# Competition among alliances of different sizes

Breno F. de Oliveira  Attila Szolnoki<EMAIL_ADDRESS>Departamento de Física, Universidade Estadual de Maringá, 87020-900 Maringá,
PR, Brazil  Institute of Technical Physics and Materials Science, Centre for
Energy Research, P.O. Box 49, H-1525 Budapest, Hungary

###### Abstract

To understand the biodiversity of an ecosystem cannot be understood by solely
analyzing the pair relations of competing species. Instead, we should consider
multi-point interactions because the presence of a third party could change
the original microscopic outcome significantly. In this way an alliance may
emerge where species, who may have biased relations otherwise, can protect
each other from an external invader. Such an alliance can be formed by two,
three or even more species. By introducing a minimal model where six species
compete for space we here study how the size of an alliance determines the
vitality of a formation. We show that in the majority of parameter space the
group of the smallest size prevails and other solutions can only be observed
in a limited parameter range. These phases are separated by discontinuous
phase transitions which can only be identified by intensive numerical efforts
due to serious finite size effects and long relaxation processes.

###### keywords:

cyclically dominated species , alliances , phase transitions

††journal: Chaos Solitons and Fractals

##  1 Introduction

It is a long standing intellectual challenge to identify the key mechanisms
which maintain biodiversity in an ecosystem formed by competing species [1, 2,
3, 4, 5]. Beside its theoretical importance, however, such knowledge also has
practical significance because an incapable human intervene into an ecological
community may result in an undesired effect. We may think of cyclically
dominated species, for instance, which is a widely accepted possibility to
explain the enigma why not a single species survives due to an evolutionary
process [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Here, if we want to weaken a
species by reducing its invasion rate towards its prey then the opposite
system reaction can be observed and the stationary portion of the target
species grows [16, 17, 18]. This, frequently called as the survival of the
weakest effect, is just a delicate example about the difficulty we face when
trying to understand a many-member system where the interactions are
characterized by a subtle graph.

Interestingly, the above mentioned cyclical dominance does not only keep all
three competing species alive, but their mutual interdependence may provide a
specific way to protect them against the invasion of an external species. If,
for instance, the latter attacks a member of the loop, but the other member is
a predator of the invader then the inner invasion can result in a situation
when the external aggressor becomes a prey, hence the stability of the
original group is maintained. In other words, the vicinity of a third party
can reverse the direction of the invasion laid down by the original
microscopic rule [19, 20]. This mechanism can be considered as a defensive
alliance which was studied intensively in the last two decades [21, 22, 23,
24, 25, 26]. It is important to stress that beside the already mentioned rock-
scissors-paper game, other strategies of evolutionary games may also show
similar non-transitive interaction and therefore they can form conceptually
identical coalitions [27, 28, 29, 30, 31, 32, 33]. Furthermore, the above
described mechanism does not restricted to a three-member formation, but
larger loops of a Lotka-Volterra-type system containing more participants may
behave similarly [34, 35]. But we can also think on a smaller group where two,
otherwise neutral, species may mutually protect each other from an external
predator.

It is reasonable to extend this concept because not only a single species can
fight against an alliance, but alliances can also compete with each other for
space [36]. It is an important question whether we can identify specific
features of an alliance which determine its vitality, hence the outcome of the
mentioned fight can be predicted in general. To give an example, when two
three-member alliances compete then the one where the inner cyclic dominance
is more intensive may dominate the other one [37]. On the other hand, the
diversity of inner invasion rates could be a detrimental feature, because
those alliance which is formed by equally strong members performs better than
the one where the simultaneous presence of a weak and more aggressive partners
form the group [38].

Figure 1: Left panel shows the food-web of our six-species model where two
different invasion rates are introduced, marked by $\alpha$ and $\delta$.
Furthermore, the relation of species 4 and 5 is neutral, but they can exchange
their next-nearest positions with probability $\beta$. The right panel shows
an alternative representation of the same model, which helps to identify
competing alliances more easily. The smallest coalition is formed by species 4
and 5 who may mutually protect their partner from an external invader. Three
species, who dominate each other cyclically, can also from an alternative
defensive partnership. For clarity we have marked these formations by
colorized triangles. In particular, species 0, 4, and 3 and alternatively,
species 1,2 and 5 have the mentioned rock-scissors-paper-like relation.
Finally, species 0,1,2, and 3 at the corner of the grey box also form a four-
species coalition, which may fight against the remaining two species
effectively. This alternative representation of the food-web also highlights
that the introduced invasion or site-exchange rates are strongly related to a
specific type of alliances.

In this work we consider another aspect and check whether the size of an
alliance is a decisive factor in its vitality or not. For this goal we
introduce a minimal six-species model where alliances with different sizes are
present and compete for space. Perhaps it is worth noting that six-species
models have already been studied previous, like in Refs. [39, 40, 41, 42], but
our ecological system is designed to give a proper answer to the raised
question.

In our model the alliance specific invasion rates are the key parameters
because, as we already noted, they could determine the strength of a specific
formation. Our principal goal is to explore the multi-dimension parameter
space and identify the dominant solution for each combination of control
parameters. As we will show, the smallest alliance is the most effective
defensive group in general and it prevails the majority of parameter space.
Therefore larger formations composed by more participants can only win in
specific ranges of control parameters. In the following we present some
typical results which help us to identify some general observations. But
before presenting them we proceed with a detailed description of the studied
spatial multi-species model.

##  2 A minimal six-species ecosystem

In our spatial system six species, marked by $0,1,\dots,5$ indexes, are
distributed randomly on a square lattice. No vacancies are allowed, all
lattice sites are occupied by one of the species. To define the microscopic
dynamics, we assume two types of interactions between nearest neighbors, which
are invasion or site exchange. If, for example, species 4 and species 5 are
nearest neighbors then they can exchange their positions with probability
$\beta$. In case of the alternative interaction species 0 beats the
neighboring species 1 with probability $\alpha$ and the empty site is occupied
by an offspring of species 0. Similar relation is defined between species 1
and 2, between species 2 and 3, and finally between species 3 and 0 in cyclic
manner.

Beside the mentioned non-transitive interactions, we also declare predator-
prey relations between species 4 (species 5) and the remaining species. For
example, species 4 invades a neighboring species 2 or 3, but becomes the prey
of a species 0 or 1. The mentioned invasions are executed with probability
$\delta$. Similar relations can be defined for species 5 who invades a
neighboring species 0 or 1, but becomes the prey of species 2 or 3.

The microscopic rules are summarized in a food-web, shown in the left panel of
Fig. 1, where the solid and dashed-line arrows depict a possible invasion
while two sided arrows with dotted line mark the possibility of site exchange
between specific species. The postulated invasion rates may seem to be
artificial at first sight, but the alternative presentation of the same food-
web, shown in the right panel of Fig. 1, reveals that they are strongly
related to a certain type of alliances. In particular, the value of $\beta$
determines the site exchange between the neutral pair of 4 and 5 which
establishes the smallest size of defensive alliance against an external
species. For example, if species 0 attacks species 4 but the endangered
species exchanges its position with a neighboring species 5 then the latter
can repel the external invasion.

A three-party alliance is established among species 0, 4, and 3 who beats each
other cyclically. If, for instance, species 1 attacks species 4 then a
neighboring species 0 can neutralize this invasion. Similar rock-scissors-
paper like alliance can be detected among species 1, 2, and 3. From the
alternative presentation of the food-web it is also clear that the invasion
rate $\delta$ characterizes partly the intensity of inner invasions in the
three-size alliances.

Last, a four-member alliance composed by the non-transitive interactions among
species 0,1,2, and 3 can also be a promising fighter in our ecosystem. Here
the intensity of the inner invasion is characterized by parameter $\alpha$.
If, for instance, an external species 4 attacks species 2, who is a member of
the larger group, then a neighboring species 1 can block the invasion of the
intruder. Similarly, species 5 may also threaten the longest loop by attacking
species 0 or 1, but neighboring species 2 or 3 becomes a guard in the
mentioned situation.

According to the model design, we have alliances with different sizes which
all can be a destination of an evolutionary process. Their relations, however,
are far from obvious because we can always suggest a method how to defend the
attack of an external species belonging to an adverse alliance. As we already
mentioned, the power of a specific formation may depend sensitively on the
inner dynamical processes, therefore the key parameters are $\alpha,\beta$,
and $\delta$. While the first two parameters can be dedicated directly to the
four-member and the two-member alliance respectively, the value of $\delta$
characterizes the interaction intensity between these formations. Therefore,
one may expect that if $\delta$ is too small then there is no proper
interaction between the mentioned alliances, hence there is no proper
competition between them. As we will show later, this is a naive expectation.

Beyond the mentioned case, the parameter $\delta$ has another role in the
microscopic dynamics. More precisely, the parameters $\alpha$ and $\delta$
together determines the inner flow in the three-member groups, which could
affect the vitality of these formations. Summing up, there is no a clear
preliminary expectation how our model should behave in dependence of the
dynamical parameters. Therefore, to answer our original question, we need to
explore the whole three-dimensional parameter space and determine for every
combination of $\alpha,\beta$, and $\delta$ values if there is a single victor
or coexistence of alliances represents the stationary state.

Technically, we carried out Monte Carlo (MC) simulations on a square grid
where the linear size was varied between $L=250$ and $5000$ lattice sites. We
note that using the necessarily large system size, where the linear size of
the square exceeds significantly the characteristic length size of the
emerging patterns, is essential. Otherwise, a small-size simulation can easily
result in misleading outcome which is not valid in the large-size limit.
During an elementary step a player and its nearest neighbor are chosen
randomly. If they represent different species then the possible elementary
interaction is defined by the food-web shown in Fig. 1. This is executed by
the given $\alpha,\beta$, or $\delta$ probabilities. If we repeat the
described step $L\times L$ times then a full MC step is executed where on
average all lattice sites have a chance to change. The necessary relaxation
time to reach the final evolutionary destination were between $10^{3}$ and
$10^{6}$ MC steps. To reach the expected accuracy we have averaged 100
independent runs starting from different initial states.

In the following we present our key observations about the battle of alliances
composed by groups of different sizes.

##  3 When the smallest is the strongest

If we consider the whole range of available parameter space, which is a
$(0,1)\times(0,1)\times(0,1)$ cubic of probability values, then we can notice
very fast that the alliance composed by the neutral pair of $4+5$ species is
the most effective formation that dominates the majority of
$(\alpha,\beta,\gamma)$ values. More precisely, if $\beta$, which controls the
intensity of their site exchanges, is high enough then this alliance beats all
the remaining groups independently of the values of $\alpha$ and $\delta$. The
latter formations have only chance to win only when $\beta$ is very small.
This case is illustrated in Fig. 2 where we present the phase diagram obtained
at a fixed $\beta=0.01$ value.

Figure 2: Phase diagram on $\alpha-\delta$ plane obtained at fixed
$\beta=0.01$ value. Here $A_{i}$ denotes the parameter region where the
alliance containing $i$ members prevails. Note that only the very small
$\beta$ value makes possible to observe alternative victors than the group of
species $4+5$ who occupy $A_{2}$ domain. Dashed lines separating different
phases mark discontinuous phase transitions.

In this diagram we marked by $A_{i}$ the parameter region where the defensive
group composed by $i$ species can win the battle of alliances. The general
feature of this diagram confirms our expectations based on previous
experiences about dynamically maintained defensive formations. Namely, the
values of $\alpha$ and $\delta$ can be a decisive factor on who wins the
competition. If $\alpha$ is small, for instance, then the four-party alliance
simply has no chance to win: their inner invasion is too slow, hence they
cannot block an external invasion effectively. More precisely, the appropriate
member of the loop may arrive too late to the place where the external
intruder attacks the territory of the four-member alliance. Therefore the
latter group can only be viable if their inner flow is intensive, in other
words $\alpha$ is high and $\delta$ is moderate simultaneously. The latter
parameter ensures that species $4$ or $5$ cannot attack the remaining four
species intensively. Indeed, the diagram of Fig. 2 confirms our argument
because the domain of $A_{4}$ can only be found in the large $\alpha$ \-
moderate $\delta$ region.

Figure 3: The stationary concentrations of species in dependence of $\delta$
obtained at $\alpha=1,\beta=0.01$. We note that in the $A_{3}$ phase the
$\rho_{i}$ values are the average of independent runs because here either
$0+4+3$ or $1+2+5$ trio wins if we wait long enough. Accordingly, the
concentration values for a specific destination are two times higher than
those shown here.

Before further discussing the phase diagram, we note that different phases are
separated by sudden or more precisely discontinuous phase transitions. This
behavior is illustrated by Fig. 3 where we made a cross section of the diagram
by keeping $\alpha=1$ fixed. For small $\delta$ values $A_{2}$ prevails hence
species $4$ and $5$ occupy the available space equally. One may argue that in
case of small $\delta$, there is no proper interaction between the two- and
the four-member alliances. Therefore they might coexist below a critical
$\delta$ value. But this is not the case here because even a very weak
interaction reveals the superiority of the species $4+5$ tandem.

Figure 4: The competition of $4+5$ and $0+1+2+3$ alliances at $\delta=0.0005$
when their interaction is extremely weak. The remaining parameters are
$\alpha=1,\beta=0.01$. The linear size $L=250$ and the color coding of species
is identical to the one we used in Fig. 1. In the initial state, shown in the
left panel, $4+5$ species occupy one third portion of the available space
randomly, while the other two thirds of space are occupied by the remaining
four species. Both alliances emerge very fast and after they start to compete
each other for space. Because of the small $\delta$ this process is very slow:
second and third panels show the stage after 50,000 and 100,000 MC steps. But
the tendency is clear and finally, not shown here, the alliance of $4+5$
species prevails.

This process is illustrated clearly in Fig. 4 where we monitor the competition
of these solutions by applying an extremely small $\delta=0.0005$(!) invasion
rate between the species belonging to different sets. At the beginning, shown
in the first panel of Fig. 4, species $4+5$ are distributed randomly in the
one third area of available space, while the other two thirds are filled by
the remaining four species. When we launched the evolution both solutions
emerge locally very fast. Namely, both the pair of $4+5$ and the quartet of
$0+1+2+3$ form a solution which would be stable in the absence of the other
formation. Eventually, however, the two-member alliance invades the other
solution. Evidently, this process is very slow because the interaction between
the competing alliances is extremely weak: note that only 5 of 10000 meetings
of competitor species would result in actual change at the frontiers!
Therefore, not really surprisingly, after 100000 MC steps the $4+5$ tandem
conquered just a small area from the $0+1+2+3$ quartet, as it is illustrated
by the right panel in Fig. 4. But the tendency is clear. If we wait long
enough, typically $10^{6}$ steps at $L=250$ system size, only species $4$ and
$5$ survive. Perhaps it is worth mentioning that this final destination can be
reached much faster if we launch the evolution from a completely random
initial state where all six species are distributed uniformly, but here our
principal goal was to illustrate the proper competition of two otherwise
stable solutions. From this phenomenon we may conclude that an alliance having
less members could be more effective than a larger coalition which is based on
the collective acts of more members.

The relation of these alliances changes if we increase $\delta$ and the four-
member group wins. Accordingly, all involved species occupies one fourth of
the available space. The characteristic snapshots of $A_{2}$ and $A_{4}$
phases are shown in the first two panels of Fig.5. Interestingly, however,
Fig. 3 suggests that by increasing $\delta$ further the triplet-based
alliances can beat the four-member coalition. Based on the above mentioned
conclusion one may ask that why $\emph{n}ot$ an $A_{3}$ solution always
dominates $A_{4}$?

Figure 5: Typical patterns obtained in different phases as we increase the
value of $\delta$ when $\beta=0.01$ and $\alpha=1$. The color code of species
are similar to we used in Fig.1. For small delta, only the two-species
alliance survives. For intermediate value of $\delta$ the four-species
formation prevails. In the range of large $\delta$ values the three-species
loops beat all other alliances. Importantly, the right panel shows just an
intermediate state of the evolutionary process because if we late long enough
then one of the triplets will occupy the whole lattice.

The explanation is based on a previous work which studied the vitality of
three-member alliances when heterogeneous inner invasion rates were used [38].
As we already mentioned, a defective alliance based on cyclic dominance of the
members may become sensitive if the inner invasion flow is too heterogeneous.
In this case one of the members forms a large homogeneous domain, which makes
the whole group vulnerable. If $\delta$ is too low compared to $\alpha=1$ then
the trio group is not efficient and the quartet can win. Figure 3 demonstrates
nicely that $A_{3}$ beats $A_{4}$ only above a threshold $\delta$ value. In
the latter case $\delta$ value becomes comparable to $\alpha=1$ invasion rate
resulting in a more balanced inner flow and a more uniform domain size
distribution. Accordingly, a fit $A_{3}$ trio can already beat an $A_{4}$
quartet in agreement with our previously declared conclusion. The phase
diagram shown in Fig.2 also underlines that $A_{3}$ domain occupies the right-
up corner of the phase space where $\alpha$ and $\delta$ values are
comparable. But the unequal $\alpha$ and $\delta$ values result in a slightly
different stationary concentrations of species, as it is illustrated in Fig. 3
[16].

We would also like to stress that the spatial distribution of species shown in
the right panel of Fig. 5 does not represent the final destination of the
evolutionary process. This plot only illustrates an intermediate stage when
three-member alliances beat the alternative defensive groups. Indeed, the trio
of $0+4+3$ and $1+2+3$ species are equally strong because both loops contain
two $\alpha$ and a $\delta$ invasion rates, but they do no coexist
permanently. Here, in the absence of symmetry breaking, both three-member
formations can be the final victor of the evolution. But the coarsening of
their domains is logarithmically slow in the lack of interfacial tension. This
dynamical process is identical to the one previous observed for voter model
[43, 44].

Figure 6: The stationary concentrations of species in dependence of $\beta$
obtained at $\alpha=1,\delta=1$ values. If the site exchange between neutral
partners becomes intensive the two-member alliance dominates the triplet
formations. Note that in the $A_{3}$ phase either $0+4+3$ or $1+2+5$ triplet
will win the competition.

Despite the diversity of phase diagram shown in Fig. 2 we should not forget
that the majority of parameter space is still dominated by the two-member
$4+5$ solution. Their tandem is efficient almost everywhere if their site
exchange capacity is not blocked. Put differently, if $\beta$ is high enough
then they always win the evolutionary contest in our ecological system. To
give an insight about the necessary intensity of site exchange we present a
cross section starting from $A_{3}$ phase when we gradually increase the value
of $\beta$. This is shown in Fig. 6. As we can see, if the intensity of site
exchange is exceeds $\beta_{c}=0.0270(2)$ critical value then the above
mentioned duo gives no chance for alternative formations. Perhaps it is worth
noting here that both $\alpha$ and $\delta$ parameter values are fixed, hence
the microscopic details which determine the vitality of the triplets are
intact. Therefore the sole change of $\beta$ influences the vitality of the
neutral pair directly.

Figure 7: The stationary concentrations of species in dependence of $\beta$
obtained at $\alpha=1,\delta=0.4$ values. The $0+1+2+3$ quartet can win the
competition only if the site exchange between the neutral $4+5$ pair is
extremely rare. Otherwise the latter formation will always win.

A similar behavior can be seen if we increase $\beta$ at a smaller
$\delta=0.4$ value. As we already shown in Fig. 2, in this case the four-
member loop of species 0, 1, 2, and 3 dominates the small $\beta$ limit. But
their kingdom is fragile because already a small increase of $\beta$ can
change this outcome. More precisely if the site exchange exceeds
$\beta_{c}=0.0190(2)$ threshold value then the neutral pair of $4+5$ will be
the victor again.

##  4 Conclusion

The success of network science in last two decades revealed the importance of
interaction graphs both in ecological and conceptually similar alternative
systems where the collective interaction of participants result in highly
complex behavior [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. A closed loop
in a food-web could be specially interesting, because it offers a sort of
higher level interaction when a group of participants, as an alliance, fight
against an external player or a group [56, 57, 58].

Our present work, however, highlights that not only the topology of the graph,
but also the intensity of dynamical process along the available invasion path
could be a decisive factor: we stress that the topology of food web is fixed,
but the final evolutionary outcome could be very different depending on the
invasion rates. We designed the graph to make possible for different alliances
to emerge and our principal question to explore which of them is the final
victor of the competition. As a general observation, we found that the
alliance with the smallest size is the most effective in a huge area of
parameter space and other competitors can win only in limited cases. More
precisely, if the site exchange of neural partners exceed $\beta>0.03$ then
their formation will always beat all other alliances independently of the
values of the remaining parameters. This conclusion seems to be reasonable
because in the mentioned optimal case only a single partner is needed to build
a protective shield against others. According to this argument a defensive
group formed by more members is always more vulnerable. But this is not
necessarily the case because we presented situations when a four-member loop
could be more efficient than a three-member group. This unexpected outcome
reveals the importance of balanced inner invasions among the members of
alliances. In this way we could provide further example and extend previously
laid conjecture that the well-known rock-scissors-paper type defending
alliance can only work properly if it is formed by fairly equal partners.

We would like to note that our model is a minimal one because several,
otherwise important, features are ignored here. For example, the migration or
the mobility of species is another factor which could modify the stable
solutions significantly [59, 60, 61, 62, 63]. To extend our model with this
feature promises further exciting observations because mobility may have
unequal impact on a small or a large-size alliance.

B.F.O. thanks Fundação Araucária, and INCT-FCx (CNPq/FAPESP) for financial and
computational support.

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