBB-GRIB),

Dr. Aiguader 88, 08003 Barcelona, Spain

Accepted 2 January 2007

Abstract

Ghost-induced delayed transitions are analyzed in high dimensional non-linear dynamical systems by means of thehypercycle model. The hypercycle is a network of catalytically-coupled self-replicating RNA-like macromolecules, andhas been suggested to be involved in the transition from non-living to living matter in the context of earlier prebioticevolution. It is demonstrated that, in the vicinity of the saddle-node bifurcation for symmetric hypercycles, the persis-tence time before extinction, T�, tends to infinity as n!1 (being n the number of units of the hypercycle), thus sug-gesting that the increase in the number of hypercycle units involves a longer resilient time before extinction because ofthe ghost. Furthermore, by means of numerical analysis the dynamics of three large hypercycle networks is also studied,focusing in their extinction dynamics associated to the ghosts. Such networks allow to explore the properties of theghosts living in high dimensional phase space with n = 5, n = 10 and n = 15 dimensions. These hypercyclic networks,in agreement with other works, are shown to exhibit self-maintained oscillations governed by stable limit cycles. Thebifurcation scenarios for these hypercycles are analyzed, as well as the effect of the phase space dimensionality in thedelayed transition phenomena and in the scaling properties of the ghosts near bifurcation threshold.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

The apparition of the so-called ghosts after a saddle-node bifurcation has been described in several non-lineardynamical systems [1–7]. Ghosts and bottlenecks and their associated square-root scaling laws are a very general prop-erty of systems close to a saddle-node bifurcation [2]. The square-root scaling law in the vicinity of the bifurcation seemsto be a universal in ghost-induced delayed transitions. Such delayed transitions involve a delay in the time in which atrajectory achieves a stable fixed point. This delay is caused because of a bottleneck region (the ghost) in phase spacewhich sucks the trajectories before allowing them to pass out towards another attracting point. This ghost is placed inthe region of phase space where two fixed points have collided and destroyed by means of the saddle-node bifurcation.As abovementioned, this delay time, s, in the vicinity of the bifurcation points can be governed by the square-root scal-ing law according to s � ðl� lcÞ

�1=2 (where l is a control paramter and lc the point at which the bifurcation occurs).

0960-0779/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2007.01.127

* Tel.: +34 933160532; fax: +34 933160550.E-mail address: [email protected]

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This scaling law has been described in models of charge density waves [1] as well as in the hypercycle dynamical systemdescribing the dynamics of autocatalytic and cross-catalytic networks under decay [3,4,6] as well as under mutation [7]processes. In these systems the ghost was the responsible of the so-called delayed transition towards extinction, foundwhen the critical parameter is overcome. As previously mentioned, the ghost in such systems involves a delay in the timebefore extinction near the bifurcation point, supposing a type of emergent behavior in the form of a concentrationmemory in phase space. Delayed transitions have been also described in several physical systems [8–11,1], as well asin the context of biological processes as in neuron firing dynamics [12,13]. Although the square-root scaling law hasbeen found in many of such systems, other scaling behaviors near bifurcation threshold have been characterized (seefor example [1,5]). Such differences in the scaling exponents have been suggested to exist because of the topologicaleffects associated to the dimensionality of the dynamical system [5]. Actually, the square-root scaling law has been clas-sically described in low dimensional (i.e. with one or two dimensions) non-linear dynamical systems [2,1,6,3,7].

A well-known example of non-linear dynamics at the molecular level is given by the so-called hypercycle dynamicalsystem, which was proposed by Eigen and Schuster in the 1970s decade [14]. These authors opened a new framework forthe understanding of the origin-of-life problem considering template-replicating systems as RNA (or RNA-like) as thepremanent precursor to life starting with nude self-replicating sequences [14,16,15]. The abstract hypercycle hypothesisstates that a complex genetic information can be achieved with the cyclic functional coupling in the form of catalyticconnections among several non physically-related sequences. It is known that as the length of a nucleotide sequenceincreases, an error threshold catastrophe occurs in a population with quasispecies distribution replicating at high muta-tion rates [14,17,15]. Hence, as the length of the sequence increases, selection becomes too weak to pull the populationof replicating sequences to a narrow mutant cloud surrounding the single best replicating sequence [14,15]. The errorcatastrophe implies that, with a given accuracy of self-replication per nucleotide and a given fitness superiority of thecurrently fittest ‘‘master sequence’’ relative to the mutant spectrum around it, there is a maximum sequence lengthwhich selection can maintain generation to generation. This maximum length is, roughly, the reciprocal of the errorrate per nucleotide [15]. Considering a plausible prebiotic accuracy, this maximum sequence length would be limitedto the size of about 50–100 nucleotides. Such a sequence length and its associated information content seems to befar from adequate to code for much of a concerted metabolism [14,15]. However a cyclic closed coupling among thesetemplate pairs in the form of a ‘‘hypercycle’’ might allow the evolution of large complex genetic information becauseseveral chemically coupled entities, each of them below the critical sequence length, could store a larger genetic messageas a whole. Hence, hypercycles might allow the overcoming of the informational threshold and thereby make possiblethe build-up of integrated genetic systems. As Kauffman stated: ‘‘The Eigen and Schuster effort is a serious and sustained

one [...]’’ [15]. However, the stability of the hypercyclic organization presents some problems (see [15] for a review). Thefirst one concerns to mutation processes [18]: if a single RNA sequence mutates improving its replication but failing incatalyzing the following hypercycle member, the replicating sequence can outgrow the system also taking all mononu-cleotide resources. These mutations are responsible of the appearance of the so-called catalytic parasites. A parasiticreplicator in a hypercycle is one unit that receives catalysis but does not catalyze any other member of the network.The general weakness in the ability of hypercycles to overcome uncooperative mutants that constantly arise from erro-neous replication of the members of the hypercycles has been pointed out by several authors [19,14,18]. Nevertheless, ithas been shown that the hypercycle is able to survive to parasites’ action by means of spatial self-organization processes[20–24]. The second problem is given by the so-called catalytic short-circuits, in which a hypercycle replicator mightcatalyze a more distant sequence of the catalytic loop. Then it might exist a shorter loop growing faster than the wholehypercycle and consequently, the hypercycle will be reduced to a less complex form. Another problem in hypercyclepersistence is given by fluctuations which could drop to zero the concentration of a given hypercycle sequence, thuspossibly collapsing the whole network [18].

The hypercycle, from a kinetic point of view, is dynamically driven by cross-catalytic interactions where the hyper-cycle species reproduce through second-order (or higher) i.e. non-linear, catalysis [16,14,25,27]. Confirmations of thehypercyclic mutual enhancement in two experimental systems has been described. The first example has been charac-terized in the infection phase of the coliphage Qb, where the viral genome, as a template, instructs its own reproduction,and, as a messenger, it expresses information which is materialized through translation. Such translation products (thereplicasa subunit and the capsid protein) provide a specific feedback loop regulating viral growth. The second examplewas provided by Lee and co-workers, who showed the hypercycle organization in two self-replicating artificial peptides[26]. The hypercycle organization has focused all the attention in the study of cooperative molecular processes. How-ever, such a kind of cooperative organization might be found at a higher scale in symbiotic processes related to ecolog-ical dynamics [28].

The aim of this work is, by means of the hypercycle model, to explore the dynamical properties of the ghosts in highdimensional dynamical systems. The persistence time before extinction for symmetric hypercycles composed of n speciesis firstly evaluated (near, from above to the bifurcation point) in the limit when n!1, by using an analytically derived

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expression for the extinction time obtained from a simplified equation which implicitly includes the number of replica-tor macromolecules forming the hypercycle. Furthermore, by means of numerical analysis three hypercycle networkswith n = 5, n = 10 and n = 15 units are also explored. The time behavior, the bifurcation scenarios as well as the prop-erties of the ghosts are explored for these networks. Hence, delayed transitions in high-dimensional non-linear dynam-ical systems can be characterized.

2. Hypercycle dynamical system

The model we use to analyze ghost-induced delayed transitions in high dimensional non-linear dynamical systems isthe well-known hypercycle model [14]. The dynamics of hypercycles can be described by means of a set of n coupledordinary differential equations. This mean-field approach describes the behavior of the hypercycle macromoleculesin the spatially homogeneous regime, thus assuming perfectly mixed populations of self-replicators that also havethe ability to catalyze the replication of other macromolecules. Such macromolecules might correspond to RNA-likereplicating strand pairs. The set of reactions describing the chemical kinetics of a hypercyclic network composed ofI i¼1...n, self-replicating species can be represented as follows:

Fig. 1.numern growspeciescatalysdenote

PleaSolit

I i þ Ij þ s!ki2I i þ Ij; ð1Þ

I i!�i s: ð2Þ

Here j ¼ i� 1þ nðdi1Þ, introduces the cyclic architecture of the hypercycle, in which the first replicator is catalyticallyassisted by the last one, thus forming a closed circular network (see Fig. 1, right). Reaction (1) describes the catalyti-cally-assisted self-replication that a given member of the hypercycle receives from its predecessor. Note that we hereonly consider, for the sake of simplicity, the catalytic i.e. non-linear, growth of replicators. The synthesis of new repli-cators is developed by using available precursor molecules named s. Reaction (2) indicates the processes of hydrolysis ofthe hypercyclic macromolecules. The reaction constants of these reactions are given by ki and �i, which denote self-rep-lication and decay rates of the hypercycle units. The previous reactions can be modeled by the following n-dimensionaldynamical system:

dxi

dt¼ kixixj 1�

Xn

r

xr

!� �ixi: ð3Þ

Here xi denotes the relative concentration of the hypercycle unit Ii. We include the logistic-like term ð1�Pn

r xrÞ, in or-der to limit the growth of the hypercycle replicators assuming a population constraint as intraspecific competition dueto limited resources. Note that our approach does not explicitly consider well-defined sequences for the hypercycle spe-cies. Moreover, the available precursors, s, are not explicitly introduced in our model. The composition of this system at

I

I1

I2

In-1

0 5×105

1×1060

2×108

4×108

(n)

ε

n

T

+

+

+

+

k

k

k

k

n

εφ

εφ

ε

φε

φ

Persistence time before extinction, T ðnÞ� , very near, from above, to the bifurcation point (with � ¼ �c þ 10�10), computedically from expression (4) by using n as control parameter with g ¼ 10�16. Note that the time spent before extinction increases ass. On the right hand side we show a schematic diagram of a symmetric, elementary hypercycle composed of n self-replicating, I i¼1...n. Each species acts as template and as a catalyst, receiving catalytic help from the previous unit as well as providingis to the following member of the network. Small circular arrows indicate self-replication processes and long curved arrowscatalytic aid. Here k and � are, respectively, self-replication and decay rates.

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any time t is given by an array of n numbers: xðtÞ ¼ ðx1ðtÞ; . . . ; xnðtÞÞ, and can therefore be represented by a point in then-dimensional concentration space Xn (i.e. phase space), defined by Xn � Rn; Xn : fx1; . . . ; xn; xi P 0; i ¼ 1 . . . ng, whereRn : fx1; . . . ; xn;�1 < x1...n <1g. If the molecular composition of the system changes, this point describes a trajectoryin Xn.

In this work the symmetric hypercycle is explored, in which the reaction constants are the same for all replicatorspecies i.e. ki � k and �i � �, "i. Under the assumption of symmetry, Eq. (3) has three fixed points [5], given by:

PleaSoli

ðx�Þ ¼ ð0Þ; ðx��Þ ¼1

2n1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4n�

k

r" #:

The point ðx� ¼ 0Þ is the extinction invariant, which is globally stable when a critical decay rate is overcome. Such acritical decay rate is �cðnÞ � k=4n. With values of � > �c the coexistence fixed point, ðx�þÞ, collides with the saddle,ðx��Þ, which is placed between the extinction and the coexistence equilibria. Such a collision involves the appearanceof a ghost in phase space responsible of delaying the time needed to achieve the extinction point [3,5,6].

The extinction delay times for symmetric hypercycles composed of n self-replicating species can be analyticallyderived from Eq. (3) (see [5] for details). Such an extinction time is the time needed for a trajectory of a given hypercyclereplicator to achieve a concentration of g << 1 from an initial condition xið0Þ. Such a time delay can be expressed, con-sidering xið0Þ ¼ 1 and g2 � 0, as:

T ðnÞ� ¼ �1

�Lngþ 1

n�n2

Ln �gþ �

nk

��� ���þ 1

barctan

1

nb� n

2Ln 1� 1

nþ �

nk

��������þ 1

barctan

ð2� 1nÞ

b

� �; ð4Þ

with b ¼ 2ffiffiffiffiffiffiffiffiffiffi1=nk

p ffiffiffiffiffiffiffiffiffiffiffiffi�� �cp

. Let us now analyze a particular case evaluating the time delay before extinction in a hyper-cycle with an infinite number of self-replicating species, thus taking the limit of T ðnÞ� as n!1 and as g! 0. This anal-ysis might not correspond to a realistic situation because the hypercycle might contain a finite number of units.Moreover, a large increase in the number of hypercycle species could involve a higher probability of appearance ofthe so-called parasites as well as the formation of catalytic short-circuits. Furthermore, the previously mentioned prob-lems as the effect of deleterious mutations as well as the effect of strong fluctuations, could also suppose a problem in thesurvival of large hypercycles. Nevertheless, the evaluation of the persistence time before extinction in the limit of n!1and g! 0, allows to conjecture about the tendency of the transient time towards extinction at increasing number ofhypercyclic units. The time delay before extinction, in the limit when n!1 and g! 0, is computed as a sum of limitsconsidering a case with � near, from above, to the bifurcation point �c, (taking � ¼ �c þ a, with �c ¼ k=4n and a < < 1),according to:

limn!1g!0

T ðnÞ� ¼ limn!1g!0

f ðnÞ1 þ f ðnÞ2 þ f ðnÞ3 þ f ðnÞ4 þ f ðnÞ5

� �; ð5Þ

with:

f ðnÞ1 ¼ � 1

�Lng; ð6Þ

f ðnÞ2 ¼ 1

2�Ln �gþ 1

4n2þ a

nk

��������; ð7Þ

f ðnÞ3 ¼ 1ffiffiffiffik

4n

q ffiffiffiapþ

ffiffiffiffi4nk

q ffiffiffiffiffia3p arctan

1ffiffiffiffi4nk

q ffiffiffiap ; ð8Þ

f ðnÞ4 ¼ � 1k

2nþ 2aLn 1� 1

nþ 1

4n2þ a

nk

��������; ð9Þ

f ðnÞ5 ¼ 1ffiffiffiffik

4n

q ffiffiffiapþ

ffiffiffiffi4nk

q ffiffiffiffiffia3p arctan

2� 1n

� 2ffiffiffiffi1nk

q ffiffiffiap ; ð10Þ

It can be shown that:

limn!1g!0

f ðnÞ3 þ f ðnÞ4 þ f ðnÞ5

� �! 0: ð11Þ

Moreover the expression:

limn!1g!0

f ðnÞ1 þ f ðnÞ2

� �; ð12Þ

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can be represented as:

Fig. 2.coexistoscillatmakesconcentrajectobottlen

PleaSolit

limn!1g!0

Ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij � gþ �=nkj

pg

!‘

; ð13Þ

with ‘ ¼ 1=�. The limit of expression (13) can be evaluated from:

limn!1g!0

‘ Ln1ffiffiffigp : ð14Þ

Note that expression (14) tends to infinity in the limit of n!1 and g! 0, because ‘ 1; 1=ffiffiffigp !1, and

Lnð1Þ ¼ 1. Hence we can conclude that the limit indicated in expression (5) tends to infinity. In this sense, the timedelay caused by the ghost might show an increasing tendency as the number of the hypercycle units increases. Such atime spent in the bottle-neck region of the ghost can be also numerically computed by means of expression (4) atincreasing values of n, considering a parameter scenario just above the extinction threshold (i.e. with � ¼ �c þ 10�10),where delayed transitions occur (we assume that the extinction concentration is g ¼ 10�8). As previously shown, T ðnÞ�increases at growing n, indicating that the extinction time might be longer when more hypercycle units are considered,although expression (4) does not gather the topological effects, which have been suggested to be of importance in de-layed transition phenomena [5,6]. In the following lines three particular cases of delayed transitions in high dimensionaldynamical systems are explored in hypercycles with n = 5, n = 10 and n = 15 self-replicating species, which actually cor-respond to the variables defining high dimensional phase space. The numerical solutions of Eq. (3) for such networks

0 0.25 0.50

0.25

0.5

1x

2x

ix ix

1x

5x

4x

2x

5x

3x

1x

2

1x

5x

4x

x

2x 3x

5x

0 1000 2000

0.25

0.5

0 1000 20000

0.3

0.6

0 0.25 0.50

0.25

0.5

0 0.25 0.50

0.25

0.5

0 0.25 0.50

0.25

0.5

0 0.4 0.80

0.4

0.8

0 0.4 0.80

0.4

0.8

0 0.4 0.80

0.4

0.8

0 0.4 0.80

0.4

0.8

timetime

coexistence extinction

bottleneck

Dynamics for the hypercycle composed of n = 5 members below and above the critical decay rate involving, respectively,ence and extinction of the whole network. The coexistence scenario (left, with � = 0.025) is governed by self-maintainedions i.e. stable limit cycle. The extinction phase (right, with � ¼ �c þ 10�7) is characterized by the apparition of the ghost, whichthe hypercycle concentration trajectories to undergo some oscillations before collapsing. The time series display the

tration of all the hypercycle units. The phase portraits show the projections in two-dimensional phase space where twories converge to the limit cycle attractor governing the coexistence phase. For the extinction scenario the orbits undergo theeck of the ghost before becoming extinct. In both cases, the arrows indicate the direction of the flow.

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1x

ix ix

5x

1x

2x

3x

7x

4x

10x

1x

8x

3x

7x

4x

10x

1x

8x

0 2500 50000

0.4

0.8

0 0.4 0.80

0.4

0.8

0 0.4 0.80

0.4

0.8

0 0.4 0.80

0.4

0.8

0 0.4 0.80

0.4

0.8

0 0.3 0.60

0.3

0.6

0 0.3 0.60

0.3

0.6

0 0.3 0.60

0.3

0.6

0 0.3 0.60

0.3

0.6

0 0.3 0.60

0.3

0.6

0 0.3 0.60

0.3

0.6

0 0.3 0.6

0.3

0.6

0 1000 2000 30000

0.3

0.6

timetime

coexistence extinction

bottleneck

Fig. 3. Same as Fig. 2 for the hypercycle composed of n = 10 members. The coexistence scenario (left, with � ¼ 0:015) is also governedby self-maintained oscillations. The extinction phase (right, also with � ¼ �c þ 10�7) shows the effect of the ghost which conserves thedynamical properties of the coalesced coexistence fixed point.

1x

2x

ix ix

1x

2x

4x

3x 6x

10x

15x 12x

4x

3x 6x

10x

15x 12x

0 5×104

1×105

0

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 3000 60000

0.4

0.8

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 1

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

timetime

coexistence extinction

bottleneck

Fig. 4. Same as Fig. 3 now with n = 15. Note that the coexistence scenario (left, with � ¼ 0:01) is also governed by a stable limit cycle.At the right (also with � ¼ �c þ 10�7) the extinction dynamics is displayed.

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Please cite this article in press as: Sardanyes J, Ghosts in high dimensional non-linear dynamical systems: ..., Chaos,Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.01.127

0.01 0.10.3

0.6

0.9

10-6

10-4

10-2

103L

ocal

max

ima

x

ε

n = 15

n = 10

n = 5

1

τ

φ

n = 5

n = 15

n = 10

Fig. 5. Left: bifurcation diagram for the hypercycles analyzed in this work, with n = 5, n = 10 and n = 15 units, by using the decay rateas control parameter (in log-linear plot). For each value of �, the local maxima of a time series on the attractor for one hypercyclemember is displayed. As the hypercycle is symmetric all the hypercycle units undergo the same concentration maxima. All thesenetworks undergo a tangent bifurcation (indicated with the vertical dashed lines) beyond � � �c. Right: extinction time, s (consideredwhen ½x1 6 10�16), on a doubly logarithmic scale, as � moves further away the bifurcation point where / is the parametric distance tothe critical decay value (i.e. / ¼ �� �c). Note that here no clear scaling region is found.

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are obtained by means of the fourth-order Runge–Kutta method using a constant time step size of dt = 0.1 [2]. For thesake of simplicity we will use in all the analyses k = 1.

Let us first analyze the hypercycle with five replicator units. The presence of a stable limit cycle governing thedynamics of this hypercycle has been described [14]. Fig. 2 shows that this network undergoes self-maintained oscilla-tions in the coexistence scenario, where all the replicator concentrations oscillate in time. The critical decay rate for sucha system is given by �c ¼ 0:05. Note that in Fig. 2 (right) we display the extinction dynamics of this network. As abovementioned, the overcoming of the bifurcation point involves the collision of the coexistence fixed point with the saddle.Note that with � ¼ �c þ 10�7, trajectories are sucked by the ghost before flowing towards extinction. Such trajectoriesfollow the orbits associated to the coalesced coexistence fixed point, thus undergoing several oscillations before achiev-ing the extinction attractor. The bifurcation diagram for this network (see Fig. 5, left) shows that below the criticaldecay rate the hypercycle members coexist. Nevertheless, the whole network becomes extinct once the critical decaythreshold is overcome. The tangent bifurcation (indicated with the dashed lines) involved in the collision among thecoexistence fixed point and the saddle is the responsible of the extinction of the hypercycle.

Similar results are obtained for the other two networks (i.e. n = 10 and n = 15, see Figs. 3 and 4). Both systems alsoexhibit periodic oscillations of the replicator species in the coexistence scenario. Note that these networks reach highermaxima concentration peaks as compared to the hypercycle with n = 5 (for all these networks we have used as initial con-ditions non-identical, near-one values). This actually means that the limit cycle has a bigger size because trajectories traveltowards higher concentration values. This effect could be of importance in the extinction dynamics because the ghost couldalso maintain this property possibly increasing the time delay of the trajectories before the collapse to the extinction point.The bifurcation diagrams of Fig. 5 show that the local maxima for the hypercycles with n = 10 and n = 15 units have higherconcentration maxima. Moreover, the critical decay rate is shown to decrease at increasing number of replicator units.

The analysis of the extinction time, s, for these three networks near bifurcation threshold is displayed in Fig. 5(right). As previously mentioned, the square-root scaling law typically arises near the bifurcation threshold. As com-mented in the Introduction section, this scaling law has been described (in the context of catalytic networks) in a singleautocatalytic replicator as well as in the two-membered hypercycle. Fig. 5 shows that no clear scaling region is found forthe hypercycles analyzed in this work. Moreover, the time delays, s, near bifurcation threshold show (for the three net-works) a plateau, which is longer at increasing n. This plateau indicates that these hypercycles are able to maintain a aconstant time before extinction as � moves further away from the bifurcation point. Moreover, this figure also showsthat the extinction time is longer as n increases. These numerical results are in agreement with the idea that a higherphase space dimensionality, given by a higher number of hypercycle units, could involve longer transients towardsextinction. In this sense, hypercycles with a high number of catalytic species could undergo this dynamic advantage.

3. Conclusions

In this paper delayed transitions towards extinction in high dimensional non-linear dynamical systems are analyzedby means of the well-known hypercycle model. By means of the study of the transient times to extinction obtained with

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a simplified expression derived from symmetric hypercycles composed of n species, it is shown that the extinction timetends to infinity as n!1, thus indicating that the ghost might undergo a stronger influence in large hypercycles. Tocomplement this unrealistic analysis three hypercyclic networks with n = 5, n = 10 and n = 15 replicator species havebeen analyzed. These networks allow the study of ghost-induced delayed transitions in high dimensional phase space.In agreement with the analysis in the limit of n!1, the extinction time is shown to increase at increasing n. Moreover,the scaling region described in smaller hypercycles as the square-root scaling law is not found for the hypercycles understudy. These networks show long plateaus in parameter space as moving further away from the bifurcation point. Inthis sense, the decrease in the persistence time before extinction as � grows beyond �c, is shown to be lower. The resultsof this work might be extended to ghost-induced delayed transitions in dynamical systems of high dimensionality gov-erned by periodic oscillations.

Acknowledgements

The author thanks the members of the Complex Systems Lab, specially Bernat Corominas, for useful suggestions. Ialso want to thank Frank Drebin for useful comments and discussions. This work has been supported by an EU PACEgrant within the 6th Framework Program under contract FP6-002035 (Programmable Artificial Cell Evolution).

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