The Hypercycle: from molecular to ecosystems dynamics

Abstract

laThe Hypercycle is a network of catalytically coupled self-replicating species in which each of the species enhances the reproduction of another one forming a closed loop. The abstract Hypercycle theory, which was initially developed in the context of prebiotic molecular evolution, is strongly tied to the origin of life problem. The Hypercycle actually models a system of entities which reproduce and cooperate between them, increasing the fitness in an altruistic way of otherwise competing species. This type of organization makes the species to behave as a single and coherent evolutionary unit, providing them with several selective advantages. For instance, benefitial mutations are spread over the whole system, species coherently grow and the system as a whole is able to store an informational content beyond the so-called error threshold. Here we review the hypercycle theory, focusing on several theoretical approaches to the dynamics of these systems, from mean field models to spatially-extended hypercycles. We discuss the potential of this theory for the study of reproducing and cooperating entities in the context of ecology and landscape ecology. We also provide several ideas for future research on this topic.

Full text

In: Landscape Ecology Research Trends
Editors: A. Dupont and H. Jacobs, pp. 1-12 ISBN 978-1-60456-672-7
c
2008 Nova Science Publishers, Inc.
Chapter 6
THE HYPERCYCLE: FROM MOLECULAR TO
ECOSYSTEMS DYNAMICS
Josep Sardany´
es∗
Complex Systems Lab (ICREA-UPF),
Barcelona Biomedical Research Park (PRBB-GRIB)
Dr. Aiguader 88, 08003 Barcelona, Spain
Abstract
The hypercycle is a network of catalytically coupled self-replicating species in
which each of the species enhances the reproduction of another one forming a closed
loop. The abstract hypercycle theory, which was initially developed in the context of
prebiotic molecular evolution, is strongly tied to the origin of life problem. The hy-
percycle actually models a system of entities which reproduce and cooperate between
them, increasing the fitness in an altruistic way of otherwise competing species. This
type of organization makes all the species to behave as a single and coherent evolu-
tionary multimolecular unit, providing them with several selective advantages. For
instance, benefitial mutations are spread over the whole system, species coherently
grow and the system as a whole is able to store an informational content beyond the
so-called error threshold. Here we review the hypercycle theory, focusing on several
theoretical approaches to the dynamics of these systems, from mean field models to
spatially-extended hypercycles. We discuss the potential of this theory for the study of
reproducing and cooperating entities in the general context of ecology and landscape
ecology. We also provide several ideas for future research on this topic.
PACS: 05.45.-a; 87.23.-n.
Keywords: cooperation, evolution, spatio-temporal phenomena, complex ecosystems,
landscape ecology, interactions and patterns
1. The Hypercycle
Introduction - The hypercycle, which was proposed in the decade of the 1970’s by Man-
fred Eigen and Peter Schuster [1], arises when a set of self-replicating molecules are able
∗E-mail address: josep.sardanes@upf.edu; Phone:+34 933160532; Fax: +34 933160550 (Author for corre-
spondence)

2 Josep Sardany´
es
to establish catalytic connections between them, therefore constituting a circularly closed
network of catalytically-coupled replicators (see Fig. 1a). Catalysis can be directly pro-
vided by the own replicators acting as genotype and phenotype at the same time, or can be
provided by a product serving as a specific replicase for the next hypercycle member. This
theory was initially developed in the context of earlier prebiotic evolution and from then
to now, a lot of theoretical works have explored the dynamics of hypercycle systems (see
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]). Hypercycle theory goes
beyond purely abstract models since some experimental works with synthetic replicators
[21] and with viral kinetics [22] have reported this type of organization in real systems.
The importance of the hypercycle theory in the origin of life problem is extremely relevant
since several selective advantages have been attributed to these kind of catalytic networks
[1, 23, 24, 16, 18], among them, the possibility to overcome the so-called error threshold
[1, 24, 15].
Although the relevant evolutionary properties of hypercycles, some problems as-
sociated to their topology and kinetic properties have been discussed [15, 25, 1, 23].
Hypercycles are vulnerable to so-called shortcuts and parasites (see Fig. 1a). Parasites
are replicators that receive catalytic help but do not catalyze the replication of any other
member of the hypercycle. Although hypercycles are sensitive to parasites because they
modify the initially functional graph structure of the catalytic network, it is known that
spatially-extended hypercycles are resistant to parasites [7, 14]. Catalytic cooperation
requires the presence of all the interacting components. The lack of a single species
destroys the hypercycle because the catalytic chain is broken [15, 26]. It is known that
large hypercycles undergo self-maintained oscillations in population numbers over time.
If one considers intrinsic or extrinsic noise in the dynamics (due to small population sizes
or to environmental perturbations, respectively), a given species of the hypercycle could
become extinct near the lowest concentration of the oscillation, involving the collapse of
the entire hypercycle.
Mean field models - The hypercycle was initially studied on the theoretical formalism of
deterministic chemical kinetics by using continuous time, autonomous ordinary differential
equations (ODEs) [1]. This mean field approach, which assumes perfectly mixed systems,
provides a statistical description of an average magnitude (e.g., replicator’s concentrations),
and assumes equiprobability of interactions, which are made proportional to the densities
of the replicators following the mass action law [28, 29]. It is known that the dynamics of
elementary hypercycles are governed by an asymptotically stable state of permanent coex-
istence between all the species [1, 15]. The closed architecture of hypercycles discards the
competitve exclusion of any replicator type [1]. Mean field models indicate that hypercy-
cles with n= 2 members have a stable node coexistence attractor in phase space [8, 30].
With n= 3 and n= 4 species, the coexistence stable state is respectively achieved by
means of spiral trajectories with fastly (n= 3) and hardly (n= 4) damped oscillations
[1, 17]. Several works also show that the dynamics of hypercycles with 5or more species
can be governed by attracting, invariant periodic orbits [1, 31].
Mean field models are extremely useful to characterize the qualitative behavior of
biological systems because one can calculate both analytically or numerically the values
of the equilibria of the system and their associated stability. This may allow us to make

The Hypercycle: From Molecular to Ecosystems Dynamics 3
0
10
20
30
40
50
0
10
20
30
40
50
0
1
2
3
+
+
+
++
x
y
P
I
I
I1
n
I
2
n−1
I1
(a)
+
species Z
species Y
species X
species A
X
Z
Y
(d)
+
+−
−
+
+
+
+−
−
+
+
+
(c)
(b)
Figure 1. (a) Elementary hypercycle of nmembers, I1...n, with a short-cut (dotted line) and
a parasite, P(dashed arrow). (b) Spiral waves in a 5member spatially extended hypercycle
model simulated with a two-dimensional reaction-diffusion model (see [11] for details). (c)
Coupling of the three-species food chain analyzed by Hastings and Powell (see [27]) with a
hypercycle-like symbiosis between the prey (species X) an the auxiliar species (species A).
Here Yrepresents a predator of species Xand Za top predator preying on Y. (d) Same
as in (c) now considering an ecosystem with eight species with symbiotic and predator-
prey interactions, where the prey and the predator, Y, cooperate with a given set of species
(cooperation and antagonistic interactions are indicated, respectively, with +and −signs).
approximate qualitative and/or quantitative predictions of the population numbers of the
studied species. However, this may not be an easy task for chaotic systems since chaos
involves sensitive dependence on initial conditions, and two nearby trajectories will diverge
exponentially over time [29, 32]. This dynamical property imposes a time horizon beyond
which prediction becomes impossible [29, 32, 28]. For instance, for a system with an stable
point attractor, any arbitray initial condition will evolve towards the same equilibrium and
measurement errors in the initial conditions will not affect the final equilibrium state. This
may not be the case for a chaotic system, which is inherently unpredictable in the long
term. We can also study the bifurcations associated to a given set of ODEs. Bifurcations are
extremely relevant phenomena because they indicate the relation between the parameters
of the model and the transition between states. For instance, from survival to extinction.
Bifurcations are also interesting because they often involve dynamical changes in the
transients towards equilibria near bifurcation thresholds [29, 17, 18, 33].
Spatially-extended hypercycles - Some works have also investigated the spatio-temporal
dynamics of hypercycles using cellular automata (CA) and reaction-diffusion models given
by partial differential equations (PDE). CA are discrete models in which each lattice site
can be empty or can contain a single molecule or group of molecules (or organisms). These
molecules may interact with others placed in a given neighbourhood according to certain
state-transition rules, which often are probabilistic events in the so-called stochastic CAs.
PDE models are approximated by finite difference equations, which are solved numerically
on a given spatial domain.
Both CA and PDE models can be applied to 2Dor 3Dmedia, depending if we want to
characterize the dynamics on a surface or in solution. Many of these models applied to the
hypercycle show the formation of large-scale spatial patterns as clusters, spots or travelling
spiral waves (see Fig. 1b) (see for example [7, 10, 11, 12, 13, 14, 16, 20]). Space is crucial

4 Josep Sardany´
es
in shaping ecological processes and spatial self-structuring in incompletely mixed media
can profoundly change the outcome of evolutionary processes [7, 34, 28, 20]. A very
good example of this phenomenon was provided by Boerlijst and Hogeweg. These authors
showed, by means of a CA model, that the emergence of spirals makes the hypercycle
resistant to a large class of parasites [7]. The resistance of the hypercycle to parasites in
spatial models thas also been afterwards demonstrated in other works [14, 19, 16, 20].
Some spatial models with hypercycles suggest the presence of chaos in these systems
[12, 20].
Hypercycles and ecology - As mentioned in the previous sections, the hypercycle is
a multimolecular system of reproducing entities which cooperate through catalysis. This
particular type of organization, and thus also the mathematical formalisms developed for
its study, can be extended to ecological systems in which species cooperate via mutual-
istic symbiosis [24]. From an ecological viewpoint, crosscatalysis and autocatalysis may
be qualitatively equivalent to interspecific (or mutualism) and intraspecific cooperation,
respectively. Mutualism is very common in nature (see [35, 36, 37, 38] and references
therein). Moreover, there are several works describing intraspecific cooperation (see [39]
for a review) in social insect colonies [39, 40], in blood sharing described in populations
of the vampire bat species Desmodus rotundus [39], as well as in primates [41]. In a wide
sense, catalysis may be generically considered as any kind of biological interaction (food
supply, protection against predators or favorable habitat modifications) between individuals
involving an increase in the fitness (e.g., higher growth rate) of one or more species of the
ecosystem. Many of the results and conclusions obtained with the abovementioned works
with the hypercycle may be extended to ecological systems.
The hypercycle can be studied only considering nonlinear growth (due to catalysis) of
the replicators [14, 20, 17, 30]. This case for an ecosystem may correspond to an example
of so-called obligate symbiosis, described between several species as the deep-sea tube-
worm Lamellibrachia luymesi and microbial symbionts [36], or in arbuscular-mycorrhizal
fungi which form obligate symbiosis with more than 80% of plant species in all terrestrial
habitats [42]. Obligate bacterial endosymbionts are also found in nearly all members of the
Aphidoidea [43]. However, there are some theoretical works that consider both malthusian
and nonlinear amplification of the reproducing species [1, 8, 12, 16], thus modeling a set
of species wich can also reproduce without cooperation in a kind of facultatitve symbiosis
where the parameter associated to the crosscatalytic interaction may denote the degree of
symbiosis. Facultative symbiosis has been also described between some species [44, 45].
By using the hypercycle theory one can modify already existing models allowing a more
realistic approach to the complexity of natural ecosystems by considering several types of
interactions between organisms at the same time. For instance, considering dynamics of
competition or antagonistic relations as predator-prey or host-parasite dynamics coupled
with mutualistic networks (see some examples in Figs. 1c,d). Ecological hypercycles may
also include parasites. Here, a given species would receive help and increase its survival
or reproductive success, but would not reciprocate such a help. For example, the so-called
interspecific brood parasitism is a widespread phenomenon in birds, insects and other taxa
(see [46, 47, 48, 49]).

The Hypercycle: From Molecular to Ecosystems Dynamics 5
2. Lanscape Ecology: Interactions in Space
Theoretical and landscape ecology - A large effort in the understanding of the popula-
tion dynamics in ecology has given place to a huge amount of scientific literature (see
[37, 50, 51] and references therein). Ecology has mainly been a descriptive science al-
though theoretical works initiated by Lotka, Volterra, Nicholson and Bailey, among others,
have provided a formal and deeper insight into the dynamics of the species in ecosystems.
Nonlinear dynamical systems theory was early used to study the dynamical behavior in
ecological systems including competition or antagonistic interactions. Actually, these type
of interactions in ecology have been much more widely studied than the dynamics of co-
operation or mutualism even though its importance is comparable to that of predator-prey
and competition interactions [37]. The interest in the dynamics of cooperation in purely
theoretical ecology could be extended to other disciplines of ecology as landscape ecology.
Landscape ecology is nowadays a scientific discipline comprising large and heteroge-
neous areas of study. This branch of ecology is clearly an applied science, having important
links to fields such as forestry and agriculture, which has always involved a strong human
impact on ecosystems [52]. Landscape ecology explicitly includes humans as entities that
cause functional changes on the landscape [53]. Both direct and indirect effects of hu-
man populations on ecosystems in many parts of the world have caused drastic changes
in ecosystem’s composition. These changes have driven endangered species to survive in
habitats that have been partially destroyed by human action thus increasing ecosystem’s
fragmentation. It has been estimated that 43% of terrestrial ecosystems are exploited by
human activity [54].
Indirect effects of human activity as climate change are also influencing ecosystem’s
characteristics. Climate change links multitude of phenomena, including the greenhouse
effect causing global warming, more atmospheric dust, jet stream movements, grater cli-
matic variablity, shift of vegetation zones, disappearance of familiar vegetation types, ocean
warming, migration of agriculture, changes in precipitation regimes, increased soil erosion,
greater fire frequency, pest outbreaks, melting glaciers, etc. [55]. Climate change is another
major component in structuring current research in landscape ecology. Ecotones, as a ba-
sic unit in landscape studies, may have significance for management under climate change
scenarios, since change effects are likely to be seen at ecotones first because of the unstable
nature of a fringe habitat [56].
The main principles underlying landscape ecology are (i) the development and the dy-
namics of spatial heterogeneity, (ii) the interactions and exchanges between heterogeneous
ladnscapes, (iii) the influence of spatial heterogeneity on biotic and abiotic processes, and
(iv) the management of spatial heterogeneity. All of these four principles mention hetero-
geneity, which marks the main difference between both landscape and traditional ecology
[57]. Heterogeneity is the measure of how different parts of a landscape are from one an-
other. Such differences can involve changes in species’ growth/death rates, or even changes
in migration or dispersal. Landscape ecology looks at how spatial structure affects organism
abundance at the landscape level, as well as the behavior and functioning of the landscape
as a whole. This includes the study of the pattern, or the internal order of a landscape, on
process, or the continuous operation of functions of organisms [58].
Environmental heterogeneity can be introduced in a simple way in theoretical models

6 Josep Sardany´
es
describing the dynamics of spatially-extended ecosystems. One way is by considering a
parameter as a function of space. For this case we will need a spatially-explicit model
(e.g., reaction-diffusion or CA models). For instance, predator-prey spatial dynamics was
studied with a one dimensional, reaction-diffusion model considering the prey growth rate
as a linear function of space [34]. The hypercycle has not been deeply analyzed considering
spatial models with environmental heterogeneities. In this sense, the study of the dynamics
of cooperation in such environments might be useful to characterize how the properties of
the habitat affect the dynamical outcome of cooperation between species. It may also be of
interest to analyze the interplay between competition, and/or antagonistic interactions with
cooperation by considering heterogeneities in the environment.
Another possible way to introduce spatial heterogeneity in theoretical models is by
using patch models in the so-called metapopulation approach. A metapopulation is a pop-
ulation consisting of spatially-separate subpopulations that are connected by the dispersal
or migration of organisms. Each subpopulation (or local population) as well as the whole
metapopulation changes in size over time, and may persist for long periods [59]. Patch, a
term fundamental to landscape ecology, is defined as a relatively homogeneous area that
differs from its surroundings [50]. Patches are the basic unit of the landscape that change
and fluctuate in a process called patch dynamics. Patches have a definite shape and spatial
configuration, and can be described compositionally by internal variables such as num-
ber of trees, number of tree species, height of trees, or other similar measurements [50].
In metapopulation theoretical models one can associate a fixed value to a given parame-
ter (e.g., intrinsic growth rate of a species) in a given patch, thus having different growth
kinetics in different patches, having environmental heterogeneities in an implicit space.
Metapopulation models include animals or plant movements between pacthes.
The abovementioned approaches to introduce environmental heterogeneities in spa-
tial models may also serve to study the role of disturbances and habitat fragmentation in
species persistence. Disturbance is an event that significantly alters the pattern of variation
in the structure or function of a system, while fragmentation is the breaking up of a habitat,
ecosystem, or land-use type into smaller parcels [50]. Disturbance is generally considered
a natural process, although human populations can also cause large distrubances in ecosys-
tems. Fragmentation causes land transformation, an important current process in landscapes
as more and more development occurs.
Landscape ecology theory includes the so-called landscape stability principle, which
emphasizes the importance of landscape structural heterogeneity in promoting total
system stability as well as in developing resistance and recovery from disturbances [60].
Integrity of landscape components helps maintain resistance to external threats, including
development and land transformation by human activity [61]. Field studies in landscape
ecology are extremely important to study and monitor the abundance and evolution of
species. However, theoretical models may also be useful to study the time evolution and
the parametrically-dependent persistence regimes of the species forming the ecosystems.
For instance, from single species or multispecies models one can study the nature of the
transitions between states (survival-extinction), which can be catastrophic and largely
irreversible [62, 63]. Spatial heterogeneity could also be studied in species undergoing
both antagonistic and cooperative interactions. Theoretical models studying cooperation in
space may be useful to determine the potential presence of absorving phase transitions or

The Hypercycle: From Molecular to Ecosystems Dynamics 7
bifurcations driving to species collapses.
Future research directions - The aim of this work is to nearer the hypercycle theory
to an audience of ecologists and landscape ecologists. Although the applicability of the
hypercycle theory to ecosystem’s dynamics has been previously suggested [24], the major-
ity of works with hypercyclic networks have been largely developed in the framework of
the molecular world in the prebiotic scenario. The important relationships between spatial
patterns and ecological processes has been continuously discussed in many works. The
linkage of time, space, and environmental change can assist land managers in applying land
management plans to solve environmental problems [61]. Modelization with hypercycles
may also play an important role in order to decide a given strategy for the conservation
of species undergoing cooperation. Some of the ideas for new research on theoretical and
landscape ecology related to the dynamics of cooperation have been discussed in the pre-
vious sections. The main ones are the integration of cooperative interactions in classical
multispecies models in ecology (e.g., Lotka-Volterra, Rosenzweigh-McArthur or Nichol-
son and Bailey). This integration might serve to increase the complexity of food webs by
considering mutualistic interactions as a way to increase the fitness of the interacting popu-
lations. The number of nonlinearities in these kind of models could involve the appearance
of interesting and novel dynamic behaviors. Another interesting research line would be
the characterization of the dynamical outcome of replicator entities undergoing coopera-
tion and competing between them. For instance, to study a given set of replicators with
hypercycle-like couplings competing for the same resources (mononucleotides, nutrients or
space) with a population of malthusian replicators.
More closely related to landscape ecology, it would be also interesting to characterize
all the previous systems considering cooperation in an explicit space with environmental
heterogeneity. This may be possible with spatially-explicit models (e.g., CA or reaction-
diffusion models) considering a space-varying parameter. Moreover, heterogeneity can also
be considered in models with implicit space as in the metapopulations studies. Here, the
parameters of growth and/or catalysis may take different values in each patch as a way to
simulate a mosaic landscape with different environmental conditions. For these approaches,
it would be interesting to explore the role of migration or dispersal in both local and global
scales. As previously mentioned, hypercycle dynamics has been widely studied with au-
tonomous ODEs. However, the analysis of the hypercycle with nonautonomous ODEs may
open a very interesting field of research to study the dynamics of cooperation in time-
dependent parameter scenarios simulating external fluctuations. In this sense, some works
have explored population dynamics in fluctuating media by using this theoretical approach
(see for example [64, 65, 66]). Such an approach may also be useful to study the interplay
between cooperation and competition and/or victim-exploiter systems with time-varying
parameters.
Acknowledgments
I want to thank Ricard V. Sol´
e for useful advices and suggestions during the preparation
of this manuscript. This work was finished in H. Collsacabra, placed in the inspiring land-
scapes of Santa Maria de Corc´
o (L’Esquirol). This work has been supported by the EU

8 Josep Sardany´
es
PACE grant within the 6th Framework Program under contract FP6-002035 (Programmable
Artificial Cell Evolution).
References
[1] M. Eigen and P. Schuster. The Hypercycle. A Principle of Natural Self-Organization.
Berlin: Springer–Verlag, 1979.
[2] C. Bresch, U. Niesert, and D. Harnasch. Hypercycles, Parasites and Packages. J.
theor. Biol.,85(3):399–405, 1980.
[3] M. Eigen, W. C. Gardiner, and P. Schuster. Hypercycles and Compartments. Com-
partment Assists - but do not replace - Hypercyclic Organization of Early Genetic
Information. J. theor. Biol.,85:407, 1980.
[4] U. M¨
uller-Herold. What is a hypercycle ? J. theor. Biol.,102:569–84, 1983.
[5] Bernd-Olaf. K¨
uppers. Molecular Theory of Evolution, chapter The Evolution of Hy-
percycles. Springer-Verlag, 1985.
[6] B. Aulbach and D. Flockerzi. The past in short hypercycles. J. Math. Biol.,27:223–
231, 1989.
[7] M. C. Boerlijst and P. Hogeweg. Spiral wave structure in pre-biotic evolution: hyper-
cycles stable against parasites. Physica D,48:17–28, 1991.
[8] J. C. Nu˜
no, F. Montero, and F. J. de la Rubias. Influence of External Fluctuations on
a Hypercycle Formed by two Kinetically Indistinguishable Species. J. theor. Biol.,
165:553–575, 1993.
[9] J. C. Nu˜
no, M. A. Andrade, F. Mor´
an, and F. Montero. A model of an autocatalytic
network formed by error-prone self-replicative species. Bull. Math. Biol.,55(2):385–
415, 1993.
[10] M. B. Cronhjort. Hypercycles versus parasites in the origin of life: model dependence
in spatial hypercycle systems. Origins Life Evol. Biosphere,25, 1995.
[11] M. B. Cronhjort and C. Blomberg. Hypercycles vs Parasites in a Two-Dimensional
Partial Differential Equations model. J. theor. Biol.,169:31–49, 1994.
[12] P. Chac´
on and J. C. Nu˜
no. Spatial dynamics of a model for prebiotic evolution. Phys-
ica D,81:379, 1995.
[13] M. C. Boerlijst and P. Hogeweg. Attractors and Spatial Patterns in Hypercycles with
Negative Interaction. J. theor. Biol.,176:199–210, 1995.
[14] M. B. Cronhjort and C. Blomberg. Cluster compartmentalization may provide resis-
tance to parasites for catalytic networks. Physica D,101:289–298, 1997.

The Hypercycle: From Molecular to Ecosystems Dynamics 9
[15] B. M. R. Stadler and P. F. Stadler. Molecular Replicator Dynamics. Adv. Complex
Systems,6:47–77, 2002.
[16] P-J. Kim and H. Jeong. Spatio-temporal dynamics in the origin of genetic information.
Physica D,203:88–99, 2005.
[17] J. Sardany´
es and R. V. Sol´
e. Delayed transitions in non-linear replicator networks:
About ghosts and hypercycles. Chaos, Solitons & Fractals,31(2):305–315, 2007.
[18] J. Sardany´
es and R. V. Sol´
e. Ghosts in the origins of life ? Int. J. of Bifurc. and Chaos,
16(9):468–482, 2006.
[19] C. S-O. Attolini and P. F. Stadler. Evolving towards the hypercycle: A spatial model
of molecular evolution. Physica D,217:134–141, 2006.
[20] J. Sardany´
es and R. V. Sol´
e. Spatio-temporal dynamics in simple asymmetric hyper-
cycles under weak parasitic coupling. Physica D,231:116–129, 2007.
[21] D. H. Lee, K. Severin, Yokobayashi. Y., and M. Reza Ghadiri. Emergence of symbio-
sis in peptide self-replication through a hypercyclic network. Nature,390:591–594,
1997.
[22] M. Eigen, C. K. Biebricher, M. Gebinoga, and W. C. Gardiner. The hypercycle. Cou-
pling of RNA and protein synthesis in the infection cycle of an RNA bacteriophage.
Biochemistry,30(46):11005–11018, 1991.
[23] Stuart A. Kauffman. The Origins of Order. Self-organization and Selection in Evolu-
tion, chapter 9: Hypercycles and coding. Oxford University Press, New York, 1993.
[24] J. Maynard Smith and E. Szathm´
ary. The major transitions in evolution, chapter The
evolution of templates. Oxford University Press, 1995.
[25] J. Maynard Smith. Hypercycles and the origin of life. Nature,280:445–446, 1979.
[26] R. V. Sol´
e and B. Goodwin. Signs of Life: how complexity pervades biology, chapter
Living at the edge of catastrophe. Basic Books, 2000.
[27] A. Hastings and T. Powell. Chaos in a three-species food chain. Ecology,72(3):869–
903, 1991.
[28] R. V. Sol´
e and J. Bascompte. Self-Organization in Complex Ecosystems, chapter Spa-
tial Self-Organization. From pattern to Process. Princeton University Press, 2007.
[29] S. H. Strogatz.
“Nonlinear Dynamics and Chaos with applications to Physics, Biol-
ogy, Chemistry and Engineering”
. Westview Press, 2000.
[30] J. Sardany´
es and R. V. Sol´
e. Bifurcations and phase transitions in spatially-extended
two-member hypercycles. J. theor. Biol.,243:468–482, 2006.
[31] J. Sardany´
es. Error threshold ghosts in a simple hypercycle with error-prone self-
replication. Chaos, Solitons and Fractals,35(2):313–319, 2008.

10 Josep Sardany´
es
[32] H. G. Schuster.
“Deterministic Chaos. An Introduction”
. physik-verlag, Weinheim,
1996.
[33] E. Fontich and J. Sardany´
es. General scaling law in the saddle-node bifurcation: a
complex phase space study. J. of Phys. A: Math. and theor.,41(015103):1–9, 2008.
[34] M. Pascual. Diffusion-induced chaos in a spatial predator-prey system. Proc. R. Soc.
Lond. B.,251:1–7, 1993.
[35] L. A. Dugatkin.
“Cooperation among animals: an evolutionary perspective”
. Oxford
University Press, New York, 1997.
[36] E. E. Cordes, M. A. Arthur, K. Shea, R. S. Arvidson, and C. R. Fisher. Modeling the
Mutualistic Interactions between Tubeworms and Microbial Consortia. PloS Biology,
3(3):e77, 0497–0506, 2005.
[37] J. D. Murray. Mathematical Biology, chapter Continuous Models for Interacting Pop-
ulations. Springer-Verlag, 1990.
[38] M. Bright and O. Giere. Microbial symbiosis in Annelida. Symbiosis,38:1–45, 2005.
[39] L. A. Dugatkin. Animal cooperation among unrelated individuals. Naturwiss-
eschaften,89:533–541, 2002.
[40] D. L. Cassill, J. Butler, S. B. Vinson, and D. E. Wheeler. Cooperation during prey
digestion between workers and larvae in the ant, Pheidole spadonia.Insect. Soc.,
52:339–343, 2005.
[41] M. K. Chen and M. Hauser. Modeling reciprocation and cooperation in primates:
evidence for a punishing strategy. J. of Theor. Biol.,235:5–12, 2005.
[42] S. E. Smith and D. J. Read.
“Mycorrhizal symbiosis”
. Academic Press, London, UK,
1996.
[43] M. A. Munson, P. Baumann, and M. A. et al Clark. Aphid-eubacterial endosymbiosis:
evidence for its establishment in an ancestor of four aphid families. J. of Bacter.,
173:6321–4, 1991.
[44] J. A. Russell, A. Latorre, N. Sabater-Mu˜
noz, A. Moya, and N. A. Moran. Side-
stepping secondary symbonts: widespread horizontal transfer across and beyond the
Aphidoidea. Mol. Ecol.,12:1061–75, 2003.
[45] R. Tumlison and S. E. Trauth. A novel facultative mutualistic relationship between
bufonid tadpoles and flagellated green algae. Herpet. Conserv. and Biol.,1(1):51–5,
2006.
[46] M. J. Lawes and S. Kirkman. Egg recognition and interspecific brood parasitism rates
in red bishops (Aves: Ploceidae). Anim. Behav.,52:553–563, 1996.
[47] Haraguchi Y. and H. Seno. Mathematical Model of the Population Dynamics of Brood
Parasitism: What is the Benefit of the Host ? J. theor. Biol.,174:281–297, 1995.

The Hypercycle: From Molecular to Ecosystems Dynamics 11
[48] M. Robert and G. Sorci. The evolution of obligate interspecific brood parasitism in
birds. Behav. Ecol.,12(2):128–133, 1995.
[49] C. Kelly. A Model to Explore the Rate of Spread of Mimicry and Rejection in Hypo-
thetical Populations of Cuckoos and their Hosts. J. theor. Biol.,125:283–299, 1987.
[50] R. T. T. Forman.
“Land Mosaics: The Ecology af Landscapes and Regions”
. Cram-
bidge University Press, UK, 1995.
[51] R. V. Sol´
e and J. Bascompte.
“Self-Organization in Complex Ecosystems”
. Princeton
University Press, 2007.
[52] L. Ryszkowski.
“Landscape Ecology in Agroecosystems Management”
. CRC press,
Boca Raton, Florida, USA, 2002.
[53] J. Sanderson and L. D. Harris.
“Landscape Ecology: A Top-down Approach”
. Lewis
Publishers, Boca Raton, Florida, USA, 2000.
[54] R. V. Sol´
e and B. Goodwin. Signs of Life: how complexity pervades biology, chapter
The Baroque of Nature. Basic Books, 2000.
[55] R. T. T. Forman. Land Mosaics: The ecology of landscapes and regions, chapter Wind
and water flows in mosaics. Crambidge University Press, UK, 1995.
[56] S. Walker, W. J. Barstow, J. B. Steel, G. L. Rapson, B. Smith, W. M. King, and Y. H.
Cottam. Properties of ecotones: evidence from five ecotones objectively determined
from a coastal vegetation gradient. J. of Vegetation Sci.,14:579–590, 2003.
[57] M. G. Turner and R. H. Gardner.
“Quantitative Methods in Landscape Ecology”
.
Springer-Verlag, NY, USA, 1991.
[58] M. G. Turner. Landscape ecology: the effect of pattern on process. Ann. Rev. Ecol.
and Systematics,20:171–197, 2005.
[59] R. T. T. Forman. Land Mosaics: The ecology of landscapes and regions, chapter
Species movement in mosaics. Crambidge University Press, UK, 1995.
[60] R. T. T. Forman and M. Godron.
“Landscape Ecology”
. John Wiley and Sons, Inc.,
NY, USA, 1996.
[61] M. G. Turner, R. H. Gardner, and R. V. O’Neil.
“Landscape Ecology in Theory and
Practice”
. Springer-Verlag, NY, USA, 2001.
[62] R. V. Sol´
e. Scaling laws in the drier. Nature,449:151–3, 2007.
[63] M. Rietkerk. Self-Organized Patchiness and Catastrophic Shifts in Ecosystems. Sci-
ence,305:1926–9, 2004.
[64] T. G. Hallam and C. E. Clark. Non-autonomous Logistic Equations as Models of
Populations in a Deteriorating Environment. J. theor. Biol.,93:303–11, 1981.

12 Josep Sardany´
es
[65] T. G. Hallam and M. Zhien. Persistence in population models with demographic
fluctuations. J. Math. Biol.,24:327–39, 1986.
[66] S. Rosenblat. Population Models in a Periodically Fluctuating Environment. J. Math.
Biol.,9:23–36, 1980.