Model of biological evolution with threshold dynamics and infinitely many absorbing states

Phase transitions in evolutionary dynamics

Sharp changes in state, such as transitions from survival to extinction, are hallmarks of evolutionary dynamics in biological systems. These transitions can be explored using the techniques of statistical physics and the physics of nonlinear and complex systems. For example, a survival-to-extinction transition can be characterized as a non-equilibrium phase transition to an absorbing state. Here, we review the literature on phase transitions in evolutionary dynamics. We discuss directed percolation transitions in cellular automata and evolutionary models, and models that diverge from the directed percolation universality class. We explore in detail an example of an absorbing phase transition in an agent-based model of evolutionary dynamics, including previously unpublished data demonstrating similarity to, but also divergence from, directed percolation, as well as evidence for phase transition behavior at multiple levels of the model system's evolutionary structure. We discuss phase transition models of the error catastrophe in RNA virus dynamics and phase transition models for transition from chemistry to biochemistry, i.e., the origin of life. We conclude with a review of phase transition dynamics in models of natural selection, discuss the possible role of phase transitions in unraveling fundamental unresolved questions regarding multilevel selection and the major evolutionary transitions, and assess the future outlook for phase transitions in the investigation of evolutionary dynamics.

Universality classes of absorbing phase transitions in generic branching-annihilating particle systems with nearest-neighbor bias

We study absorbing phase transitions in systems of branching annihilating random walkers and pair contact process with diffusion on a one-dimensional ring, where the walkers hop to their nearest neighbor with a bias ε. For ε=0, three universality classes-directed percolation (DP), parity-conserving (PC), and pair contact process with diffusion (PCPD)-are typically observed in such systems. We find that the introduction of ε does not change the DP universality class but alters the other two universality classes. For nonzero ε, the PCPD class crosses over to DP, and the PC class changes to a new universality class.

Multichain models of conserved lattice gas

Conserved lattice-gas models in one dimension exhibit absorbing state phase transition (APT) with simple integer exponents β=1=ν=η, whereas the same on a ladder belong to directed percolation (DP) universality. We conjecture that additional stochasticity in particle transfer is a relevant perturbation and its presence on a ladder forces the APT to be in the DP class. To substantiate this we introduce a class of restricted conserved lattice-gas models on a multichain system (M×L square lattice with periodic boundary condition in both directions), where particles which have exactly one vacant neighbor are active and they move deterministically to the neighboring vacant site. We show that for odd number of chains, in the thermodynamic limit L→∞, these models exhibit APT at ρ_{c}=1/2(1+1/M) with β=1. On the other hand, for even-chain systems transition occurs at ρ_{c}=1/2 with β=1,2 for M=2,4, respectively, and β=3 for M≥6. We illustrate this unusual critical behavior analytically using a transfer-matrix method.

Multicritical absorbing phase transition in a class of exactly solvable models

We study diffusion of hard-core particles on a one-dimensional periodic lattice subjected to a constraint that the separation between any two consecutive particles does not increase beyond a fixed value n+1; an initial separation larger than n+1 can however decrease. These models undergo an absorbing state phase transition when the conserved particle density of the system falls below a critical threshold ρ_{c}=1/(n+1). We find that the ϕ_{k}, the density of 0-clusters (0 representing vacancies) of size 0≤k<n, vanish at the transition point along with activity density ρ_{a}. The steady state of these models can be written in matrix product form to obtain analytically the static exponents β_{k}=n-k and ν=1=η corresponding to each ϕ_{k}. We also show from numerical simulations that, starting from a natural condition, ϕ_{k}(t)s decay as t^{-α_{k}} with α_{k}=(n-k)/2 even though other dynamic exponents ν_{t}=2=z are independent of k; this ensures the validity of scaling laws β=αν_{t} and ν_{t}=zν.

Pair contact process with a particle source

We study the phase diagram and critical behavior of the one-dimensional pair contact process (PCP) with a particle source using cluster approximations and extensive simulations. The source creates isolated particles only, not pairs, and so couples not to the order parameter (the pair density) but to a nonordering field, whose state influences the evolution of the order parameter. While the critical point p(c) shows a singular dependence on the source intensity, the critical exponents appear to be unaffected by the presence of the source, except possibly for a small change in beta. In the course of our paper, we obtain high-precision values for the critical exponents of the standard PCP, confirming directed-percolationlike scaling.

Criticality of natural absorbing states

We study a recently introduced ladder model that undergoes a transition between an active and an infinitely degenerate absorbing phase. In some cases the critical behavior of the model is the same as that of the branching-annihilating random walk with N>/=2 species both with and without hard-core interaction. We show that certain static characteristics of the so-called natural absorbing states develop power-law singularities that signal the approach of the critical point. These results are also explained using random-walk arguments. In addition to that we show that when dynamics of our model is considered as a minimum-finding procedure, it has the best efficiency very close to the critical point.

Criticality in a model with absorbing states

We study a one-dimensional model that undergoes a transition between an active and an absorbing phase. Monte Carlo simulations supported by some additional arguments prompted us to predict the exact location of the critical point and critical exponents in this model. The exponents delta=0.5 and z=2 follows from random-walk-type arguments. The exponents beta=nu( perpendicular) are found to be nonuniversal and encoded in the singular part of reactivation probability, as recently discussed by H. Hinrichsen (cond-mat/0008179). A related model with quenched randomness is also studied.

First-order phase transition with a logarithmic singularity in a model with absorbing states

Recently, Lipowski [Phys. Rev. E 62, 4401 (2000)] investigated a stochastic lattice model which exhibits a discontinuous transition from an active phase into infinitely many absorbing states. Since the transition is accompanied by an apparent power-law singularity, it was conjectured that the model may combine features of first- and second-order phase transitions. In the present work it is shown that this singularity emerges as an artifact of the definition of the model in terms of products. Instead of a power law, we find a logarithmic singularity at the transition. Moreover, we generalize the model in such a way that the second-order phase transition becomes accessible. As expected, this transition belongs to the universality class of directed percolation.

Systems with superabsorbing states

We report on some extensive analyses of a recently proposed model [A. Lipowski, Phys. Rev. E 60, 6255 (1999)] with infinitely many absorbing states. By performing extensive Monte Carlo simulations, we have determined critical exponents and shown strong evidence that this model is not in the directed percolation universality class. The conjecture that this two-dimensional model exhibits a dimensional reduction (behaving as one-dimensional directed percolation) is firmly disproven. The reason for the model not exhibiting standard directed percolation scaling behavior is traced back to the existence of what we call superabsorbing sites, i.e., absorbing sites that cannot be directly activated by the presence of neighboring activity in one or more than one direction. Supporting this claim we present two strong evidences: (i) in one dimension, where superabsorbing sites do not appear at the critical point, the system behaves as directed percolation, and (ii) in a modified two-dimensional variation of the model, defined on a honeycomb lattice, for which superabsorbing sites are very rarely observed, directed percolation behavior is recovered. Finally, a parallel updating version of the model exhibiting a nonequilibrium first-order transition is also reported.

First-order transition with power-law singularity in models with absorbing states

We study one- and two-dimensional models which undergo a transition between active and absorbing phases. The transition point in these models is of a novel type: jump of the order parameter coincides with its power-law singularity. Some arguments supported by Monte Carlo simulations prompted us to predict the exact location of the transition point. Both models possess gaugelike symmetry.

Generic criticality in a model of evolution

Using Monte Carlo simulations, we show that for a certain model of biological evolution, which is driven by nonextremal dynamics, active and absorbing phases are separated by a critical phase. In this phase both the density of active sites rho(t) and the survival probability of spreading P(t) decay as t(-delta), where delta approximately 0.5. At the critical point that separates the active and critical phases delta approximately 0.29, which suggests that this point belongs to the so-called parity-conserving universality class. Such a classification is also supported by finite-size analysis. The model has infinitely many absorbing states and, except for a single point, has no apparent conservation law.

Dimensional reduction in a model with infinitely many absorbing states

Using the Monte Carlo method, we study a two-dimensional model with infinitely many absorbing states. Our estimation of the critical exponent beta is approximately equal to 0.273(5) suggests that the model belongs to the (1+1) rather than to (2+1) directed-percolation universality class. We also show that for a large class of absorbing states the dynamic Monte Carlo method leads to spurious dynamical transitions.