Figure 1

Model with $D=3$ active states or species that attack each other across an intermediate inactive state. At each time step, the system is updated by a recruitment step $r$ and a noise step $n$: $r$) At each time step, a random site $j$ is selected. If it is in one of the active species, it attempts a conversion of another randomly selected site $k$: If $k$ is one of $j$’s antagonistic species, it is reset to the IS. If $k$ is IS, then it is converted to the same state as $j$. If the state of $k$ is equal the state of $j$, then no change is made. $n$) Select a random site $l$ with probability $\beta$ and change its state: If $l$ is active, it becomes IS, and if $l$ is IS, it becomes one of the randomly chosen active species.Reuse & Permissions

Figure 2

Dynamical simulation of the $D=3$ model in a system of size $N=1000$. The grey dots refer to the IS. Time is counted in number of attempted pair interactions per site. Panel (C) examines a 10 times longer time interval than (A) and (B), showing that stability of states depends strongly on $\beta$. Panel (B) shows occasional appearance of the MMS.Reuse & Permissions

Figure 3

Deterministic dynamics for the $D=10$ model. (A, B) Trajectories starting from uniformly distributed $n_0$, $n_1$, $n_2$, with all other $n_i$ equal, shown for $\beta =0.20$ in (A) and $\beta =0.25$ in (B). Trajectories are colored after their final fixed point. Fixed points are shown as solid red and black dots. Initial conditions fix $n_i$, $i\ge 3$ to be equal and thus exclude trajectories ending in DS with $i\ge 3$. (C) Population size of two of the stable states as a function of $\beta$ for $D=10$. The dashed “gold”-colored curve is the unstable fixed point that separates the MMS and the DS. (D) The parameter region with MSS (blue region) and DS ($\mathrm{\text{green}}+\mathrm{\text{blue}}$ region), with varying D and $\beta$.Reuse & Permissions

Figure 4

Dynamics of a variant model where conversion from IS is local and the general “kill” global [8], as illustrated in (A): Although noise ($n$) is unchanged, the recruitment ($r$) reads as follows: ($r$) At each time step, a random site $j$ is selected, and if it is in one of the active species, then one of two moves is attempted: (i) With probability $\frac{1}{2}$, a neighbor $k$ to $j$ is selected, and if $k$ is in IS, it is converted to a species of $j$; (ii) else, a random site $k$ is selected, and if $k$ is one of $j$’s antagonistic species, it is changed to the IS. If the selected $j$ is not active or $k$ is not fulfilling the corresponding requirements, then no change is made. (B, C) show the dynamics for $D=3$, $\beta =0.098$, and $N=100$. Note the MMS for time $t\in [4000,4800]$. (D) Dynamics of a corresponding $D=20$ model where the interactions are randomly assigned a strength $r(i,j)\in [0,1]$ at the start of the simulations. The recruitment of a species $j$ by an active species $i$, tried in step ($r$-ii), is then accepted with probability $r(i,j)$. (E) As in (D), but with evolutionary reshuffling of its interactions $r_{i,j}$ each time a population of a species vanishes. Different colors represent different species, with grey dots referring to the IS.Reuse & Permissions

Figure 5

Local $D=5$ model, where species at positions $x$, $y$ can only convert or recruit at positions $\in [x-1,x+1]\times [y-1,y+1]$. Simulations are for $N=100\times 100$ and $\beta =0.136$. (A) Population dynamics, with IS shown as grey dots. (B, C) Snapshots of the system separated by $\Delta t=20$. At a slightly smaller $\beta$, the DS dominates and the dynamics show punctuated switching between the dominating species.Reuse & Permissions