Expanding spatial domains and transient scaling regimes in populations with local cyclic competition

We investigate a six-species class of May-Leonard models leading to the formation of two types of competing spatial domains, each one inhabited by three species with their own internal cyclic rock-paper-scissors dynamics. We study the resulting population dynamics using stochastic numerical simulations in two-dimensional space. We find that as three-species domains shrink, there is an increasing probability of extinction of two of the species inhabiting the domain, with the consequent creation of one-species domains. We determine the critical initial radius beyond which these one-species spatial domains are expected to expand. We further show that a transient scaling regime, with a slower average growth rate of the characteristic length scale $L$ of the spatial domains with time $t$, takes place before the transition to a standard $L\propto t^{1/2}$ scaling law, resulting in an extended period of coexistence.

Figure 1

Predation rules in our model: bidirectional predation (solid arrows) and cyclic predation (dashed arrows). The predation interaction probability of species $i$ with species $i\pm 1$, and $i\pm 3$ is equal to $p$ (solid arrows). The predation interaction probability of species $i$ with species $i+2$ is equal to $p_{\mathrm{PRS}}$ (dashed arrows). Except for the labeling of the different species, this figure is invariant under rotation by an angle of $2k\pi /6$, where $k$ is an integer, thus leading to a $Z_6$ symmetry.

Figure 2

The upper and lower panels show snapshots of the spatial distribution of the different species on the ${2000}^2$ lattice at various moments for one single realization of our model (with $p=0.25$, $p_{\mathrm{PRS}}=p$, $m=0.5$, $r=0.25$), whereas the central panel shows the fractional density of the different species ${\rho }_i$ for the entire time span of the simulation. The arrows highlight the instants of time corresponding to the snapshots indicated in the lower and upper panels.

Figure 3

The initial stage of the single simulation presented in Fig. 2. The four snapshots show the spatial distribution of the individuals at the instants of time indicated in the figure. As in Fig. 2, the solid lines represent the evolution of the density of individuals of the different species as a function of time.

Figure 4

The probability $P$ that the whole lattice becomes dominated by the species 2 (initially confined to a circular spatial domain of radius $R$) as a function of $R$, assuming that $p=0.25$, $p_{\mathrm{PRS}}=0.25p$, $m=0.5$, $r=0.25$. The results for each $R$ are taken from an average over 1000 simulations, considering different initial conditions for the outer spatial domain containing species $1,3,5$. The critical radius, defined by $P\left(R_c\right)=1/2$ is approximately equal to $R_c=45$ (grid points). The left and right inset panels show two snapshots of runs with initial radius $R=30$ and $R=60$, respectively. The times required for the circle to collapse (left inset panels) or to invade all the territory (right inset panels) are displayed between snapshots (the lower and upper inset panels represent the initial and final configurations).

Figure 5

The critical radius $R_c$ as a function of the density ${\rho }_{\otimes }^{★}$ of empty sites in the outer spatial domain (top panel) and of $p_{\mathrm{PRS}}$ (bottom panel), assuming that $p=0.25$, $m=0.5$, and $r=0.25$. The best fits represent the power laws $R_c\propto {\left({\rho }_{\otimes }^{★}\right)}^{{\beta }_2}$ and $R_c\propto p_{\mathrm{PRS}}^{{\beta }_1}$, with exponents ${\beta }_1=-1.16$ and ${\beta }_2=-0.98$, respectively.

Figure 6

Evolution of the density ${\rho }_{\otimes }^{-}$ of empty sites between the enemy partnership spatial domains as a function of the simulation time (or, equivalently, the number of generations) assuming that $p=0.25$, $m=0.5$, $r=0.25$, and either $p_{\mathrm{PRS}}=0$ (top panel), $p_{\mathrm{PRS}}=0.25p$ (middle panel), or $p_{\mathrm{PRS}}=p$ (bottom panel). Both the points and the scaling exponents were obtained from an average over a set of 100 realizations with different initial conditions. The empty sites between spiral arms inside the spatial domains have not been considered in this analysis.

Figure 7

The density ${\rho }_{\otimes }^{-}$ of empty sites between enemy partnership spatial domains as a function of the simulation time (or, equivalently, the number of generations) for a single realization, assuming that $p=0.25$, $m=0.5$, $r=0.25$, and $p_{\mathrm{PRS}}=p$.