On Long-Term Species Coexistence in Five-Species Evolutionary Spatial Cyclic Games with Ablated and Non-Ablated Dominance Networks

Abstract

I present a replication and, to some extent, a refutation of key results published by Zhong, Zhang, Li, Dai, & Yang in their 2022 paper "Species coexistence in spatial cyclic game of five species" (Chaos, Solitons and Fractals, 156: 111806), where ecosystem species coexistence was explored via simulation studies of the evolutionary spatial cyclic game (ESCG) Rock-Paper-Scissors-Lizard-Spock (RPSLS) with certain predator-prey relationships removed from the game's "interaction structure", i.e. with specific arcs ablated in the ESCG's dominance network, and with the ESCG run for 100,000 Monte Carlo Steps (MCS) to identify its asymptotic behaviors. I replicate the results presented by Zhong et al. for interaction structures with one, two, three, and four arcs ablated from the dominance network. I then empirically demonstrate that the dynamics of the RPSLS ESCG have sufficiently long time constants that the true asymptotic outcomes can often only be identified after running the ablated ESCG for 10,000,000MCS or longer, and that the true long-term outcomes can be markedly less diverse than those reported by Zhong et al. as asymptotic. Finally I demonstrate that, when run for sufficiently many MCS, the original unablated RPSLS system exhibits essentially the same asymptotic outcomes as the ablated RPSLS systems, and in this sense the only causal effect of the ablations is to alter the time required for the system to converge to the long-term asymptotic states that the unablated system eventually settles to anyhow.

Figures (49)

• Figure 1: Dominance network, a directed graph or digraph, for the three-species Rock-Paper-Scissors (RPS) game. Species $S_i$ are denoted by nodes numbered by index $i\in\{1, 2, 3\}$, with directed edges running from the dominator ("predator") species to the dominated ("prey"') species. There are multiple labelings of this graph (e.g.: $(S_1$$=$$R, S_2$$=$$P, S_3$$=$$S); (S_1$$=$$P, S_2$$=$$S, S_3$$=$$R); \ldots$) but if one graph can be turned into another purely by rearranging the node labels then those two graphs are topologically equivalent or isomorphic. All possible labelings of the RPS digraph are isomorphic with each other, so there is only one isomorphically unique RPS digraph.
• Figure 2: Network Z0 (left) is the Rpsls dominance network as presented in Figure 1(a) of Zhong et al. (2022): the outer pentagon subnetwork formed by triangle-headed arrows is referred to by Zhong et al. as the spontaneous competition dominance interactions while the inner pentagram subnetwork formed by diamond-headed arrows is referred to by Zhong et al. as the alternative competition dominance interactions. Network Z0' (right) is topologically equivalent to Z0, despite appearing superficially different. The rules of this game are: scissors cut paper; paper covers rock; rock blunts scissors; scissors decapitates lizard; lizard eats paper; paper disproves Spock; Spock vaporizes rock; rock crushes lizard; lizard poisons Spock; and Spock smashes scissors.
• Figure 3: The ablated dominance networks explored in Zhong et al. (2022). Network Z1 is Zhong et al.'s Figure 1(b); Network Z2a is Zhong et al.'s Figure 1(c)-upper; Network Z2b is Zhong et al.'s Figure 1(c)-lower; Network Z3a is Zhong et al.'s Figure 1(d)-upper; Network Z3b is Zhong et al.'s Figure 1(d)-lower; and Network Z4 is Zhong et al.'s Figure 1(e).
• Figure 4: Graphs of $F(n_s(t_\text{max})$$=$$c)$ for $c \in \{1, \dots, 5\}$ (i.e., frequency of outcome of number of species surviving at the end of the experiment) vs. $M$ (mobility) -- referred to as $FvsM$ plots -- in my replication of the $N_a$$=$$1$ (dominance network Z1) experiments reported by Zhong et al.: upper graph is Zhong et al.'s Figure 3(a) $FvsM$ results; lower graph shows corresponding $FvsM$ results from my replication of the same experiments. The legend in Zhong et al.'s Figure 3(a) uses the abbreviations $R$ for 'Rock', $Sc$ for 'Scissors', $L$ for 'Lizard', and $Sp$ for 'Spock'. Note that Zhong et al. combine the results for $F(n_s(t_\text{max})$$=$$2)$ and $F(n_s(t_\text{max})$$=$$1)$ into a single class of outcome labelled "two or extinction". The legend in the lower graph of outcomes from my replication uses the numeric node-labels introduced in Figure \ref{['fig:RPSLSDomNets']}: each row of the legend shows the color of the line and marker for the given value of $n_s$ followed by, for $F(n_s(t_\text{max})$$>$$2)$, in parentheses and separated by '+' symbols, the node-numbers of that set of $n_s$ surviving species.
• Figure 5: Replication of one of the two $N_a$$=$$2$ (dominance network Z2a) experiments results reported by Zhong et al.: Upper graph is Zhong et al.'s Figure 5a (edited to include the legend); $N_\text{\sc iid}$$=$$500$; lower graph shows corresponding results from my replication of the same experiments; $N_\text{\sc iid}$$=$$200$. Format and legend labels as for Figure \ref{['fig:ZhongFig3']}.