Narrow Results

It is shown that the parallel Bak–Sneppen model proposed by Sornette and Dornic can be used as an extinction model. Yet, to include the environmental effect, it may be better to augment it by a coherent noise rule. The resulting model simulates both the evolution and the extinction phenomena.

In this work, we study the critical behavior of an epidemic propagation model that considers individuals that can develop drug resistance. In our lattice model, each site can be found in one of the four states: empty, healthy, normally infected (not drug resistant) and strain infected (drug resistant) states. The most relevant parameters in our model are related to the mortality, cure and mutation rates. This model presents two distinct stationary active phases: a phase with co-existing normal and drug resistant infected individuals, and an intermediate active phase with only drug resistant individuals. We employed a finite-size scaling analysis to compute the critical points and the critical exponents, β/ν and 1/ν, governing the phase transitions between these active states and the absorbing inactive state. Our results are consistent with the hypothesis that these transitions belong to the directed percolation universality class.

We analyze the absorbing state phase transition exhibited by two distinct unidimensional delayed contact process (CP). The first is characterized by the introduction of an infection period and the second by an immune period in the dynamics of the original model. We characterize these CP by the quantities t_d (infection or disease period) and t_i (immune period). The quantity t_d corresponds to the period interval an individual remains infected after being contaminated, while the period t_i is the time interval an individual remains immune after being cured. We used Monte Carlo simulations to compute the critical parameters associated with the absorbing state phase transition exhibited by these models. We find two distinct power-law scale relations for the critical infection rate and the critical cure rate . For the CP delayed by the minimum infection period we find μ_d = 0.98, while we obtained μ_i = 0.80 for the case of a delay due to immunity. In addition, we used a finite-size scaling analysis to estimate the critical exponents β/ν and ν, and found that these models belong to the universality class of directed percolation irrespective to the time delay.

We present an alternative finite-size scaling (FSS) of the contact process on scale-free networks compatible with mean-field scaling and test it with extensive Monte Carlo simulations. In our FSS theory, the dependence on the system size enters the external field, which represents spontaneous contamination in the context of an epidemic model. In addition, dependence on the finite size in the scale-free networks also enters the network cutoff. We show that our theory reproduces the results of other mean-field theories on finite lattices already reported in the literature. To simulate the dynamics, we impose quasi-stationary states by reactivation. We insert spontaneously infected individuals, equivalent to a droplet perturbation to the system scaling as $N^{-1}$. The system presents an absorbing phase transition where the critical behavior obeys the mean-field exponents, as we show theoretically and by simulations. However, the quasi-stationary state gives finite-size logarithmic corrections, predicted by our FSS theory, and reproduces equivalent results in the literature in the thermodynamic limit. We also report the critical threshold estimates of basic reproduction number $R_0={\lambda }_c$ of the model as a linear function of the network connectivity inverse $1∕z$, and the extrapolation of the critical threshold function for $z\to \infty$ yields the basic reproduction number $R_0=1$ of the complete graph, as expected. Decreasing the network connectivity increases the critical $R_0$ for this model.

In contact processes, the population can have heterogeneous recovery rates for various reasons. We introduce a model of the contact process with two coexisting agents with different recovery times. Type A sites are infected with probability $p$, only if any neighbor is infected independent of their own state. The type $B$ sites, once infected recover after $\tau$ time-steps and become susceptible at $(\tau +1)\mathrm{\text{th}}$ time-step. If susceptible, type $B$ sites are infected with probability $p$, if any neighbor is infected. The model shows a continuous phase transition from the fluctuating phase to the absorbing phase at $p=p_c$. The model belongs to the directed percolation universality class for small $\tau$. For larger values of $\tau =8,16$, the model belongs to the activated scaling universality class. In this case, the fraction of infected sites of either type shows a power-law decay over a range of infection probability $p<p_c$ in the absorbing phase. This region of generic power laws is known as the Griffiths phase. For $p>p_c$, the fraction of infected sites saturates. The local persistence $P_l(t)$ also shows a power-law decay with continuously changing exponent for either type of agent. Thus, the quenched disorder in timescales can lead to the temporal Griffiths phase in models that show a transition to an absorbing state.

We review the critical behavior of nonequilibrium systems, such as directed percolation (DP) and branching-annihilating random walks (BARW), which possess phase transitions into absorbing states. After reviewing the bulk scaling behavior of these models, we devote the main part of this review to analyzing the impact of walls on their critical behavior. We discuss the possible boundary universality classes for the DP and BARW models, which can be described by a general scaling theory which allows for two independent surface exponents in addition to the bulk critical exponents. Above the upper critical dimension d_c, we review the use of mean field theories, whereas in the regime d<d_c, where fluctuations are important, we examine the application of field theoretic methods. Of particular interest is the situation in d=1, which has been extensively investigated using numerical simulations and series expansions. Although DP and BARW fit into the same scaling theory, they can still show very different surface behavior: for DP some exponents are degenerate, a property not shared with the BARW model. Moreover, a "hidden" duality symmetry of BARW in d=1 is broken by the boundary and this relates exponents and boundary conditions in an intricate way.

We investigate the critical behavior of a stochastic lattice model describing a predator–prey system. By means of Monte Carlo procedure we simulate the model defined on a regular square lattice and determine the threshold of species coexistence, that is, the critical phase boundaries related to the transition between an active state, where both species coexist and an absorbing state where one of the species is extinct. A finite size scaling analysis is employed to determine the order parameter, order parameter fluctuations, correlation length and the critical exponents. Our numerical results for the critical exponents agree with those of the directed percolation universality class. We also check the validity of the hyperscaling relation and present the data collapse curves.

The spread of infectious disease, virus epidemic, fashion, religion and rumors is strongly affected by the nearest neighbor hence underlying morphologies of the colonies are crucial. Likewise, the morphology of naturally grown patterns ranges from fractal to compact with lacunarity. We analyze the contact process on the fractal clusters simulated by generalized Diffusion-limited Aggregation (g-DLA) model. In g-DLA model, randomly walking particle is added to the cluster with sticking probability $p$ depending on the local density of occupied sites in the neighborhood of radius $r$ from the center of active site. It takes values $1$, $\alpha$ and ${\alpha }^2$ ($0<\alpha \le 1$) for highly dense, moderately dense and sparsely occupied regions, respectively. The corresponding morphology varies from fractal to compact as $\alpha$ varies from $1$ to $0$. Interestingly, the contact process on the g-DLA clusters shows clear transition from active phase to absorbing phase and the exponent values fall between 1-d and 2-d in directed percolation (DP) universality class. The local persistence exponents at transition are studied and are found to be much smaller than that for 1-d and 2-d DP cases. We conjecture that infection in the fractal cluster does not easily reach far-flung or remote areas at the periphery of the cluster.

We consider a directed percolation process on an rectangular lattice whose vertical edges are directed upward with an occupation probability y and horizontal edges directed toward the right with occupation probabilities x and 1 in alternate rows. We deduce a closed-form expression for the percolation probability P(x,y), the probability that one or more directed paths connect the lower-left and upper-right corner sites of the lattice. It is shown that P(x,y) is critical in the aspect ratio at a value α_c(x, y) = [1 - y² - x(1 - y)²]/2y² where P(x, y) is discontinuous, and the critical exponent of the correlation length for α < α_c(x, y) is ν = 2.