Apex predator and the cyclic competition in a rock-paper-scissors game of three species

This work deals with the effects of an apex predator on the cyclic competition among three distinct species that follow the rules of the rock-paper-scissors game. The investigation develops standard stochastic simulations but is motivated by a procedure which is explained in the work. We add the apex predator as the fourth species in a system that contains three species that evolve following the standard rules of migration, reproduction, and predation, and study how the system evolves in this new environment, in comparison with the case in the absence of the apex predator. The results show that the apex predator engenders the tendency to spread uniformly in the lattice, contributing to destroy the spiral patterns, keeping biodiversity but diminishing the average size of the clusters of the species that compete cyclically.

Figure 1

In the left panel the diagrams show how predation works for three and four species. The black arrows indicate unidirectional predation and the gray dashed arrows indicate bidirectional predation. In the right panel are typical snapshots of stochastic simulations that run for ${10}^4$ generations. Panel (d) shows the system where the fourth species constitutes the apex predator.

Figure 2

Density of species as a function of time. The colored curves represent the respective species, and the black curve stands for the empty sites. The (a) and (b) panels represent the $X3$ and $X4$ systems, and the (c) panel stands for the last system, $SX3$, which contains the apex predator. The long-time evolution shows that all the systems evolve ensuring coexistence of species.

Figure 3

The black dots represent the histograms of the number of maxima in the interval of ${10}^3$ generations and the red dashed lines show the fits as Gaussian curves. The cases (a) and (b) represent the systems $X3$ and $X4$, in which all the species compete cyclically. Panel (c) is for the three species $A$, $B$, and $C$ of the system $SX3$, and the case (d) represents the species $D$, the apex predator.

Figure 4

The autocorrelation functions for the systems $X3$ (a), $X4$ (b), and $SX3$ (c). The colors identify the species, and the black dotted curves stand for the empty sites. In (c) the three species compete cyclically in the presence of an apex predator, represented by yellow.

Figure 5

The autocorrelation function for the systems $X3$ (red curve), $X4$ (green curve), and $SX3$ (blue and yellow curves). The yellow curve represents the apex predator. The inset shows perfect agreement between the numerical simulation and the approximations used to fit the curves, shown as the empty-circle curves.

Figure 6

Average number of clusters $\langle N\left(t\right)\rangle$ as a function of time. The red, green, and blue lines represent the species in cyclic competition in the systems $X3$, $X4$, and $SX3$, respectively. The yellow curve stands for the apex predator.

Figure 7

The mono-log behavior of the average size of the clusters $\langle S\left(t\right)\rangle$ as a function of time. The colors are as in Fig. 6.