When faster rotation is harmful: The competition of alliances with inner blocking mechanism

Competitors in an intransitive loop of dominance can form a defensive alliance against an external species. The vitality of this superstructure, however, is jeopardized if we modify the original rock-scissors-paper-like rule and allow that the vicinity of a predator blocks stochastically the invasion success of its neighboring prey towards a third actor. To explore the potential consequences of this multipoint interaction we introduce a minimal model where two three-member alliances are fighting but one of them suffers from this inner blocking mechanism. We demonstrate that this weakness can be compensated by a faster inner rotation which is in agreement with previous findings. This broadly valid principle, however, is not always true here because the increase of rotation speed could be harmful and results in a series of reentrant phase transitions on the parameter plane. This unexpected behavior can be explained by the relation of the blocked triplet and a neutral pair formed by a triplet member with an external species. Our results provide novel aspects to the fundamental laws which determine the evolutionary process in multistrategy ecological systems.

Figure 1

Food-web characterizing the microscopic dynamics in our six-member ecological system. Arrows indicate the direction of invasion between species. Two triplets form defending alliances, whose members are in a rock-scissors-paper-like relation. While the inner invasion probability in the 1$\to$3$\to$5$\to$1 loop is $\alpha$, the same probability in the 0$\to$2$\to$4$\to$0 circle is $\alpha +\delta$. Importantly, the members of the latter triplet can block each other with probability $\gamma$. If, for instance, there is a species 4 in the neighborhood of species 0 then the 0$\to$2 or the 0$\to$1 invasion is blocked with probability $\gamma$. The invasion probability between the members of competing triplets is $\beta$.

Figure 2

Stationary pattern of a three-species rock-scissors-paper system with blocking mechanism. In this sub-system only even-labeled species are present where the color code is identical to those we used in Fig. 1. (a) to (c) show spatially distribution of competing strategies obtained at $\gamma =0$, 0.5, and 1, respectively. The blocking mechanism effectively increases the typical domain size of competing species. (d) shows the two-point probability of identical species in dependence of $\gamma$ blocking rate.

Figure 3

Phase diagrams for strong interaction between rival triplets ($\beta =1$). Horizontal axis shows the $\gamma$ blocking strength while vertical axis denotes the $\delta$ extra inner invasion rate in the even-labeled group. (a) to (c) respectively show cases when the inner invasion strength in the odd group is strong ($\alpha =0.8$), medium ($\alpha =0.4$), or weak ($\alpha =0.2$). Light blue (orange) marks the parameter area where even (odd) species prevail, while pink shows the area where all six species survive. Dashed (solid) lines denote the position of discontinuous (continuous) phase transition points.

Figure 4

(a) Time dependence of the order parameter obtained for different $\gamma$ values as indicated in the legend at fixed $\beta =1$, $\alpha =0.8$, and $\delta =0.1$ parameter values. The system evolves into either $(\mathit{0}+\mathit{2}+\mathit{4})$ or $(\mathit{1}+\mathit{3}+\mathit{5})$ state, and the change is sudden between $\gamma =0.184$ and 0.185. The linear system size is $L=800$ and we averaged over 100 independent runs. (b) Stationary state of the order parameter in dependence of $\gamma$ obtained at fixed $\beta =1$, $\alpha =0.4$, and $\delta =0.12$. These values are independent of system size in the large-size limit. It suggests that there is an intermediate phase where all six species coexist. Line is just to guide the eye.

Figure 5

Phase diagrams when interaction between rival triplets is intermediate ($\beta =0.5$). Horizontal axis shows the $\gamma$ blocking strength while vertical axis denotes the $\delta$ extra value of inner invasion in the even group. (a) to (c) respectively show cases when the inner invasion strength in the odd-labeled group is strong ($\alpha =0.7$), moderate ($\alpha =0.2$), or very weak ($\alpha =0.1$). Light blue (orange) marks the parameter area where even (odd) species prevail, while pink shows the area where all six species coexist. Dashed (solid) lines denote the position of discontinuous (continuous) phase transition points.

Figure 6

Reentrant phase transitions when rotation intensity is changed in the blocked triplet. Panel in the top-left corner shows the order parameter in dependence of $\delta$ at $\beta =0.5$, $\alpha =0.2$, and $\gamma =0.25$. Arrows show the positions of the $\delta$ values where we launched the evolution from a specific initial state shown in (a). The diverse evolution of patterns in cases I, II, and III are shown in the rows. Final destinations, which are $(\mathit{1}+\mathit{3}+\mathit{5})$, $(\mathit{0}+\mathit{2}+\mathit{4})$, and $(\mathit{1}+\mathit{3}+\mathit{5})$ phase again, are not shown. The color code of species is identical those used in Fig. 1. The applied linear system size is $L=800$. Further details can be found in the main text.

Figure 7

Phase diagrams for weak interaction between the rival triplets ($\beta =0.2$). Horizontal axis shows the $\gamma$ blocking strength while vertical axis denotes the $\delta$ extra value of inner invasion in the even group. Panels (a) to (c) respectively show cases when the inner invasion strength in the odd-labeled group is strong ($\alpha =0.7$), moderate ($\alpha =0.3$), or weak ($\alpha =0.1$). Light blue (orange) marks the parameter area where even (odd) species prevail, while pink shows the area where all six species coexist. Dashed (solid) lines denote the position of discontinuous (continuous) phase transition points.