Figure 1

(Color online) Coexistence probability $p_{coex}$ as a function of the radius $R$ of interaction range in the absence of individual mobility for $N=3000$, $L=1$, where $p_{coex}$ is defined to be the ratio of the number of the survival cases to the total number (200) of independent simulation realizations. The probability is calculated after a transient time $T\propto N$ from random initial configurations with equal densities of three species. The inset shows the dependence of $R_T$, $R_{Min}$, and $R_{Max}$ on the population size $N$.Reuse & Permissions

Figure 2

(Color online) For $N=10000$ and $R=4.25$, pattern transition between plane waves and spiral waves (top row), (a) the corresponding time series of species density ${\rho }_a$, (b) average species density $⟨{\rho }_a⟩$, $⟨{\rho }_b⟩$, $⟨{\rho }_c⟩$, typical patterns about the transition point (two left panels, bottom row), and plane wave traveling along diagonal directions (two right panels, bottom row). Red (gray), blue (dark gray), and green (light gray) represent species $A$, $B$, and $C$, respectively. The rapid oscillations in the density are due to the fact that, in the spiral-wave phase, the curved boundaries between different species generate relatively large interaction areas so that many more individuals can interact with others in a random manner. The average species density $⟨\rho ⟩$ is taken by down sampling the original time series and the average values are calculated over 100 time steps.Reuse & Permissions

Figure 3

(Color online) The effect of the size of the square cell on the coexistence probability $p_{coex}$ with respect to the rescaled selection range $R_L$. We compute coexistence curves for five different combinations of the cell size $L$ and the total population $N$. The nonmonotonic behavior of $p_{coex}\sim R_L$ is independent of the cell size, as indicated by the collapse of the three curves into a single one for $N=2500,3000,6000$. The peaks for $N=1500$ and 2000 are lower, due to a relatively stronger stochastic effect for smaller values of $N$.Reuse & Permissions

Figure 4

(Color online) Spatial patterns at physical selection range $r=0.0913$ for (a) cell size $L^2=2$ and (b) $L^2=1/2$, and at $r=0.0365$ for (c) cell size $L^2=5/6$ and (d) $L^2=2$. For all these patterns, the density $N/L^2$ of individuals is fixed to be 3000. In (a), the rescaled selection range is $R_L=3.82$, which corresponds to $R_{max}$ in Fig. 3. Spiral waves and plane waves can switch between each other at this $R_L$ value. In (b), species go extinct for smaller cell size than (a), because $R_L=6.66$ is in the extinction region. In (c), $R_L=2.10$, close to $R_{Min}$ in Fig. 3; while in (d), by increasing $L$, $R_L$ decreases to 1.71, which is outside of the pattern switch region (the peak). The color (grayscale) scheme is the same as in Fig. 2.Reuse & Permissions

Figure 5

(Color online) For $N=3000$, coexistence probability $p_{coex}$ as a function of the individual mobility $M$ for different values of interaction range $R$. (a) For $R=R_{Min}=2.5$, a proper value of $M$ can considerably enhance the coexistence probability $p_{coex}$. (b) In other region of $R$, as is $M$ increased, coexistence is always inhibited.Reuse & Permissions

Figure 6

(Color online) Comparison of the coexistence curves for $M=0$ and $M=3.3\times {10}^{-4}$ $\left(u=s\right)$, where the population size is $N=3000$. When mobility is present, in the region between two dashed lines, mobility promotes coexistence. Other parameters are the same as in Fig. 1.Reuse & Permissions

Figure 7

(Color online) Pattern transition from a plane to a spiral wave as predicted by the theoretical PDE models. Red (gray), green (light gray), and blue (dark gray) represent species $A$, $B$, and $C$, respectively.Reuse & Permissions

Figure 8

(Color online) For $N=10000$, dependence of spatial correlation $C_{AA}$ (a) and correlation length $L_{corr}$ (b) on the interaction range $R$ from both stochastic simulation $\left(R=2.5\right)$ and the PDE model $\left(D=0.00045R^2\right)$, where $C_{AA}$ as a function of $R$ is obtained from Eqs. (3, 4). Here, we select a spatial position $\mathbf{r}$ and calculate $C_{AA}$ as a function of $\left|\mathit{r}-{\mathit{r}}^{\prime }\right|$ according to Eq. (3). We then count all $C_{AA}$’s with the same $\left|\mathit{r}-{\mathit{r}}^{\prime }\right|$ and calculate an average of them, yielding $C_{AA}$ as a function of the normalized distance $R$.Reuse & Permissions

Figure 9

(Color online) Coexistence probability $p_{coex}$ as a function of normalized interaction radius $R$ in model (6) in the absence of individual mobility. Other parameters are the same as in Fig. 1.Reuse & Permissions

Figure 10

(Color online) For model (6) with $N=3000$ and $L=1$, (a) spatial patterns and the corresponding time series of species density ${\rho }_a$ and (b) time series of the overlapping distance $D_o$ in the interaction range. The color (grayscale) scheme for spiral and plane waves is the same as in Fig. 2.Reuse & Permissions