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Long-term evolutionary dynamics of continuous phenotypes is ultimately driven by birth and death rates of individual organisms. In general, these birth and death rates are determined in a complicated way by many different phenotypic properties, which could be as diverse as the molecular efficiency of photosynthesis and... |
III Model and Results We use adaptive dynamics theory Geritz et al. 1998 ); Dieckmann & Law ( 1996 to study the long-term evolutionary dynamics in a large class of multidimensional single-species competition models. The starting point is the widely used logistic model Doebeli ( 2011 |
[EQUATION] Here [MATH] is the density of individuals of phenotype [MATH] at time [MATH] , and [MATH] is the carrying capacity of a monomorphic population consisting entirely of [MATH] -individuals. The competitive impact between individuals of phenotypes [MATH] and [MATH] is given by the competition kernel [MATH] , so ... |
Doebeli ( 2011 . Here we assume more generally that [MATH] is a [MATH] -dimensional vector describing [MATH] scalar phenotypic properties. We also assume that [MATH] for all [MATH] , and that the intrinsic growth rate [MATH] is independent of the phenotype [MATH] and is equal to 1. To derive the adaptive dynamics of th... |
[EQUATION] The selection gradient [MATH] is derived from the invasion fitness as [EQUATION] Finally, the adaptive dynamics of the trait [MATH] is |
[EQUATION] where [MATH] is a [MATH] -matrix describing the mutational process in the [MATH] phenotypic components Leimar ( 2009 ); Doebeli ( 2011 (and where [MATH] and [MATH] are column vectors). In general, the entries of [MATH] depend on the current population size, and hence implicitly on [MATH] , but for simplicity... |
Complicated dynamics in the form of oscillations can already occur if the selection gradient [MATH] in (4) is linear. In fact, with randomly chosen coefficients, the probability of oscillatory behaviour is [MATH] , and hence rapidly approaches 1 as the dimension of phenotype space is increased Edelman ( 1997 . To study... |
[EQUATION] For the carrying capacity, we assume a simple symmetric form: [MATH] , which, together with the competition kernel gradient ( ), ensures that the trajectories of the adaptive dynamics (4) are confined to a finite region of phenotype space. With these assumptions, the adaptive dynamics (4), describing the evo... |
[EQUATION] The parameters [MATH] and [MATH] reflect the epistatic interactions among the [MATH] phenotypic components. If [MATH] for [MATH] and [MATH] for [MATH] , there is no epistasis between the phenotypic components, and in that case, the adaptive dynamics (6) always converges to an equilibrium in phenotype space. ... |
Devaney ( 1986 (see Methods ). Based on Lyapunov exponents, we classified the evolutionary trajectories into three groups: fixed points ( [MATH] ), periodic or quasi-periodic attractors ( [MATH] ), and chaotic attractors ( [MATH] ). Examples are shown in Fig. 1. |
Our main result is that the probability of chaos increases with the dimensionality [MATH] of the evolving system, approaching 1 for |
[MATH] (Fig. 2). Moreover, already for [MATH] , the majority of chaotic trajectories become ergodic Devaney ( 1986 , and hence essentially fill out the available phenotype space over evolutionary time. The size of the filled phenotype space scales approximately as [MATH] for each phenotypic dimension [MATH] Methods ), ... |
It is important to note that because these ergodic trajectories fill out large areas of phenotype space, they are very different from noisy equilibrium points. Finally, we observe that the largest Lyapunov exponent converges to the universal asymptotic [MATH] (Fig. 4). |
In Methods we provide qualitative analytical explanations for these numerical results. In particular, we derive an analytical approximation for the probability of chaos as a function of the dimension of phenotype space (Fig. 9) by arguing that a trajectory is chaotic if all fixed points of system (6) have at least one ... |
IV Discussion Ergodic chaos in long-term evolutionary dynamics offers two main conceptual perspectives. First, frequency-dependent ecological interactions can generate complicated evolutionary trajectories that visit all feasible regions of phenotype space in the long run even if the external environment (given by syst... |
Second, chaotic evolutionary trajectories are intrinsically unpredictable. At the very basic level, biological evolution is stochastic, because the single-molecule events that correspond to spontaneous mutations are subject to fundamental, quantum mechanical randomness. The adaptive dynamics models considered here are ... |
Our results are also relevant for the general problem of the prevalence of chaos in multidimensional dynamical systems. Chaos is well studied in high-dimensional Hamiltonian systems Zaslavsky et al. 1991 , as well as in discrete-time systems of coupled oscillators with many degrees of freedom Kaneko ( 1989 ); Ishihara ... |
Because the likelihood of chaotic evolutionary dynamics in our models is strongly influenced by the dimensionality of phenotype space, the biological relevance of our results hinges on the number of phenotypic properties affecting ecological interactions in real systems, and on the potential for epistasis between these... |
It would seem to be an important empirical endeavour to gain a general understanding of the number of different phenotypic properties that can be expected to affect ecological interactions, and of the degree of epistasis between them. |
Even if one accepts the premise of high-dimensional phenotype spaces, one could question the realism of the logistic competition models used here. While it is true that our models do not derive from an underlying mechanistic model for ecological interactions between individual organisms, our statistical approach examin... |
For now, our results warrant at least a critical re-examination of the generality of simple equilibrium and optimization dynamics in evolution. 40 years ago, the realization that simple ecological models can have very complicated dynamics revolutionized ecological thinking May ( 1976 . Our high-dimensional models are n... |
Methods Here we describe how the largest Lyapunov exponent is calculated, illustrate the divergence of chaotic trajectories, and provide approximate analytical explanations for the numerical results reported for the size of chaotic attractors (Figure 2), the probability of chaos as a function of the dimension [MATH] of... |
V.1 Calculation of Lyapunov exponents For each trajectory obtained through numerical integration of the adaptive dynamics (6), the time average of the largest Lyapunov exponent [MATH] was calculated as follows. Every [MATH] time units the trajectory was slightly perturbed, [MATH] , by a vector with a constant magnitude... |
[EQUATION] and subsequently averaged over the trajectory. Visual inspection of trajectories led us to the following selection criteria: Trajectories with the [MATH] were usually chaotic, trajectories with the [MATH] were quasi-periodic, and trajectories with [MATH] |
converged to fixed points. Choosing suitable time intervals [MATH] for the numerical calculations of the largest Lyapunov exponent is constrained on both sides. On the one hand, values of [MATH] that are too small do not leave sufficient time for a randomly chosen direction of perturbation to align itself with the dire... |
V.2 Divergence of evolutionary trajectories Consider two trajectories initially separated by a small distance, say [MATH] . Evolution of both trajectories is given by the adaptive dynamics (6). For high dimensions of phenotype space ( [MATH] in this example), the adaptive dynamics is almost certainly chaotic, so one wo... |
V.3 Size of chaotic attractors and magnitude of Lyapunov exponents. First we consider the scaling of the spatial coordinates, [MATH] , illustrated in Fig. 3 and in Figs. 6, 7 and the scaling of the Lyapunov exponent, [MATH] (Fig. 4). |
If we make the reasonable assumption that each phenotypic coordinate has a similar scale, [MATH] , the dynamical system (6) becomes |
[EQUATION] for [MATH] . Here the [MATH] and [MATH] are identically distributed random terms with zero mean and unit variance, and a typical value of the sum of [MATH] such terms is the standard deviation [MATH] , which yields |
[EQUATION] Introducing new variables, [EQUATION] we convert ( ) into the differential equation [EQUATION] with two universal, [MATH] -independent terms and a linear term that vanishes in the limit of large [MATH] The transformation ( 10 ) explains the observed scaling of the size of chaotic attractors, [MATH] (Figs. 3,... |
[MATH] in (6) does not produce any significant effect on the probability of chaos and on the form of the attractor for large [MATH] |
Taking into account ( 10 ), it is possible to rescale the coefficients [MATH] and [MATH] in such a way that the total “strength” of epistatic interactions between the phenotypic components, as well as the size of the area of phenotype space filled out by ergodic trajectories, do not depend on the dimension [MATH] of ph... |
[MATH] V.4 Probability of chaos. Second, we provide an explanation for the increase in the occurrence of chaos with the dimension [MATH] of phenotype space. Stationary points of the adaptive dynamics (6) are defined as solutions of the corresponding system of algebraic equations where the right-hand side is set equal t... |
[MATH] has a positive real part. If [MATH] is the probability that the real part of an eigenvalue is negative, and assuming that all Jacobian eigenvalues are statistically independent, the probability that at least one out of |
[MATH] eigenvalues of the Jacobian at a stationary point has a positive real part is [MATH] . Hence the probability of chaos is [EQUATION] |
If [MATH] is a stationary point of (6), the elements of the Jacobian matrix [MATH] consist of two terms, [EQUATION] where [MATH] is the identity matrix. Here we ignored the linear term [MATH] |
which we argued above to be negligible for increasing [MATH] We assume that the distribution of [MATH] is the same as for the coordinates [MATH] themselves and is given by the universal invariant measure shown in Fig. 3. This assumption allows us to consider the two terms [MATH] and [MATH] as statistically independent.... |
[MATH] -matrix with Gaussian-distributed elements with zero mean and unit variance are uniformly distributed on a disk in the complex plane with radius [MATH] . Thus, the eigenvalues of |
[MATH] are uniformly distributed on a disk with radius [MATH] . The probability for an eigenvalue to have real part [MATH] , with [MATH] , is then proportional to the length of the chord intersecting the radius of the disk at the point |
[MATH] [EQUATION] (The factor [MATH] normalizes the distribution to one.) Considering the second, diagonal, term of the Jacobian, [MATH] , we rely on the numerical observation that the distribution function [MATH] |
has a universal form, which is independent of [MATH] , and whose scaled form [MATH] , with [MATH] , is given in Fig. 3. Both [MATH] and [MATH] contribute terms of order [MATH] to the eigenvalues of the Jacobian. The contribution from [MATH] may have a positive or a negative real part, and the probability that it has a ... |
[EQUATION] Here the 1/2 term reflects the probability that the eigenvalue of [MATH] has a negative real part, [MATH] , and the double integral gives the probability that the positive real part of [MATH] is smaller than the contribution [MATH] from [MATH] Integration on [MATH] produces |
[EQUATION] Using the numerical data for [MATH] shown in Fig. 3, we calculate [MATH] and perform numerical integration of [MATH] to obtain [MATH] . Substituting this value into Eq. ( 16 ) above provides a reasonable fit for the observed probability of chaos, as illustrated in Fig. 9. |
It is important to note that while the details of the calculations of [MATH] depend on the particular form of the dynamical system (6) and its Jacobian matrix, the conclusion that the probability of chaos increases with the dimension [MATH] is general: If each eigenvalue has a non-vanishing probability to have a positi... |
V.5 Scaling of Lyapunov exponents. Finally, we show that the slow convergence of the largest Lyapunov exponent [MATH] to its scaling asymptotic, as shown in Fig. 4, can be explained as a general consequence of extreme value statistics. We again use the arguments provided above for the fact that the eigenvalues of [MATH... |
[EQUATION] Here the term [MATH] is the probability that the largest eigenvalue is equal to [MATH] , and the integral term gives the probability that the remaining [MATH] eigenvalues are less than [MATH] . The factor [MATH] |
reflects the fact that any of [MATH] eigenvalues could be the largest. To calculate the average value of the largest eigenvalue, |
[EQUATION] we substitute ( 14 ) into ( 17 ) and integrate 18 ) by parts, obtaining [EQUATION] where [MATH] is a constant representing the upper limit of the rescaled largest eigenvalue. Above we ignored the scaling coefficient for [MATH] and the contribution of the diagonal part of Jacobian |
[MATH] , thus were unable to explain the numerical value for this upper limit, [MATH] However, our simple estimate based on the extreme value statistics provides a reasonable description of how the largest eigenvalue [MATH] |
approaches its asymptotic value [MATH] as [MATH] , Fig. 10 Acknowledgements. I. I. was supported by FONDECYT (Chile) project #1110288. M. D. was supported by NSERC (Canada). Both authors contributed equally to this work. |
# Source: arxiv 1404.6267 # Title: Social Evolution: New Horizons # Sections: all # Downloaded: 2026-03-03T01:59:22.759028+00:00 |
More than one Author with different Affiliations Social Evolution: New Horizons Abstract Cooperation is a widespread natural phenomenon yet current evolutionary thinking is dominated by the paradigm of selfish competition. Recent advances in many fronts of Biology and Non-linear Physics are helping to bring cooperation... |
Introduction Cooperation is everywhere but ecological and evolutionary theories are firmly grounded on competition. Cooperation is so common and overwhelming in nature that a simple turn of our head will spot it around immediately, appearing in multiple ways and forms. It is so widespread, so much widespread, that it i... |
or bacteria or even as emergent phenomena in artificial societies of robots or other creatures of the cyberspace . Why are we so late in acknowledging this fact? What is the reason of so many years in which biology has lacked a good evolutionary theory of cooperation and social emergence? Charles Darwin was already awa... |
It was short after the publication of On the Origin of the Species that Herbert Spencer first used the phrase ``survival of the fittest''. The phrase was quickly incorporated into the Darwinian views of biological evolution alongside another masterpiece of ideology uncritically converted into science:``the struggle for... |
But it needed not to be this way. As lucidly stated by S.J. Gould: ``struggle is often a metaphorical description and need not be viewed as overt combat, guns blazing. Tactics of reproductive success include a variety of nonmartial activities such as […] better cooperation with partners in raising offspring." In fact, ... |
Cooperation at the dawn of life It is unknown how the very first living organisms and their ecosystems on earth looked like. However it is known that the most ancient fossilized organisms were cooperative and social. This is the case of the 3.45 billion years-old Cyanobacteria estromatolites (see Figure ). Cyanobacteri... |
Stromatolites aside, the most ancient remains of an ecosystem activity currently known have an estimated age of 3.48 billion years-old. These are mineral structures known as Microbially Induced Sedimentary Structures (MISS) and are thought to have been formed by biofilms of single-cell organisms, likely bacteria |
. It is worth noting that present day biofilms are well-known paradises for the emergence of complex social interactions and cooperative phenomena among microorganisms (Figure ). In fact, it is remarkable that Horizontal Gene Transfer (HGT) was initially discovered in bacteria and that this mechanism of gene transferri... |
. In order for HGT to work there are at least one essential requirement: that the cells involved in the process come and stay together for a while, socially interacting. This is the reason why biofilms are so important in the early evolution of social life and cooperation. The finding of HGT has triggered many fundamen... |
, for example, is the prokaryote diversity and evolution the result of the horizontal exchange of genetic material that allows for the sharing and incorporation of new encoding possibilities, i.e. genetic novelties that are more accessible to selection? or should we stick to the old idea of speciation through random mu... |
? Let's remember just one thing: HGT is a gene mixing process that occurs between different prokaryotic species and even genera Closely related prokaryote species show individual genomes that are highly diverse in terms of gene content. As Cordero and Polz reviewed |
, much of this variation is associated with social and ecological interactions, which have an important role in the biology of wild populations of bacteria and archaea. Genetic diversity requires the delineation of populations according to cohesive gene flows (social interactions) and ecological factors, as micro-evolu... |
In the evolutionary history, shortly after the emergence of the prokaryote, single-cell and multi-cell eukaryote emerged but not outside of a cooperative scenario. At a stage somewhere between grouped-individuals and complex multicellular organisms, the Colonial Invertebrates emerged. This is the case of the siphonopho... |
Cooperation played a crucial role in the emergence of multicellularity , regarded by John Maynard-Smith and colleges as one of the major transitions in the evolution of life |
. It is also interesting to notice that Maynard-Smith regarded the origin of social groups (for example ants, bees, wasps and termites) as another major transition in evolution. He has, however, failed to remark that cooperation is implicated in most of his identified major evolutionary transitions to the point that th... |
Modern post-Darwinian evolutionary theory sees natural selection and randomness as important mechanisms in evolution but argues that these are not the only sources of the extraordinary creativity of nature that we see around |
, something else is missing . Biological evolution did not strictly begin when the first life forms appeared on earth billions of years ago. It is part of a continuum unstopped evolution of matter that started with the big bang and where atoms, molecules and abiotic complex molecules have been built up under the action... |
Social evolution: the past 3.1 The old uncrossed frontier for the ideas on sociality In contrast to HGT, Vertical Gene Transfer (VGT) is the mechanism where transmission of genes occurs from the parental generation to offspring via sexual or asexual reproduction. It is under this mechanism that most of the genetic hypo... |
As mentioned previously, social insects and their eusociality have always been a challenge for the theory of evolution in Biology. Social colonies are composed of cooperative individuals, most of them subfertile or even sterile, which would not succeed in a world ``red in tooth and claw'' where only the strongest and s... |
this is far from settled, being perhaps the highest mountain pass, a formidable barrier we still need to cross in order to fully understand not simply sociality in insects but the very heart of the theory of evolution. |
Examples abound of organisms exhibiting a behaviour in which sterile offspring cohabits with and cooperatively helps their parents to raise fertile offspring, the so-called ``eusociality''. It is found among bees, wasps, ants, termites |
, aphids , thrips , ambrosia beetles , shrimp and naked mole rats . If this definition is relaxed a bit, allowing senile sterility of parents (as opposed to offspring sterility) and keeping the idea of group members containing multiple generations and prone to perform altruistic acts as part of their division of labor |
40 , p. 22] , then we may well add even humans to the list of eusocial animals The past: puzzles, solutions, and more doubts Darwin himself, dedicated a whole chapter to this subject in ``On the Origin of Species'' |
. Describing the puzzle of the existence of cooperative, sterile individuals in social insects, Darwin comments that they represent one special difficulty, which at first appeared to me insuperable, and actually fatal to the whole theory. He circumvented this doubt proposing that queens which are able to produce altrui... |
Darwin's solution for the evolution of cooperation prevailed for nearly one hundred years, until 1964 when William D. Hamilton advanced an elegant mathematical formalism aimed as an attempt to solve the riddle |
. It consisted of the so-called ``kin selection'', which differs from –but does not conflict with– Darwin's proposition by establishing that each member of the colony is targeted by selection individually, rather than together with its parents and siblings. Kin selection predicts that individuals cooperate with family ... |
Box 1 . Relatedness in haplo-diploid systems: suppose a fully heterozygous haplodiploid cross: AB aB In Hymenoptera, all offspring produced from this cross are female (males are produced parthenogenetically). Let's take a look at the degree of relatedness between these sisters: |
sisters AB aB AB 1.0 0.5 aB 0.5 1.0 That is, on average, sisters are related to each other by: [MATH] Haplo-diploidy in Hymenoptera (bees, ants, wasps and sawflies), where males are haploid and females are diploid, was proposed by Hamilton to be the key to the puzzle (see Box 1). A hymenopteran female, by virtue of hap... |
Haplo-diploidy, however, is not sufficient to explain the evolution of eusociality: a significant portion of hymenopteran species, while haplo-diploid, are solitary. Maybe more striking, there are many diplo-diploid organisms (having both, males and females, diploid) which are eusocial (Box 2): all the nearly 3 thousan... |
It was indeed eusociality in termites –consistently overlooked by texts focusing kin selection– that always kept alive the challenge, and even more now when the list of eusocial diplo-diploids is frequently updated. Much effort has been made to conciliate termites with kinship selection |
but, as noted by Thorne et al. , a convincing explanation on why they are eusocial despite their full diploidy is still needed . An important step in this direction was taken by Korb and collaborators |
, who presented a broad overview of the ecology of social evolution across large parts of the animal kingdom, including termites |
and other diplo-diploids, thereby expanding the study beyond haplo-diploids. Box 2 . Relatedness in diplo-diploid systems: in a fully heterozygous diplo-diploid cross we would observe the following offspring: |
AB Ab aB ab This will imply in the following degree of relatedness between each of the siblings: siblings AB Ab aB ab AB 1.0 0.5 |
0.5 0.0 Ab 0.5 1.0 0.0 0.5 aB 0.5 0.0 1.0 0.5 ab 0.0 0.5 0.5 1.0 In such case, the average relatedness between siblings is: [MATH] |
Contempts Meanwhile, it has been argued that the right question has been not posed! In his heavy criticism of the way research has been conducted on kin selection, E.O. Wilson |
claims to have spotted a philosophical fault in such studies: we have been busy trying to accommodate exceptions to the theory, rather than searching for a better theory that accommodates it all. That is, rather than asking how to conform termites and other diploids to kin selection theory, we should have kept Darwin's... |
. In Wilson's (2011, pag. 166) own words: ``[…] unwarranted faith in the central role of kinship in social evolution has led to the reversal of the usual order in which biological research is conducted. The proven best way in evolutionary biology, as in most of science, is to define a problem arising during empirical r... |
Stating that Hamilton's rule `` almost never holds '', Martin Nowak and collaborators brought recent upheaval to the community of scientists supporting kin selection. It attracted immediate reaction in the form of contentious papers |
, readily counteracted by supporting ones . In an attempt to perform neutral analysis of the debate Birch identifies ambiguities in Hamilton's defenders and supporters and offers a common vocabulary to help their communication. In short, he states that while kin selection supporters' argument is based on a general form... |
As an urgent alternative to kin selection as an explanation for the emergence of sociality, Nowak et al. , followed by Wilson , proposed that the full theory of eusocial evolution would include the following stages (taken almost ipsis litteris from |
): 1. The formation of groups. 2. The occurrence of a combination of pre-adaptive traits causing the groups to be tightly formed. Such a combination would include a valuable and defensive nest, they stress. |
3. The appearance of mutations that prescribe the persistence of the group, most likely by the silencing of dispersal behaviour. |
4. Emergent traits caused by the interaction of group members are shaped through natural selection by environmental forces. 5. Multilevel selection drives changes in the colony life cycle and social structures, often to elaborate extremes. |
In essence, these authors consider groups as an additional unit of selection with selection simultaneously occurring at different levels, e.g., between individuals in the group and between groups. They also remark the importance of spatio-temporal mechanisms that cause individuals to come and stay together. Such ideas ... |
(Figure ). Social evolution: the future 4.1 Emergent properties of grouping behaviour Interestingly, some authors view Nowak and colleagues' proposition as complementary rather than alternative to kin selection theory. Better stated, they would claim that this ``new'' group-selection theory is in fact a more general pr... |
, despite strong refutation by Wilson . Based on the empirical evidence compiled in the various chapters of their book for a broad range of animals (both vertebrates and invertebrates; full diploids or haplo-diploids), Korb and Heinze |
agree with Wilson that the newly re-discovered group-selection framework is a promising way to investigate the evolution of social phenomena. |
This view would sustain that while in kin selection models relatedness is paramount, the new group-selection models emphasize between-group versus within-group selection, thereby opening an avenue for studies of group level phenomena. Group level phenomena, in which simple repeated interactions between individuals can ... |
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