State Differentiation by Transient Truncation in Coupled Threshold Dynamics

Dynamics with a threshold input--output relation commonly exist in gene, signal-transduction, and neural networks. Coupled dynamical systems of such threshold elements are investigated, in an effort to find differentiation of elements induced by the interaction. Through global diffusive coupling, novel states are found to be generated that are not the original attractor of single-element threshold dynamics, but are sustained through the interaction with the elements located at the original attractor. This stabilization of the novel state(s) is not related to symmetry breaking, but is explained as the truncation of transient trajectories to the original attractor due to the coupling. Single-element dynamics with winding transient trajectories located at a low-dimensional manifold and having turning points are shown to be essential to the generation of such novel state(s) in a coupled system. Universality of this mechanism for the novel state generation and its relevance to biological cell differentiation are briefly discussed.Comment: 8 pages. Phys. Rev. E. in pres

Correlation of Positive and Negative Reciprocity Fails to Confer an Evolutionary Advantage: Phase Transitions to Elementary Strategies

Economic experiments reveal that humans value cooperation and fairness. Punishing unfair behavior is therefore common, and according to the theory of strong reciprocity, it is also directly related to rewarding cooperative behavior. However, empirical data fail to confirm that positive and negative reciprocity are correlated. Inspired by this disagreement, we determine whether the combined application of reward and punishment is evolutionarily advantageous. We study a spatial public goods game, where in addition to the three elementary strategies of defection, rewarding, and punishment, a fourth strategy that combines the latter two competes for space. We find rich dynamical behavior that gives rise to intricate phase diagrams where continuous and discontinuous phase transitions occur in succession. Indirect territorial competition, spontaneous emergence of cyclic dominance, as well as divergent fluctuations of oscillations that terminate in an absorbing phase are observed. Yet, despite the high complexity of solutions, the combined strategy can survive only in very narrow and unrealistic parameter regions. Elementary strategies, either in pure or mixed phases, are much more common and likely to prevail. Our results highlight the importance of patterns and structure in human cooperation, which should be considered in future experiments

Conditional strategies and the evolution of cooperation in spatial public goods games

The fact that individuals will most likely behave differently in different situations begets the introduction of conditional strategies. Inspired by this, we study the evolution of cooperation in the spatial public goods game, where besides unconditional cooperators and defectors, also different types of conditional cooperators compete for space. Conditional cooperators will contribute to the public good only if other players within the group are likely to cooperate as well, but will withhold their contribution otherwise. Depending on the number of other cooperators that are required to elicit cooperation of a conditional cooperator, the latter can be classified in as many types as there are players within each group. We find that the most cautious cooperators, such that require all other players within a group to be conditional cooperators, are the undisputed victors of the evolutionary process, even at very low synergy factors. We show that the remarkable promotion of cooperation is due primarily to the spontaneous emergence of quarantining of defectors, which become surrounded by conditional cooperators and are forced into isolated convex "bubbles" from where they are unable to exploit the public good. This phenomenon can be observed only in structured populations, thus adding to the relevance of pattern formation for the successful evolution of cooperation.Comment: 7 two-column pages, 7 figures; accepted for publication in Physical Review

Competing associations in six-species predator-prey models

We study a set of six-species ecological models where each species has two predators and two preys. On a square lattice the time evolution is governed by iterated invasions between the neighboring predator-prey pairs chosen at random and by a site exchange with a probability Xs between the neutral pairs. These models involve the possibility of spontaneous formation of different defensive alliances whose members protect each other from the external invaders. The Monte Carlo simulations show a surprisingly rich variety of the stable spatial distributions of species and subsequent phase transitions when tuning the control parameter Xs. These very simple models are able to demonstrate that the competition between these associations influences their composition. Sometimes the dominant association is developed via a domain growth. In other cases larger and larger invasion processes preceed the prevalence of one of the stable asociations. Under some conditions the survival of all the species can be maintained by the cyclic dominance occuring between these associations.Comment: 8 pages, 9 figure

Two-population replicator dynamics and number of Nash equilibria in random matrix games

We study the connection between the evolutionary replicator dynamics and the number of Nash equilibria in large random bi-matrix games. Using techniques of disordered systems theory we compute the statistical properties of both, the fixed points of the dynamics and the Nash equilibria. Except for the special case of zero-sum games one finds a transition as a function of the so-called co-operation pressure between a phase in which there is a unique stable fixed point of the dynamics coinciding with a unique Nash equilibrium, and an unstable phase in which there are exponentially many Nash equilibria with statistical properties different from the stationary state of the replicator equations. Our analytical results are confirmed by numerical simulations of the replicator dynamics, and by explicit enumeration of Nash equilibria.Comment: 9 pages, 2x2 figure

Statistics of Certain Models of Evolution

In a recent paper, Newman surveys the literature on power law spectra in evolution, self-organised criticality and presents a model of his own to arrive at a conclusion that self-organised criticality is not necessary for evolution. Not only did he miss a key model (Ecolab) that has a clear self-organised critical mechanism, but also Newman's model exhibits the same mechanism that gives rise to power law behaviour as does Ecolab. Newman's model is, in fact, a ``mean field'' approximation of a self-organised critical system. In this paper, I have also implemented Newman's model using the Ecolab software, removing the restriction that the number of species remains constant. It turns out that the requirement of constant species number is non-trivial, leading to a global coupling between species that is similar in effect to the species interactions seen in Ecolab. In fact, the model must self-organise to a state where the long time average of speciations balances that of the extinctions, otherwise the system either collapses or explodes. In view of this, Newman's model does not provide the hoped-for counter example to the presence of self-organised criticality in evolution, but does provide a simple, almost analytic model that can used to understand more intricate models such as Ecolab.Comment: accepted in Phys Rev E.; RevTeX; See http://parallel.hpc.unsw.edu.au/rks/ecolab.html for more informatio

Evolutionary Dynamics of Populations with Conflicting Interactions: Classification and Analytical Treatment Considering Asymmetry and Power

Evolutionary game theory has been successfully used to investigate the dynamics of systems, in which many entities have competitive interactions. From a physics point of view, it is interesting to study conditions under which a coordination or cooperation of interacting entities will occur, be it spins, particles, bacteria, animals, or humans. Here, we analyze the case, where the entities are heterogeneous, particularly the case of two populations with conflicting interactions and two possible states. For such systems, explicit mathematical formulas will be determined for the stationary solutions and the associated eigenvalues, which determine their stability. In this way, four different types of system dynamics can be classified, and the various kinds of phase transitions between them will be discussed. While these results are interesting from a physics point of view, they are also relevant for social, economic, and biological systems, as they allow one to understand conditions for (1) the breakdown of cooperation, (2) the coexistence of different behaviors ("subcultures"), (2) the evolution of commonly shared behaviors ("norms"), and (4) the occurrence of polarization or conflict. We point out that norms have a similar function in social systems that forces have in physics

Spontaneous Symmetry Breaking in Directed Percolation with Many Colors: Differentiation of Species in the Gribov Process

A general field theoretic model of directed percolation with many colors that is equivalent to a population model (Gribov process) with many species near their extinction thresholds is presented. It is shown that the multicritical behavior is always described by the well known exponents of Reggeon field theory. In addition this universal model shows an instability that leads in general to a total asymmetry between each pair of species of a cooperative society.Comment: 4 pages, 2 Postscript figures, uses multicol.sty, submitte

Complexity Bounds for Ordinal-Based Termination

`What more than its truth do we know if we have a proof of a theorem in a given formal system?' We examine Kreisel's question in the particular context of program termination proofs, with an eye to deriving complexity bounds on program running times. Our main tool for this are length function theorems, which provide complexity bounds on the use of well quasi orders. We illustrate how to prove such theorems in the simple yet until now untreated case of ordinals. We show how to apply this new theorem to derive complexity bounds on programs when they are proven to terminate thanks to a ranking function into some ordinal.Comment: Invited talk at the 8th International Workshop on Reachability Problems (RP 2014, 22-24 September 2014, Oxford

Stochastic Ballistic Annihilation and Coalescence

We study a class of stochastic ballistic annihilation and coalescence models with a binary velocity distribution in one dimension. We obtain an exact solution for the density which reveals a universal phase diagram for the asymptotic density decay. By universal we mean that all models in the class are described by a single phase diagram spanned by two reduced parameters. The phase diagram reveals four regimes, two of which contain the previously studied cases of ballistic annihilation. The two new phases are a direct consequence of the stochasticity. The solution is obtained through a matrix product approach and builds on properties of a q-deformed harmonic oscillator algebra.Comment: 4 pages RevTeX, 3 figures; revised version with some corrections, additional discussion and in RevTeX forma