# Impact of parity in rock-paper-scissors type models

P.P. Avelino  Instituto de Astrofísica e Ciências do Espaço, Universidade do
Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal  Departamento de
Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do
Campo Alegre 687, PT4169-007 Porto, Portugal  B.F. de Oliveira  Departamento
de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900
Maringá, PR, Brazil  R.S. Trintin  Departamento de Física, Universidade
Estadual de Maringá, Av. Colombo 5790, 87020-900 Maringá, PR, Brazil

###### Abstract

We investigate the impact of parity on the abundance of weak species in the
context of the simplest generalization of the rock-paper-scissors model to an
arbitrary number of species — we consider models with a total number of
species ($N_{S}$) between 3 and 12, having one or more (weak) species
characterized by a reduced predation probability (by a factor of
${\mathcal{P}}_{w}$ with respect to the other species). We show, using lattice
based spatial stochastic simulations with random initial conditions, large
enough for coexistence to prevail, that parity effects are significant. We
find that the performance of weak species is dependent on whether the total
number of species is even or odd, especially for $N_{S}\leq 8$, with odd
numbers of species being on average more favourable to weak species than even
ones. We further show that, despite the significant dispersion observed among
individual models, a weak species has on average a higher abundance than a
strong one if ${\mathcal{P}}_{w}$ is sufficiently smaller than unity — the
notable exception being the four species case.

##  I Introduction

Non-hierarchical predator-prey models are an important tool for the
understanding of the dynamics of complex biological systems involving a large
number of species. Several models of this type have been proposed and
investigated in the literature [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], most of them considering
three basic interactions — predator-prey, reproduction and mobility — usually
assumed to happen with the same probability for all species (reproduction and
mobility) or pairs of non-minimally interacting species (predator-prey).
Parity — the even or odd nature of the total number of species — has been
observed to play an important role in many of these models. It can affect not
only the dynamics of the network, and consequently coexistence, but also the
properties of the geometrical patterns associated to the resulting dynamical
structures or the symmetric/asymmetric evolution of the interface profiles
separating different domains [1, 2, 3, 4, 5, 6, 8, 13].

In [13] a generalization of the rock-paper-scissors (RPS) model to an
arbitrary number of species has been developed. This family of models
generates dynamical spiral structures with a number of arms equal to the
number of species. If the predation, reproduction and mobility rates do not
vary from species to species, the resulting average density is the same for
all species, assuming that the simulation box is large enough for coexistence
to prevail — for smaller simulation boxes, the survival probabilities are also
equal for all species, assuming unbiased initial conditions.

In realistic biological systems the species strength is not expected to be the
same for all species, which can affect both the population sizes and the
chances of survival of the different species. Species with a reduced predation
probability are usually referred to as weak. Nevertheless, it has been shown,
in the context of three species RPS models, that weak species may often have a
strong performance both in terms of population abundance and survival
probability [27, 28, 29, 30] (see also [31]). Recent research [32] on the
dynamics of three strains of E. coli interacting cyclically also concluded for
the dominance of the weakest strain. Still, no systematic difference in the
global performance of weak and strong species has been found in the context of
RPS models with four species [33]. This result raises the following question:
is parity a key factor on the performance of weak and strong species?

In this paper we consider the simplest generalization of the spatial
stochastic RPS model to an arbitrary number of species proposed in [13], but
shall relax the assumption that all species have equal strength. We
investigate the impact that a reduction of the predation probability of some
of the species — the weak ones — has on the overall dynamics of the network
and, in particular, on the abundance of such species. We shall consider models
with a total number of species $N_{S}$ between 3 and 12, paying particular
attention to the impact of parity on the overall abundance of weak and strong
species.

Figure 1: Predator-prey interactions of a sample of models with $5$, $6$, $7$
and $8$ species studied in the present paper (corresponding the first three
models listed in Table I with that number of species). Filled and open circles
represent weak and strong species, respectively.

##  II RPS type models with $N_{S}$ species

Here, we shall briefly describe the simplest generalization of the spatial
stochastic RPS model to $N_{S}$ species (May-Leonard formulation) [13]. In
this model the different species are labelled by $i$ (or $j$) with
$i,j=1,...,N_{S}$, and modular arithmetic, where integers wrap around upon
reaching $1$ or $N_{S}$, is used (the integers $i$ and $j$ represent the same
species whenever $i=j\ {\rm mod}\ N_{S}$, where mod denotes the modulo
operation). We shall perform spatial stochastic simulations on a square
lattice with $N^{2}$ sites and periodic boundary conditions, employing a May-
Leonard formulation in which every site is either occupied by a single
individual of one of the $N_{S}$ species or empty (an empty site is
represented by a ‘$0$’). The number of individuals of the species $i$ and the
number of empty sites is denoted by $I_{i}$ and $I_{0}$, respectively — the
density of individuals of the species $i$ and the density of empty sites are
defined respectively by $\rho_{i}=I_{i}/N^{2}$ and $\rho_{0}=I_{0}/N^{2}$. The
allowed interactions are predation [$i\ (i+1)\to i\ 0$], reproduction [$i\
0\to i\ i$], and mobility [$i\ \odot\to\odot\ i$], where $\odot$ represents
either an individual of any species or an empty site. Reproduction and
mobility interactions occur, respectively, with probabilities $r$ and $m$
(assumed to be the same for all the species). In our baseline model the
predation probability $p$ is the same for all possible predator-prey
interactions. However, in this paper we shall investigate the dynamical impact
of a reduction of the predation probability the weak species by a factor of
$\mathcal{P}_{w}\in\ ]0,1[$ (the other species are sometimes referred to as
strong).

The predator-prey interactions of a sample of models with $5$, $6$, $7$ and
$8$ species studied in the present paper (corresponding to the first three
models listed in Table I with that number of species) are represented in Fig.
1, with filled and open circles representing weak and strong species,
respectively. The one-sided arrows represent one-directional predator-prey
interactions between species $i$ and $i+1$, while the double sided arrows
represent the possible bi-directional predator-prey interactions between
species $i$ and $i+2$, $\dots$, $i+N_{S}-2$. The list of all the different
models with $N_{S}=5,6,7,8$ is displayed in Table I, where the digits
identifying each model represent the weak species. For each value of $N_{S}$,
the models listed in Table I have been ordered in decreasing order (from left
to right, and then top to bottom) of the value of ${\mathcal{A}}_{w}$ for
$\mathcal{P}_{w}=0.5$ shown in Fig. 6 (the relevance of this ordering will
become clear later on, when discussing the results shown in Fig. 6). Notice
that there are in general several combinations of digits corresponding to a
single model. In particular, any permutation among weak or among strong
species results in an equivalent model. Also, relabeling all the species $i$
as $i+n$, where $n$ is an integer, does not lead to a different model. Hence,
only one equivalent combination is listed in Table I. For example, models
$21$, $23$, $32$, $34$, $43$, $45$, and $54$ are all equivalent to $12$ for
any $N_{S}\geq 5$, which is the digit combination which appears in table I
(also notice that, for $N_{S}=5$, models $15$ and $51$ would also both be
equivalent to $12$).

$N_{S}=5$ |  12 |  123 |  1 |  1234 |  134
---|---|---|---|---|---
13 |  |
$N_{S}=6$ |  12 |  123 |  125 |  14 |  1234
1245 |  1 |  12345 |  124 |  12355
13 |  135 |  |
$N_{S}=7$ |  12 |  123 |  125 |  1245 |  1236
126 |  1234 |  12356 |  1 |  12345
124 |  14 |  1235 |  123456 |  12346
1246 |  13 |  135 |  |
$N_{S}=8$ |  12 |  123 |  1234 |  127 |  1256
1245 |  125 |  1236 |  12356 |  12456
12347 |  1237 |  126 |  1257 |  123467
1 |  14 |  124 |  12457 |  12345
1247 |  123456 |  123567 |  1235 |  12346
1234567 |  136 |  123457 |  12357 |  1246
15 |  13 |  135 |  1357 |
Table 1: List of all the different models with $N_{S}=5,6,7,8$. The digits
identifying each model represent the weak species. The models have been
ordered in decreasing order (from left to right, and then top to bottom) of
the value of ${\mathcal{A}}_{w}(\mathcal{P}_{w}=0.5)$ shown in the four panels
.

At every simulation step, the algorithm randomly picks an occupied site to be
the active one, randomly selects one of its adjacent neighbour sites to be the
passive one, and randomly chooses an interaction to be executed by the
individual at the active position: predation, mobility or reproduction with
probabilities $p$, $m$ and $r$, respectively — in this paper we use the von
Neumann neighbourhood (or 4-neighbourhood) composed of a central cell (the
active one) and its four non-diagonal adjacent cells (it has been shown in
[29], in the context of a three species model, that a Moore neighbourhood
leads to the same qualitative results). These three actions are repeated until
a possible interaction is selected — note that the interaction cannot be
carried out whenever predation is selected and the passive is not a prey of
the active, or if reproduction is selected and the passive is not an empty
site. A generation time (our time unit) is defined as the time necessary for
$N^{2}$ successive interactions to be completed.

Figure 2: The panels display snapshots of the spatial distribution of the
different species on a $1000^{2}$ lattice at $t=1.1\times 10^{4}$ for
realizations of the spatial stochastic RPS type models with: (a) $N_{S}=5$,
(b) $N_{S}=6$, (c) $N_{S}=7$ and (d) $N_{S}=8$ (the other model parameters are
$m=0.2$, $p=0.4$, $r=0.4$, and $\mathcal{P}_{w}=1$). Notice the appearance of
spiral patterns with a number of arms equal to the number of species $N_{S}$.

##  III Results

In this section we present the results of $1000^{2}$ spatial stochastic
numerical simulations, considering models with a number of species in the
interval $[3,12]$ and different values of $\mathcal{P}_{w}$. The parameters
$m=0.2$, $p=0.4$, $r=0.4$ are assumed in all simulations.

Fig. 2 displays the distribution of the different species on a square lattice
after $1.1\times 10^{4}$ generations for a model in which all the species have
the same strength ($\mathcal{P}_{w}=1.0$). The number of species are (a)
$N_{S}=5$ (top left panel), (b) $N_{S}=6$ (top right panel), (c) $N_{S}=7$
(bottom left panel) and (d) $N_{S}=8$ (bottom right panel). Spiral patterns
with a number of arms equal to the number of species are present in all the
simulations. Also, no clear predominance of one species over the others is
observed in any of the snapshots (as expected, since $\mathcal{P}_{w}=1.0$).

Figure 3: The same as in Fig. 2c and Fig. 2d (RPS type models with seven and
eight species), but now considering $\ \mathcal{P}_{w}=0.5$: $(a)$ $N_{S}=7$,
model $1$; $(b)$ $N_{S}=7$, model $12$; $(c)$ $N_{S}=8$, model $1$; $(d)$
$N_{S}=8$, model $12$. Notice the significant differences between the shapes
and characteristic sizes of the domains associated to the different species.

Fig. 3 is analogous to the bottom panels of Fig. 2 (Figs. 2c and 2d, which
contemplate cases with seven and eight species, respectively), but now
considering models $1$ and $12$, and $\mathcal{P}_{w}=0.5$: $(a)$ $N_{S}=7$,
model $1$; $(b)$ $N_{S}=7$, model $12$; $(c)$ $N_{S}=8$, model $1$; $(d)$
$N_{S}=8$, model $12$. Contrarily to Fig. 2, Fig. 3 shows that some species
are more dominant than others. Also, although the spiral patterns are still
easily recognizable, significant differences exist between the shapes and
characteristic sizes of the domains associated to the different species. This
is true both for the models with one weak species (left panels) and two weak
species (right panels), but more so in the latter.

Fig. 4 displays the evolution of the densities of the different species and
empty spaces for the realizations of spatial stochastic RPS type models with
$7$ and $8$ species considered in 3: $(a)$ $N_{S}=7$, model $1$ (1st panel);
$(b)$ $N_{S}=7$, model $12$ (2nd panel); $(c)$ $N_{S}=8$, model $1$ (3rd
panel); $(d)$ $N_{S}=8$, model $12$ (4th panel). The impact that the reduced
predation probability of one or two weak species has on the abundance of the
different species is qualitatively similar for models with seven and eight
species. In model $1$, with a single weak species, the most abundant species
is the prey of the weak species (represented by a blue filled circle),
followed by the weak species (represented by a red open circle). In model
$12$, with two weak species, the most abundant species is one of the weak
species (the one represented by a blue open circle), followed by its prey
(represented by a yellow filled circle), and then by the other weak species
(represented by a red open circle).

Figure 4: The evolution of the density of the different species and empty
sites ($\rho_{i}$ and $\rho_{0}$) over time for realizations of spatial
stochastic RPS type models with $7$ and $8$ species considered in Fig. 3a,
Fig. 3b, Fig. 3c and Fig. 3d (from top to bottom, respectively). Notice the
impact that the inclusion of one or two weak species has on the abundance of
the different species.

In order to quantify the impact of the species strength on their overall
abundance, we define the average density of weak and strong species as

$\langle\rho_{w}\rangle=\frac{1}{\\#W}\sum_{i\in
W}\langle\rho_{i}\rangle\,,\quad\langle\rho_{s}\rangle=\frac{1}{\\#S}\sum_{i\in
W}\langle\rho_{i}\rangle\,,$ (1)

where $W$ and $S$ are, respectively, the sets whose elements are the weak and
the strong species, and $\\#$ is used to represent the number of elements of
each set. Let us also define the parameter

${\mathcal{A}}_{w}=\frac{\langle\rho_{w}\rangle-\langle\rho_{s}\rangle}{\max(|\langle\rho_{w}\rangle|,|\langle\rho_{s}\rangle|)}\,,$
(2)

whose absolute value represents the relative advantage (if
${\mathcal{A}}_{w}>0$) or disadvantage (if ${\mathcal{A}}_{w}<0$) in being a
weak species.

Figure 5: The relative advantage in being a weak species ${\mathcal{A}}_{w}$
(or disadvantage if $\mathcal{A}_{w}<0$) as a function of $\mathcal{P}_{w}$
and $N_{S}$ for models with one (top panel) or two (bottom panel) weak
species, considering a total number of species $N_{S}$ between 3 and 12 and
$\mathcal{P}_{w}$ between $0.5$ and $1$ (in the bottom panel the average value
of $\mathcal{A}_{w}$ among models with two weak species is considered). Notice
the impact of parity, specially for $N_{S}\leq 8$. Figure 6: The relative
advantage in being a weak species ${\mathcal{A}}_{w}$ (or disadvantage if
$A_{w}<0$) as a function of $\mathcal{P}_{w}$ for the cases with $N_{S}=5$,
$N_{S}=6$, $N_{S}=7$ and $N_{S}=8$. The dashed black line represents
$\mathcal{A}_{w}=0$. The different colors and line types represent models with
a different number of weak species. Figure 7: The average relative advantage
in being a weak species ${\mathcal{A}}_{w}$ as a function of $\mathcal{P}_{w}$
and $N_{S}$, for models with a total number of species between 3 and 8. Again,
the impact of parity is significant for $N_{S}\leq 8$, with odd numbers of
species being more favourable to weak species than even ones.

The top panel of Fig. 5 shows the relative advantage in being a weak species
${\mathcal{A}}_{w}$ (or disadvantage if $\mathcal{A}_{w}<0$) as a function of
$\mathcal{P}_{w}$ and $N_{S}$, for models with one weak species, considering a
total number of species $N_{S}$ between 3 and 12 and $\mathcal{P}_{w}$ between
$0.5$ and $1$. The bottom panel displays the average relative advantage among
all two weak species models with $N_{S}$ species (also represented by
$\mathcal{A}_{w}$ for simplicity of notation) as a function of
$\mathcal{P}_{w}$ and $N_{S}$. In both models the impact of parity is
significant at least up to $N_{S}=8$, with odd numbers of species being more
favourable to weak species than even ones. In these models the weak species
generally have a significant an advantage if ${\mathcal{P}}_{w}$ is
sufficiently smaller than unity, the exception being the models with $N_{S}=4$
(this particular case has been investigated in detail in [33]).

Taking into account that parity effects are most noticeable for $N_{S}\leq 8$,
we shall now consider the models with a number of species $N_{S}$ between $5$
and $8$ one by one (the models with $3$ and $4$ species have already been
investigated in detail in [29, 30] and [33], respectively). Fig. 6 displays
the relative advantage in being a weak species ${\mathcal{A}}_{w}$ (or
disadvantage if $A_{w}<0$) as a function of $\mathcal{P}_{w}$ for all models
with $N_{S}=5$, $N_{S}=6$, $N_{S}=7$ and $N_{S}=8$ (the dashed black line
represents $\mathcal{A}_{w}=0$). The different colors and line types represent
models with a different number of weak species. In order to allow for a better
identification of the models in Fig. 6, the models in Table I have been
ordered in decreasing order (from left to right, and then top to bottom) of
the value of ${\mathcal{A}}_{w}$ obtained for $\mathcal{P}_{w}=0.5$.

Fig. 6 shows that for any particular $N_{S}$ between $5$ and $8$ there is a
significant dispersion of the curves of
${\mathcal{A}}_{w}({\mathcal{P}}_{w})$. This is true even if only models with
a fixed number of weak species are selected. Fig. 6 also shows that, for any
value of $N_{S}$ between $5$ and $8$, the average abundance of a weak species
is higher than that of a strong one in most models. Still, there are a few
models with an even number of species which are highly adverse for weak
species (e.g. models $135$ and $1357$, respectively for $N_{S}=6$ and
$N_{S}=8$) — this does not happen if the number of species is odd.

Fig. 7 shows the average relative advantage as a function of the number of
species and a total number of species $N_{S}$ between 3 and 8. Again the
impact of parity is noticeable, with odd numbers of species being on average
more favourable to weak species than even ones if ${\mathcal{P}}_{w}$ is
sufficiently smaller than unity. The results displayed in Fig. 7 for
$N_{S}\leq 8$ are qualitatively similar to those of Fig. 5.

##  IV Conclusions

In this paper we investigated the dynamics of RPS type models with a total
number of species between 3 and 12, in the presence of one or more weak
species. We showed that parity effects are significant, with the abundance of
weak species having a significant dependence on whether the number of species
is even or odd for $N_{S}\leq 8$.

We have shown that, unlike in the case of RPS models with three different
species [29, 30], the relative advantage in being a weak species may vary
significantly from model to model in models with more than four species and a
fixed value of the reduced predation probability — a significant dispersion is
observed even among models with a fixed number of weak species. These results
are in agreement with the findings of [33] where only models with four species
were considered.

Notwithstanding the large dispersion among models, we found that a weak
species has on average a significant advantage over a strong one if
${\mathcal{P}}_{w}$ is sufficiently smaller than unity, the only exception
being the four species case. Still, we have shown that parity plays a key
role, with odd numbers of species being on average more favourable to weak
species than even ones in terms of the overall abundance.

###### Acknowledgements.

P.P.A. acknowledges the support Fundação para a Ciência e a Tecnologia (FCT)
through the research grants UIDB/04434/2020, UIDP/04434/2020. B.F.O. and
R.S.T. thank CAPES - Finance Code 001, Fundação Araucária, and INCT-FCx
(CNPq/FAPESP) for financial and computational support.

## References

  * Kobayashi and Tainaka [1996] K. Kobayashi and K. Tainaka, Critical phenomena in cyclic ecosystems: Parity law and selfstructuring extition patter, Journal os the Physical Society of Japan 66, 38 (1996).
  * Tainaka and Araki [1999] K. Tainaka and N. Araki, Press perturbation in lattice ecosystems: Parity law and optimum strategy, Journal of Theoretical Biology 197, 1 (1999).
  * Sato  _et al._ [2002] K. Sato, N. Yoshida, and N. Konno, Parity law for population dynamics of n-species with cyclic advantage competitions, Applied Mathematics and Computation 126, 255 (2002).
  * Szabó  _et al._ [2007] G. Szabó, A. Szolnoki, and G. A. Sznaider, Segregation process and phase transition in cyclic predator-prey models with an even number of species, Phys. Rev. E 76, 051921 (2007).
  * Szabó and Szolnoki [2008] G. Szabó and A. Szolnoki, Phase transitions induced by variation of invasion rates in spatial cyclic predator-prey models with four or six species, Phys. Rev. E 77, 011906 (2008).
  * Peltomäki and Alava [2008] M. Peltomäki and M. Alava, Three- and four-state rock-paper-scissors games with diffusion, Phys. Rev. E 78, 031906 (2008).
  * Szabó  _et al._ [2008] G. Szabó, A. Szolnoki, and I. Borsos, Self-organizing patterns maintained by competing associations in a six-species predator-prey model, Phys. Rev. E 77, 041919 (2008).
  * Nakagiri  _et al._ [2010] N. Nakagiri, Y. Sakisaka, T. Togashi, S. Morita, and K. Tainaka, Effects of habitat destruction in model ecosystems: Parity law depending on species richness, Ecological Informatics 5, 241 (2010).
  * Allesina and Levine [2011] S. Allesina and J. M. Levine, A competitive network theory of species diversity, PNAS 108, 5638 (2011).
  * Avelino  _et al._ [2012a] P. P. Avelino, D. Bazeia, L. Losano, and J. Menezes, von neummann’s and related scaling laws in rock-paper-scissors-type games, Phys. Rev. E 86, 031119 (2012a).
  * Li  _et al._ [2012] Y. Li, L. Dong, and G. Yang, The elimination of hierarchy in a completely cyclic competition system, Physica A: Statistical Mechanics and its Applications 391, 125 (2012).
  * Roman  _et al._ [2012] A. Roman, D. Konrad, and M. Pleimling, Cyclic competition of four species: domains and interfaces, Journal of Statistical Mechanics: Theory and Experiment 2012, P07014 (2012).
  * Avelino  _et al._ [2012b] P. P. Avelino, D. Bazeia, L. Losano, J. Menezes, and B. F. Oliveira, Junctions and spiral patterns in generalized rock-paper-scissors models, Phys. Rev. E 86, 036112 (2012b).
  * Lütz  _et al._ [2013] A. F. Lütz, S. Risau-Gusman, and J. J. Arenzon, Intransitivity and coexistence in four species cyclic games, Journal of Theoretical Biology 317, 286 (2013).
  * Roman  _et al._ [2013] A. Roman, D. Dasgupta, and M. Pleimling, Interplay between partnership formation and competition in generalized may-leonard games, Phys. Rev. E 87, 032148 (2013).
  * Cheng  _et al._ [2014] H. Cheng, N. Yao, Z.-G. Huang, J. Park, Y. Do, and Y.-C. Lai, Mesoscopic interactions and species coexistence in evolutionary game dynamics of cyclic competitions, Scientific Reports 4, 7486 (2014).
  * Szolnoki  _et al._ [2014] A. Szolnoki, M. Mobilia, L.-L. Jiang, B. Szczesny, A. M. Rucklidge, and M. Perc, Cyclic dominance in evolutionary games: a review, Journal of The Royal Society Interface 11, 20140735 (2014).
  * Kang  _et al._ [2016] Y. Kang, Q. Pan, X. Wang, and M. He, A five species cyclically dominant evolutionary game with fixed direction: A new way to produce self-organized spatial patterns, Entropy 18, 284 (2016).
  * Roman  _et al._ [2016] A. Roman, D. Dasgupta, and M. Pleimling, A theoretical approach to understand spatial organization in complex ecologies, Journal of Theoretical Biology 403, 10 (2016).
  * Brown and Pleimling [2017] B. L. Brown and M. Pleimling, Coarsening with nontrivial in-domain dynamics: Correlations and interface fluctuations, Phys. Rev. E 96, 012147 (2017).
  * Park  _et al._ [2017] J. Park, Y. Do, B. Jang, and Y.-C. Lai, Emergence of unusual coexistence states in cyclic game systems, Scientific Reports 7, 7465 (2017).
  * Bazeia, D.  _et al._ [2017] Bazeia, D., Menezes, J., de Oliveira, B. F., and Ramos, J. G. G. S., Hamming distance and mobility behavior in generalized rock-paper-scissors models, EPL 119, 58003 (2017).
  * Souza-Filho  _et al._ [2017] C. A. Souza-Filho, D. Bazeia, and J. G. G. S. Ramos, Apex predator and the cyclic competition in a rock-paper-scissors game of three species, Phys. Rev. E 95, 062411 (2017).
  * Esmaeili  _et al._ [2018] S. Esmaeili, B. L. Brown, and M. Pleimling, Perturbing cyclic predator-prey systems: How a six-species coarsening system with nontrivial in-domain dynamics responds to sudden changes, Phys. Rev. E 98, 062105 (2018).
  * Avelino  _et al._ [2019a] P. P. Avelino, J. Menezes, B. F. de Oliveira, and T. A. Pereira, Expanding spatial domains and transient scaling regimes in populations with local cyclic competition, Phys. Rev. E 99, 052310 (2019a).
  * Szolnoki  _et al._ [2020] A. Szolnoki, B. F. de Oliveira, and D. Bazeia, Pattern formations driven by cyclic interactions: A brief review of recent developments, EPL (Europhysics Letters) 131, 68001 (2020).
  * Frean and Abraham [2001] M. Frean and E. R. Abraham, Rock-scissors-paper and the survival of the weakest, Proc. R. Soc. Lond. B 268, 1323 (2001).
  * Berr  _et al._ [2009] M. Berr, T. Reichenbach, M. Schottenloher, and E. Frey, Zero-one survival behavior of cyclically competing species, Phys. Rev. Lett. 102, 048102 (2009).
  * Avelino  _et al._ [2019b] P. P. Avelino, B. F. de Oliveira, and R. S. Trintin, Predominance of the weakest species in lotka-volterra and may-leonard formulations of the rock-paper-scissors model, Phys. Rev. E 100, 042209 (2019b).
  * Avelino  _et al._ [2021] P. P. Avelino, B. F. de Oliveira, and R. S. Trintin, Weak species in rock-paper-scissors models (2021), arXiv:2104.14276 .
  * Menezes  _et al._ [2019] J. Menezes, B. Moura, and T. A. Pereira, Uneven rock-paper-scissors models: Patterns and coexistence, EPL (Europhysics Letters) 126, 18003 (2019).
  * Liao  _et al._ [2020] M. J. Liao, A. Miano, C. B. Nguyen, L. Chao, and J. Hasty, Survival of the weakest in non-transitive asymmetric interactions among strains of e. coli, Nature Communications 11, 6055 (2020).
  * Avelino  _et al._ [2020] P. P. Avelino, B. F. de Oliveira, and R. S. Trintin, Performance of weak species in the simplest generalization of the rock-paper-scissors model to four species, Phys. Rev. E 101, 062312 (2020).