PHYSICAL REVIEW E 83, 011125 (2011) Phase diagram and critical behavior of a forest-fire model in a gradient of immunity Nara Guisoni, 1,2,* Ernesto S. Loscar, 1 and Ezequiel V. Albano 2,3 1 Instituto de Investigaciones Fisicoqu´ ımicas Te´ oricas y Aplicadas (INIFTA), Facultad de Ciencias Exactas, Universidad Nacional de La Plata, CONICET CCT-La Plata, Sucursal 4, CC 16 (1900) La Plata, Argentina 2 Instituto de F´ ısica de L´ ıquidos y Sistemas Biol ´ ogicos (IFLYSIB), Universidad Nacional de La Plata, CONICET CCT-La Plata, CC 565 (1900) La Plata, Argentina 3 Departamento de F´ ısica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, La Plata, Argentina (Received 26 July 2010; revised manuscript received 12 December 2010; published 25 January 2011; publisher error corrected 1 February 2011) The forest-fire model with immune trees (FFMIT) is a cellular automaton early proposed by Drossel and Schwabl [Physica A 199, 183 (1993)], in which each site of a lattice can be in three possible states: occupied by a tree, empty, or occupied by a burning tree (fire). The trees grow at empty sites with probability p, healthy trees catch fire from adjacent burning trees with probability (1 − g), where g is the immunity, and a burning tree becomes an empty site spontaneously. In this paper we study the FFMIT by means of the recently proposed gradient method (GM), considering the immunity as a uniform gradient along the horizontal axis of the lattice. The GM allows the simultaneous treatment of both the active and the inactive phases of the model in the same simulation. In this way, the study of a single-valued interface gives the critical point of the active-absorbing transition, whereas the study of a multivalued interface brings the percolation threshold into the active phase. Therefore we present a complete phase diagram for the FFMIT, for all range of p, where, besides the usual active-absorbing transition of the model, we locate a transition between the active percolating and the active nonpercolating phases. The average location and the width of both interfaces, as well as the absorbing and percolating cluster densities, obey a scaling behavior that is governed by the exponent α = 1/(1 + ν ), where ν is the suitable correlation length exponent (ν ⊥ for the directed percolation transition and ν for the standard percolation transition). We also show that the GM allows us to calculate the critical exponents associated with both the order parameter of the absorbing transition and the number of particles in the multivalued interface. Besides, we show that by using the gradient method, the collapse in a single curve of cluster densities obtained for samples of different side is a very sensitive method in order to obtain the critical points and the percolation thresholds. DOI: 10.1103/PhysRevE.83.011125 PACS number(s): 05.70.Fh, 02.50.−r, 64.60.ah, 82.20.Wt I. INTRODUCTION A wide class of far-from-equilibrium systems exhibits irreversible phase transitions (IPTs) between an active phase and an inactive (absorbing) regime, so that the system becomes irreversibly trapped into the absorbing phase when a suitable control parameter is finely tuned across the transition point. In fact, IPTs have been reported in models for heterogeneously catalyzed reactions [1], prey-predator systems [2], epidemic propagation [3], as well as in different models proposed to mimic biological systems, such as the immune system [4], the spreading of virus propagation [5], and calcium propagation inside the cells [6–8]. In contrast, self-organized criticality (SOC) describes the way in which some nonequilibrium systems develop power-law correlations in a steady state without any tuning of parameters to a given value. The concept of SOC has attracted much interest since it might explain the spontaneous onset of scale-free distributions in nature, economy, social sciences, etc. [9]. Within this broad context, we focus our attention on a forest- fire model with immune trees, which is a variant of the model early introduced by Bak, Chen, and Tang [10]. In general, forest-fire models are a cellular automaton, so that each site of a d -dimensional hypercubic lattice can be in three different * naraguisoni@conicet.gov.ar states: occupied by a tree, occupied by a burning tree, or empty. The system is updated in parallel as follows: (i) A burning tree becomes an empty site. (ii) A tree grows with probability p at empty sites. (iii) A green tree becomes a burning tree with probability 1 − g if at least one of its nearest neighbors is burning. (iv) A tree becomes a burning tree with probability f ≪ 1 if no neighbor is burning. In the original version [10] the forest-fire model was presented with only one control parameter, the tree growth probability p (that is, f = 0 and g = 0), and shows regular spiral-shaped fire fronts in the limit of p → 0[11,12]. The most studied version of the model, proposed by Mossner, Drossel, and Schwabl [12], included the nonzero lightning parameter f (with g = 0). For this version in the limit f/p → 0, the model has been shown to exhibit SOC in a nonconservative system for the first time. Self-organization is found in the model of Mossner, Drossel, and Schwabl because the steady state is independent of both the initial conditions and the exact values of the parameters, as long as f/p is small enough. The system is critical because there are power-law correlations over long distances and long time intervals [13]. Later, the inclusion of the immunity g = 0 in the model [14,15] by means of rule (iii) was proposed. When the immunity is considered (with f = 0), depending on the value of g, the model exhibits a second-order IPT between an active phase and an absorbing phase, in which all sites are occupied by green trees. This version of the model is known as the forest-fire 011125-1 1539-3755/2011/83(1)/011125(9) © 2011 American Physical Society