Symbiotic contact process: Phase transitions, hysteresis cycles, and bistability

We performed Monte Carlo simulations of the symbiotic contact process on different spatial dimensions ($d$). On the complete and random graphs (infinite dimension), we observe hysteresis cycles and bistable regions, what is consistent with the discontinuous absorbing-state phase transition predicted by mean-field theory. By contrast, on a regular square lattice, we find no signs of bistability or hysteretic behavior. This result suggests that the transition in two dimensions is rather continuous. Based on our numerical observations, we conjecture that the nature of the transition changes at the upper critical dimension ($d_c$), from continuous ($d<d_c$) to discontinuous ($d>d_c$).

Figure 1

Hysteretic cycle for $\mu =1/4$, obtained from the mean-field calculation. Symbols represent simulations performed in complete graphs (circles) and random graphs (rectangles). The highlighted arrows indicate the direction of the cycle. The solution $\rho =0$ (the red continuous line) corresponds to the stable absorbing state. The solution $\rho =2p^{-}+2p_{AB}^{-}$ (the dashed green line) is unstable for any value of $\lambda$. Finally, $\rho =2p^{+}+2p_{AB}^{+}$ represents the stable active solution (the dotted blue line).

Figure 2

The $(\tau ,\mathrm{\Delta })$ stability diagram of the mean-field solutions for $\mu =1/4$. The regions $I,II,III,IV,$ and $V$ correspond to, respectively, the stable nodes, unstable nodes, stable spirals, unstable spirals, and saddle points. The solution $\rho =0$ (continuous red line) is stable if $0\le \lambda \le 1$, since, from this condition, the absorbing solution lies in region $I$. For $\lambda >1$ the absorbing solution lies in region $V$, being therefore unstable. The active solution (dotted blue line) $\rho =2p^{+}+2p_{AB}^{+}$, lies in the region $I$ if $\lambda \ge {\lambda }_c\left(\mu =1/4\right)=\sqrt{3/4}$. The solution $\rho =2p^{-}+2p_{AB}^{-}$ (dashed green line) is unconditionally unstable, since for any value of $\lambda$ this solution lies in region $V$.

Figure 3

The phase diagram of the symbiotic contact process. Three phases can be identified: Active, Bistable, and Absorbing. The continuous, dashed, and dotted lines represent solutions of the mean-field equations. For $\mu >1/2$, the system undergoes a continuous phase transition between the active and absorbing phase. For $\mu <1/2$, the system describes a discontinuous phase transition, defining a bistable region where the active and absorbing phases are both stable. The highlighted arrow indicates the direction of the transition for the case $\mu =0.25$, where the bistable region is identified in the hysteretic cycle shown in Fig. 2. The open and solid symbols represent the critical reproduction rate ${\lambda }_c$ obtained from simulations performed in complete graphs, with $N=5\times {10}^4$. For the open symbols, the initial configuration is in the absorbing state, while for the solid symbols the fully occupied system defines the initial configuration.

Figure 4

Snapshots for ordinary and symbiotic contact processes. (a) Ordinary contact process at the distance 0.01 (absolute) or 0.006 (relative) from the critical point. (b) 2SCP at the same absolute distance of Fig. 4. (c) 2SCP at the same relative distance of Fig. 4.

Figure 5

Hysteresis cycles of the order parameter $\left(\rho \right)$ in terms of the creation rate $\lambda$, for $\mu =0.25$, on a square lattice of linear size $L=200$. Here we consider $t_{\max }=5\times {10}^4$ MCS (red triangles) and $t_{\max }=10\times {10}^4$ MCS (blue circles) as time increments. For each cycle, the control parameter $\lambda$ is increased and decreased in the range $1.0\le \lambda \le 1.20$ at the constant intervals $\mathrm{\Delta }\lambda =0.001$.

Figure 6

The dependence of the initial density ${\rho }_0$ of particles on stationary state. Accordingly, we have fixed the values of $\lambda$ and $\mu$ to identify a possible bistable behavior. (a) Complete graph, $\mu =0.25$ and $\lambda -{\lambda }_c=0.035$. (b) Random graph, $\mu =0.25$ and $\lambda -{\lambda }_c=0.025$. (c) Regular square lattice, $\mu =0.25$ and $\lambda -{\lambda }_c=0.0127$.