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

Attila Szolnoki<EMAIL_ADDRESS>Institute of Technical Physics and
Materials Science, Centre for Energy Research, P.O. Box 49, H-1525 Budapest,
Hungary  Xiaojie Chen<EMAIL_ADDRESS>School of Mathematical
Sciences, University of Electronic Science and Technology of China, Chengdu
611731, China

###### Abstract

Competitors in an intransitive loop of dominance can form a defensive alliance
against an external species. The vitality of this super-structure, 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 multi-point 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 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 multi-strategy ecological systems.

##  I Introduction

A stronger actor in an ecological system cannot be safe because a third type
of competitor may dominate it. Ironically, the originally “weaker” player
could also be the predator of the last one, which establishes a cyclic
dominance among these three rival participants. Such intransitive relation,
that is captured in a rock-scissors-paper game Szolnoki  _et al._ (2014);
Dobramysl  _et al._ (2018), can be detected in several living systems,
starting from bacteria Shibasaki and Shimada (2018); Garde  _et al._ (2020);
Liao  _et al._ (2020) and plants Cameron  _et al._ (2009); Lankau and Strauss
(2007) to animals Guill  _et al._ (2011); Sinervo and Lively (1996), or even
among humans who follow different strategies in a social dilemma situation
Hauert  _et al._ (2002); Lee  _et al._ (2023); Canova and Arenzon (2018);
Szolnoki and Chen (2021); Li  _et al._ (2022); Szolnoki and Chen (2015). Over
the last decade several excellent works have studied the behavior of such
systems where cyclical dominance has a central role on the emerging pattern
formation Avelino  _et al._ (2019); Mir  _et al._ (2022); Kei-ichi Tainaka
_et al._ (2021); Yoshida  _et al._ (2022); Yang and Park (2023); Baker and
Pleimling (2020); Mobilia  _et al._ (2016); Avelino  _et al._ (2023); Park and
Jang (2019); Serrao and Täuber (2021).

If we step onto a higher level, such triplet Szabó (2005); Szabó  _et al._
(2007), or even a larger loop Mir  _et al._ (2022); Park  _et al._ (2023);
Brown  _et al._ (2019), may also be considered as an entity because their
members can form a defensive alliance against an external species or
alternative group Mao  _et al._ (2021). It is an interesting question whether
we can identify general dynamical rules which determine the vitality of such
formation, especially in the case when equally strong groups compete for space
in an evolutionary process. For example, a group formed by less members is
generally more effective than a larger loop Szolnoki and Perc (2023); de
Oliveira and Szolnoki (2022); Szolnoki and Chen (2023). Or, a loop having
faster inner rotation is more powerful than the alternative formation where
the average invasion is less intense Perc  _et al._ (2007); Szolnoki and Perc
(2015a). The latter, however, is not necessarily a decisive factor because a
members of a loop are not always equally strong Tainaka (1995); Avelino  _et
al._ (2021); Juul  _et al._ (2013). As a consequence, a group which is formed
by equally strong partners can surpass a rival group of diverse members where
the average inner invasion is faster Szolnoki  _et al._ (2020).

The efficiency of a defensive alliance may be affected by alternative
mechanisms, too. In particular, the actual invasion strength between a
predator-prey pair could also be influenced by the close vicinity of a third
party Szolnoki and Perc (2015b). In a microbial community, as a specific
example, an antibiotic-degrading species attenuates the inhibitory
interactions between two other species Kelsic  _et al._ (2015). The
consequence of such multi-point interaction Palombi  _et al._ (2020) could be
specially interesting in a cyclically dominated loop because it does not
simply modify the inner dynamics of the loop-members, but it can also change
the invasion strength toward external species.

To explore how such kind of extension varies the vitality of a defensive
alliance, we introduce a minimal model where two three-member loops compete.
One of them still represents the traditional rock-scissors-paper-like
interaction where during a potential invasion only the state of involved pair
counts. In the alternative triplet, however, we introduce a blocking
mechanism. More precisely, staying at the rock-scissors-paper example, a rock
will only beat scissors with a limited probability if there is a paper in the
nearest neighborhood of the actor representing rock state. Similar inhibition
is also applied when a rock tries to beat an external species, but the
neighboring paper blocks it. Summing up, the blocking mechanism is valid for
not just the inner dynamics, but also hurts the external invasion capacity of
the affected triplet.

At first glance, the additional blocking rule has only negative consequence on
the vitality of the mentioned triplet, but this seemingly reasonable
conclusion is not always valid. As we show in this work, a faster inner
invasion is capable to compensate the shortage of the alliance. Interestingly
enough, a faster rotation among triplet members could also be harmful, which
has never been observed. This unexpected system behavior can be explained by
the relation of the blocked triplet and a pair formed by an inner member and
an external species. As a consequence, the concept of defecting alliance is
more subtle then we thought earlier.

The rest of this paper is organized as follows. In Sec. II we define the
details of the minimal six-species model which is capable to test the
potential consequence of the blocking mechanism. Here we also present the
first observation obtained in a limited three-species subsystem which
establishes the preliminary expectations about the modified dynamical rule.
Our main results are summarized in Sec. III where we systematically visit the
full parameter space of the minimal model. This involves to survey series of
phase diagrams obtained at some representative combination of model
parameters. We here also give arguments which explain the unexpected system
behavior at specific parameter values. Finally, we conclude with the summary
of the results and a discussion of their implications in Sec. IV.

##  II Blocking the invasion by a neighbor

In our minimal ecosystem species, marked by 0, $\dots$, 5, are distributed on
a $L\times L$ square lattice where periodic boundary conditions are used. Each
point is occupied by one of the species, no empty site is allowed. During an
elementary step we select two neighboring sites randomly and if they are
occupied by a predator-prey pair then the predator invades the position of the
prey with a certain probability. To execute a full Monte Carlo (MC) step we
repeat this procedure $N=L^{2}$ times.

The possible predator-prey interactions are summarized in Fig. 1. As the food-
web illustrates, practically we have two three-member loops whose members are
in a rock-scissors-paper relation with each other. For instance, the odd-
labeled species beat each other cyclically with probability $\alpha$. The
inner invasion rate in the alternative loop, formed by even-labeled species,
is higher and characterized by probability $\alpha+\delta$. Here parameter
$\delta$ describes the excess invasion level which provides a faster inner
invasion to these species. On the other hand, there is a three-point
interaction in the latter loop because the presence of a third party can block
the invasion between the original predator-prey pair. As an example, in a
normal case species 0 invades a neighboring species 2 with probability
$\alpha+\delta$. But this invasion is blocked with probability $\gamma$ if
there is a species 4 in the neighborhood of species 0. Importantly, this
blocking mechanism is valid not only for the inner invasions of the triplet,
but also for the process between the competing triplets. Staying at the
previously mentioned example, the invasion between species 0 and 1 is also
blocked with probability $\gamma$ in the vicinity of species 4. Last, the
invasion strength between the rival groups is equally strong which is
characterized by parameter $\beta$.

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 $\beta$.

The above described blocking mechanism may explain why we use faster invasion
in the even-labeled triplet because the external blocking is disadvantageous
to them. There is, however, another aspect which also justifies why we need a
faster invasion for a chance of surviving in the circle where blocking in the
inner invasions may happen. To understand it, we present some representative
patterns obtained at three different $\gamma$ values in a subsystem where only
the even-labeled species are present. These are shown in the first three
panels of Fig. 2 where the $\gamma$ values are 0, 0.5, and 1 respectively.
Evidently, the symmetry between the species are not broken, but these plots
suggest that the average domain size increases for larger $\gamma$ values.
This effect can be confirmed quantitatively if we measure the stationary value
of two-point probability of identical species which is proportional to the
characteristic length of domain sizes. More precisely, we measure
$\sum_{i}p_{2}(i,i)$ for $i=0,2,4$ in the stationary state obtained at a
specific value of $\gamma$. Panel (d) in Fig. 2 shows that this value, hence
the average domain size, grows practically linearly with $\gamma$.

Based on the above described observations we can conclude that stronger inner
blocking mechanism produces larger domains among the species who are involved
in the modified dynamics. According to our previous experience obtained for
the competition of unequal triplets, larger mono-domains could be the
Achilles-point of a defending alliance because a large homogeneous domain is
always an easy target for a hostile alliance based on cyclic rotation Szolnoki
_et al._ (2020). Therefore, we expect that the group of even-labeled species,
who may block each other via multi-point interactions, is generally weaker
than the alternative blocking-free alliance. In this way we need to “support”
the former group by allowing faster inner invasion because speedy exchange
among group members generally helps an alliance to survive Perc  _et al._
(2007).

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. Panels
(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. Panel (d) shows the
two-point probability of identical species in dependence of $\gamma$ blocking
rate.

To explore the phase diagrams for all model parameters we need to use linear
system size ranging from $L=400-6000$ because finite-size problems can easily
be detected at the vicinity of phase transition points. For instance, when the
stationary fraction of a certain species is low then the extinction of these
species may easily happen at small system sizes. Therefore only those states
should be considered valid at specific parameter values where the portions of
species become independent of system size as $L$ is increased. Beside random
starting configurations we also use prepared initial states where different
patches of sub-solutions are separated in space at the beginning and after
their proper fight is monitored Szolnoki and Perc (2016); Szolnoki and Chen
(2018). In this way we can reach the stationary solution that is valid in the
large-size limit more reliably. The relaxation time depends on the system
size, the initial configuration, and also on the distance from transition
point. The typical relaxation time is between $2\cdot 10^{4}-5\cdot 10^{6}$ MC
steps.

##  III Results

Our model has four parameters, $\alpha,\beta,\gamma,$ and $\delta$, therefore
we need a systematic study to explore the typical system behavior in the sea
of parameter space. Since probability $\beta$ characterizes the interaction
strength between competing alliances, we can distinguish three major cases
from this aspect. Namely, we study the case when the interaction between the
competing triplets is strong, intermediate, or weak. The other criterion which
separates system behavior onto different subsections is the inner invasion
strength characterizing the group of odd species. Here we also distinguish
three classes when the $\alpha$ value is low, intermediate or high. The
typical and representative system behavior are presented on the plane of the
remaining two parameters, $\delta$ and $\gamma$. As noted, $\gamma$
characterizes the intensity of blocking mechanism in the alliance formed by
even-labeled species, while $\delta$ denotes the excess rate in their inner
invasion.

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. Panels (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.

Our first results, shown in Fig. 3, were obtained when the interaction between
the triplets are strong. The three panels summarize the system behavior for
$\alpha=0.8,0.4$, and 0.2, where the inner invasion in odd-labeled loop is
strong, intermediate or weak. Evidently, the probability $\alpha+\delta$
cannot exceed 1, which explains the different maximal $\delta$ values for each
panels.

Panel (a) shows the case when interactions in general are strong. As expected,
when blocking mechanism is reasonable then the weakened even-labeled triplet
cannot compete with the rival alliance and the group of (1+3+5) species
dominate the large-$\gamma$ region. Interestingly enough, however, the
handicap of even-labeled group can be compensated by increasing their inner
invasion. As we increase $\gamma$ blocking strength, we need larger $\delta$,
faster rotation among (0+2+4) species, to beat the stronger group. There is a
discontinuous phase transition as we increase $\gamma$ which separates the
domains where either even-labeled or odd-labeled triplet dominates the whole
system. The character of this phase transition will be discussed in detail
later.

At intermediate $\alpha$, shown in panel (b), the border between the (0+2+4)
and (1+3+5) phases become less sharp and there is a $\gamma$ interval when all
six species survive. We marked this phase as “all” in the phase diagram.
Finally, when the inner rotation is low in the odd-labeled loop, the orange
domain of this state completely disappears, no mater the invasion of their
members is never blocked. This cannot be said about the (0+2+4) group, still,
they can dominate in the low $\gamma$ \- high $\delta$ corner in panel (c)
because fast inner rotation recovers their individual weakness. For all other
parameter values all six species coexist represented by the pink “all” domain.
The latter can be understood, because the low general invasion due to small
$\alpha$ makes the three-member loops hazardous. The large $\beta$ value, on
the other hand, makes the 0$\to$1$\to\dots\to$5$\to$0 loop viable. In this way
the latter formation prevails no matter it requires the simultaneous presence
of more members.

Next we present two panels to illustrate the diverse character of phase
transitions observed in the diagrams of Fig. 3. The sum of the fractions of
even-labeled species is a good order parameter to characterize the actual
state of the system. When it is 1 then only (0+2+4) species survives. When it
is zero then alliance formed by (1+3+5) species prevails. In between these
extreme values the system is in a state where all six species coexist.

Figure 4: Panel (a): Time dependence of the order parameter obtained for
different $\gamma$ values as indicated in the legend at fixed $\beta=1$,
$\alpha=0.8$, $\delta=0.1$ parameter values. The system evolves into either
(0+2+4) or (1+3+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. Panel (b): Stationary state of the order parameter in dependence of
$\gamma$ obtained at fixed $\beta=1$, $\alpha=0.4$, $\delta=0.12$. 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 4(a) illustrates a situation when we launch the evolution from a state
where initially half of the space is occupied by (0+2+4) species while the
other half is filled by (1+3+5) species. In this way we can monitor the fight
of the alliances directly. As the panel shows, there is a sharp difference
between the destinations as we gradually increase the $\gamma$ value while all
other parameters are fixed. For smaller $\gamma$, when the blocking mechanism
is moderate, the (0+2+4) triplet can enjoy the advantage of faster inner
invasion, while above a threshold $\gamma_{c}=0.1845(3)$ the negative
consequence of blocking cannot be compensated anymore. This explains why we
have discontinuous transition between (0+2+4) and (1+3+5) phases. As a
technical note, this phenomenon can be observed at any system size, the only
difference is the requested relaxation time to reach the final destination
grows by increasing $L$.

Figure 4(b) shows a completely different situation. Here the order parameter
remains between zero and one in the stationary state stably signaling the
emergence of a new phase where all six species are present. More precisely, if
the system size is large enough then the value of order parameter becomes
size-independent. Evidently, even smaller system size, like $L=400$, is enough
to avoid finite-size effects if $\gamma$ is far from the critical transition
point, while we need larger system sizes in the vicinity of $\gamma_{c}$. The
results shown in Figure 4(b) are valid in the large-size limit. As a
conclusion, the transitions between the “all” and three-member phases are
continuous.

In the following we survey the situation when the interaction between the
competing triplets is intermediate. Again, as previously, we distinguish three
cases where the inner invasion of odd-labeled triplet is fast, moderate, or
very weak. The representative phase diagrams are shown in Fig. 5. When the
inner rotation is fast, displayed in panel (a), the system behavior is similar
to those we observed for large $\beta$. Namely, (1+3+5) species can form a
solid winning alliance almost everywhere, the only exception is the low
$\gamma$ \- large $\delta$ corner where the faster rotation in the even-
labeled triplet can compensate the shortage of blocking. When the invasion
strength in the sound triplet is weaker, shown in panel (b), we can observe a
similar behavior previously found in Fig. 3(b). Namely, there is a parameter
region where neither (0+2+4) nor (1+3+5) alliance is strong enough to beat the
rival group hence all six species survive. There is, however, a qualitative
difference compared to the previous case. In particular, the system behavior
depends on the dynamical parameters on a more subtle way. If we fix $\gamma$,
for instance, and only increase the value of $\delta$ then we can detect some
non-monotonous behavior: starting from (1+3+5) phase we can enter to “all”,
after (0+2+4), followed by “all” again, and finally arriving back to (1+3+5)
phase again.

This system behavior has serious consequence. Previous results obtained in
other models and those we presented above for fast triplet rotations confirmed
each other and suggest the general finding that faster inner invasion in a
triplet is always beneficial to the alliance. This is partly true here because
we can reach the (0+2+4) phase by solely increasing $\delta$. But increasing
this parameter further has a negative effect on the fitness of this loop. In
short, a faster rotation becomes harmful to this alliance here. Before
discussing this curious behavior more deeply, we first present the remaining
low-rotation case, shown in panel Fig. 5(c). Since we decreased $\alpha$, the
general rotation speed decayed in the (1+3+5) group. As a consequence, the
domain of this phase has shrunk in the phase diagram by giving space for “all”
phase. But in contrast to Fig. 3(c), the orange phase does not disappear fully
because we have smaller $\beta$, hence the large loop containing all six
species is not as effective anymore as for large $\beta$ values.

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. Panels (a) to (c) respectively show cases when the
inner invasion strength in the odd-labeled group is strong ($\alpha=0.7$),
moderate ($\alpha=0.2$), 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 $\beta=0.5$, $\alpha=0.2$, $\gamma=0.25$. Arrows
show the positions of the $\delta$ values where we launched the evolution from
a specific initial state shown in panel (a). The diverse evolution of patterns
in Cases $I$, $II$, and $III$ are shown in the rows. Final destinations, which
are (1+3+5), (0+2+4), and (1+3+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.

The top-left panel of Fig. 6 illustrates quantitatively the above mentioned
curious re-entering phenomenon. Here we plotted the
$\rho_{0}+\rho_{2}+\rho_{4}$ order parameter in dependence of $\delta$ at
fixed $\beta=0.5$, $\alpha=0.2$, $\gamma=0.25$ parameter values. It is an
important observation when we monitor the spatial battle of triplets that the
front separating the competing domains is always fluctuating, hence patches
containing only two neutral species can emerge easily. As a result, triplets
do not simply fight each other, but they also compete with other solutions.
The latter has particular role in understanding the unexpected phenomenon. To
demonstrate it we present the evolution of spatial pattern in three different
cases as marked by arrows in the mentioned panel. In all three cases the
evolution starts from a prepared initial state where triplets are separated by
domains containing neutral species. This common starting state is shown in
Fig. 6(a) where we indicate those species who are present in a specific
stripe. As a technical note, for better visibility we only present two of the
possible neutral pairs here, but similar pattern formation could be observed
in the presence of (2+5) pair solution. We first discuss Case I obtained at
$\delta=0.05$. In agreement with the standard expectation about defending
alliances, the sound (1+3+5) triplet beat both (0+3) and (1+4) duplets. Here
both species 0 or species 4 can be considered as an intruder which is defeated
by the rotating triplet. The same is not true for (0+2+4) triplet because the
blocking mechanism prevents them to defend effectively against (0+3) or (1+4)
domains. After 1200 MC steps, as it is illustrated in panel (b), yellow-blue
and green-white patches emerge in the bulk of (0+2+4) domain, signaling the
negative consequence of blocking. After 5800 MC steps, shown in panel (c), the
latter formation disappears and the sound (1+3+5) triplet gradually prevails.
We stress that the latter formation are capable to dominate the alternative
triplet directly, as it was illustrated in all previous case when $\delta$ is
too small. As expected, the relation of triplets changes by increasing
$\delta$ because a faster inner rotation makes blocked (0+2+4) formation
viable. It happens in Case II shown in the middle row. As panel (e)
illustrates, taken after 5000 MC steps, both triplets can grow on the expense
of domains formed by duplets. Panel (f) depicts the event, after 8500 MC
steps, when triplets meet and the superiority of even-labeled triplet over
odd-labeled triplet becomes evident. The last row of Case III shows when the
inner rotation in the blocked triplet is almost maximal, which has a fatal
consequence on the vitality of this even-labeled alliance. Panel (h), obtained
after 2000 MC steps, illustrates that duplet of neutral pairs becomes stronger
again. Importantly, the strength of blocking mechanism is identical to all
three cases, the only difference is the inner invasion strength in the blocked
group. Let us consider, for instance, the battle of (0+2+4) and (0+3). While
3$\to$4 invasion remains intact, the 2$\to$3 invasion is partly blocked. In
this way the the former process has a larger probability if the inner rotation
in the triplet is fast enough. It simply means that too fast rotation could be
harmful for blocked triplet against two-member domains and they go extinct
after 5000 MC steps, as shown in panel (i). On the other hand, the sound
triplet still dominates the two-member domains, hence odd-labeled group will
be the final victor, no matter they would be weaker against the even-labeled
group in a direct interaction. Summing up, the final destinations are
identical in Case I and in Case III, but the temporarily emerging two-member
patches have a critical role on defeating the even-labeled triplet in Case
III. While the odd-labeled trio is capable to beat the even-labeled trio
directly in Case I, this is not true in the alternative case. Here it needs
the help of neutral pairs who can beat the blocked trio because of their too
fast inner rotation. But later the neutral pair solutions give space for the
rival triplet. This dynamics recalls “the Moor has done his duty, the Moor may
go” effect previously observed in a different system modeling a social dilemma
situation Szolnoki  _et al._ (2011).

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.

To complete our study we briefly summarize our results obtained for the case
when interaction between the competing triplets is weak. As previously, panels
of Fig. 7 outline the typical behaviors for cases when the general rotation
speed among triplet members is large, intermediate or slow. Based on our
previous observations the system behavior can be interpreted in the following
way. Similarly to those we detected for other $\beta$ values, when the general
inner rotation is high, shown in panel (a), the competition between the
triplets is always clear: either even- or odd-labeled group win. The only
difference is the consequence of blocking is more serious now. Consequently,
the drawback of (0+2+4) group can be compensated just for very small $\gamma$
values. The re-entrance phenomenon can be observed both in panel (b) and panel
(c), respectively representing intermediate and slow inner rotation. The area
of “all” phase, however, is significantly smaller than for other $\beta$ cases
discussed previously. The latter feature can be explained by the small $\beta$
value which generally makes the large six-member loop fragile. We must note,
however, that the $\beta\to 0$ limit provides a qualitatively different
situation where all six species coexist again. In the latter case there is no
interaction between the competing triplets anymore, hence they can form two
there-member stable solutions independently. But this phase is different from
the one we call as “$all$” here because the latter is based on a cyclic
dominance among all six competitors.

##  IV Conclusion

Different aspects of cyclically dominant dynamical systems have been studied
intensively in the last years Menezes and Barbalho (2023); Park (2022);
Szolnoki and Chen (2020); Serrao and Täuber (2021); Avelino  _et al._ (2018);
Roman  _et al._ (2016). By choosing a similar system, the key question in our
present work is to clarify how a blocking mechanism affects the vitality of
the involved formation. Importantly, such blocking, when the interaction
between two actors is influenced by the vicinity of a third partner, is a
common phenomenon in microbiological systems or even in human societies
Alvarez-Rodriguez  _et al._ (2021).

While this kind of blocking may stabilize a coexistence in a three-member
system Bergstrom and Kerr (2015), there are two reasons why such blocking
could be harmful for a cyclic loop in a more complex situation. The first one
is plausible because members may impede each other to invade successfully an
external party, which can be detrimental for the whole alliance. The other one
is the consequence of retarded inner rotation. In this case the average size
of domains formed by the members of the loop grows. And the increase of local
homogeneity is always disadvantageous for a defending alliance which is based
on cyclic dominance.

By using a minimal model, where two three-member loops compete, we
demonstrated that the above described drawbacks can be compensated by a faster
inner rotation. In this case the blocked triplet can dominate the sound
formation where blocking is not applied. Naturally, if the blocking
probability is too large then the untouched formation will be the victor and
the transition between these solutions is discontinuous. This observation
remains valid independently of the interaction strength between the triplets.
If the general invasion withing the alliance is weak then a higher loop,
involving all six species, can win resulting in the stable coexistence of all
competitors.

When the interaction between the triplets is moderate or weak then we could
detect interesting reentrant phase transitions by solely changing the inner
invasion speed in the blocked triplet. First, if rotation becomes intensive
then blocked triplet become dominant. If we increase the rotation speed
further, however, the original block-free triplet wins again. The explanation
of this unexpected behavior is based on an special “Moor-effect” where a third
type of solution brings the final victory to the block-free triplet. In a
“Moor-effect”, when originally two solutions compete, a third solution can
beat one of them and later it gives space to the another competitor Szolnoki
_et al._ (2011). In our present case when two triplets compete, their frontier
serves as a birthplace of alternative solutions. Namely, there is always a
chance that a two-member solution of neutral species emerges. If the rotation
speed is too high in the blocked triplet then such two-member solution, formed
by a triplet member and an external species, can be more effective: while the
invasion of external party is not blocked, it can meet more easily with its
internal predator who cannot beat it due to blocking. As a consequence, the
two-member formations invade the blocked triplet, leaving only the alternative
triplet alive. Importantly, the defensive mechanism still working in the sound
triplets, hence they eventually prevail. No matter they would be weaker in a
direct comparison against the blocked triplet.

As a general conclusion, our system provides a nice example that the
consequences of dynamical rules determining the vitality of a solution are
more subtle than we originally thought. Therefore further research is
necessary to establish solid principles to understand such ecological systems
more deeply.

###### Acknowledgements.

A.S. was supported by the National Research, Development and Innovation Office
(NKFIH) under Grant No. K142948. X.C. was supported by the National Natural
Science Foundation of China (Grant Nos. 62036002 and 61976048).

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