Type of Article: Perspectives for Special Issue
Keywords : Biodiversity; Coexistence; Rock-paper-scissors game;
Intransitive Competition; Eco-Evolutionary Feedbacks; Stabilizing
Mechanism
Statement of authorship: G.L.D. conceived the theoretical
framework with guidance from T.C and J.J. D.M conducted the mathematical
simulations, described the methods, and created the corresponding
figure. G.L.D led the writing of the manuscripts. All authors
contributed substantially to the drafts and gave final approval for
publication.
Conflicts of Interest: The authors declare that they have no
known competing financial interests or personal relationships that could
have appeared to influence the work reported in this paper.
Data Accessibility Statement: No new data was used in this study.
Upon acceptance, all code to reproduce the simulation results will be
deposited in a dedicated Github repository, and a corresponding Zenodo
DOI will be included.
Abstract: 150 Words Main Text: 4596 Words References: 43 4 Figures and 1
Table
Giacomo L. Delgado
Ohmstrasse 24 Zürich, 8050 Switzerland
+41768194048;
gidelgado@student.ethz.ch
Abstract
Rock-paper-scissors (RPS) dynamics have been shown to affect the
evolutionary relationships within populations. These processes are
analogous to the ways in which intransitive competition modifies
ecological outcomes and the co-existence between species within
communities. Here we explore the similarities between rock-paper-scissor
dynamics and intransitive competition and how this link opens new
avenues of research into eco-evolutionary processes. Intransitivity can
drive the stable coexistence of phenotypes within species, as well as
the diversity of species within communities. In addition, the links
between these dynamics highlight possible feedback mechanisms that might
operate across these evolutionary and ecological scales. Using
simulations, we show that greater intraspecific intransitivity within a
population can lead greater levels of intransitivity at the
community-level, with direct implications for community diversity and
stability. As such, RPS dynamics and intransitivity can serve as an
ideal conceptual framework to understand the feedback mechanisms that
drive diversity across evolutionary and ecological scales.
INTRODUCTION
In nature, systems that lack a purely hierarchical structure can mimic
the children’s game Rock-Paper-Scissors (RPS). In this analogy, there
are three possible “strategies” each of which “beats” one other;
rock beats scissors, scissors beat paper and paper beats rock (Figure
1a). A system like this is said to be perfectly intransitive and lacks a
single best competitor. Due to this inherent lack of hierarchy, RPS
dynamics and intransitive interactions can lead to stable (albeit
dynamic) co-existence in a population of phenotypes of a single species
(Sinervo and Lively 1996) and in communities of different species (May
and Leonard 1975, Buss and Jackson 1979).
In the late 20th century, evolutionary biologists
described a unique oscillating population dynamic in a natural system
(Sinervo and Lively 1996). Coined “Rock-Paper-Scissors” (RPS) dynamics
these systems give rise to the development of distinct evolutionary
trajectories (Sinervo and Lively 1996, Gray and McKinnon 2006). Within a
population, RPS dynamics have the potential to increase genetic
diversity (San-Jose et al. 2014, Huang et al. 2012), genetic stability
(Liao et al. 2019), and even lead to speciation (Gray and McKinnon
2006). At the same time, ecologists noticed that competitive
interactions among species can also be highly intransitive (May and
Leonard 1975, Buss and Jackson 1979) and that these interactions
increase the stability and even diversity of their communities (Gallien
et al. 2017, Maynard et al. 2017). By ensuring that no single species
dominates, such intransitive interactions can help to maintain
co-existence of competitors (Buss and Jackson 1979, Soliveres et al.
2015, Soliveres et al. 2018, Gallien et al. 2017), encourage parasite
and host co-evolution (Cameron et al. 2009, Liao et al. 2019) and give
insights into entire community assemblages (Soliveres and Allan 2018,
Levine et al. 2017). As extensions of both game theory and negative
frequency density-dependent selection, these patterns have broad
implications for the maintenance of diversity at both evolutionary and
ecological scales. Moreover, it is likely that these ecological and
evolutionary processes may be working synergistically to shape and
stabilize natural systems in a wide variety of contexts.
Ecology and evolution have largely remained two separate disciplines
with experts in either field leveraging unique explanatory models,
invoking their own theories, and publishing in separate journals.
However, with the development of increasingly sophisticated genetic
analysis tools at the end of the 20th century, a
growing body of research suggested that evolutionary dynamics can take
place over ecologically relevant timescales (Thompson 1998, Hairston et
al. 2005, Hendry and Kinnison 1999). This “rapid evolution” suggested
that genetic changes could happen quickly enough to affect ecological
outcomes, potentially driving feedbacks to influence evolutionary
dynamics again in turn (Pelletier et al. 2009, Garant et al. 2007).
Despite a large amount of interest in these “Eco-evolutionary
feedbacks” serious questions, issues and knowledge gaps remain (Hendry
2019). As such, there remains a need for the development of conceptual
frameworks that leverage both ecological and evolutionary theory to
understand the diversity and stability of natural systems under current
and future climate scenarios. Given the highly dynamic, integrative, and
stabilizing nature of intransitive systems across both ecology and
evolution, we propose that intransitivity may provide a valuable
framework to study eco-evolutionary dynamics in natural systems.
At their most basic level RPS dynamics and intransitive interactions,
whether working within populations or across communities, help to
explain the incredible diversity of life, a goal that ecologists and
evolutionary biologists both share. However, perhaps even more
importantly, they represent unique mathematical modeling approaches that
go beyond pair-wise interactions and hierarchical models. In addition,
the potential for eco-evolutionary feedbacks that operate to promote
diversity within and across species mean that intransitive dynamics
represent a wealth of possible new approaches to studying stability
mechanisms within natural systems. Here, we propose that RPS dynamics
and intransitive competition work in tandem to shape populations and
communities. In this context, we review the development of mathematical
frameworks (Allesina and Levine 2011, May and Leonard 1975, Vandermeer
2011, Laird and Champ 2018) and ecological tools to study intransitivity
across populations and communities, highlighting that the similarities
across fields may represent an ideal opportunity for establishing
feedbacks between ecology and evolution. We use model simulations
adapted from Maynard et al. (2019) to evaluate the potential for
feedbacks between intransitivity within and between species. Such
feedbacks may potentially be linked to the stability of entire
communities suggesting that intransitive relationships play a critical
role in maintaining stability within natural systems.
EVOLUTIONARY DYNAMICS: EXPLAINING POLYMORPHISM
Genetic diversity within populations is confounding in the face of
classic evolutionary theory. Many mechanisms of natural selection and
genetic drift should work to decrease the amount of genetic diversity by
moving populations towards a “most fit” genotype (Brisson 2018, Gray
and McKinnon 2006). As such, models of balancing selection must be
introduced to accommodate for the large genetic diversity seen in
nature. Such models include host-parasite coevolution, niche theory,
spatial or temporal habitat heterogeneity, heterozygote advantage and
negative frequency dependent selection (NFDS). NFDS is particularly
powerful as it can work alongside other mechanisms of selection such as
host-parasite coevolution (Koskella and Lively 2009), sexual selection
and reproductive investment strategies (Iserbyt et al. 2013, Takahashi
et al. 2010), plant-pollinator interactions (Gigord et al. 2001),
competition (Le Gac et al. 2012, Huang et al. 2012) and predator-prey
dynamics (Brisson 2018).
RPS dynamics can maintain genetic diversity and polymorphisms through a
modification of the same logic. Critically, rather than a rare allele
(say allele C) enjoying a fitness advantage due to its low absolute
frequency (as in NFDS) its advantage comes from its relative frequency
compared to a dominant allele (allele A) which it outcompetes. Adding in
a third allele (allele B) leads to a scenario in which no allele holds a
competitive advantage in the population for long enough to cause a
competitive exclusion. In other words, alleles or phenotypes that exist
at low frequencies can be “rescued” from extinction time and time
again. Thus, this mechanism can maintain a stable coexistence between
the three alleles or phenotypes through a dynamic cycling of allele
frequencies. Sinervo and Lively (1996) showed this exact scenario plays
out in a population of the common side-blotched lizard (Uta
stansburiana ). In this system, RPS dynamics have led to the evolution
of three throat-color morphs (orange, blue and yellow) in males of this
species, each employing its own mating strategy. The high heritability
of throat color and the reproductive advantage that each mating strategy
enjoys over one another leads to stable cycling between the three morphs
(Figure 1c-d).
Similar patterns of stable coexistence of phenotypes resulting from RPS
dynamics have been shown in populations of Zootoca vivipara(San-Jose et al. 2014), Escherichia coli (Kerr et al. 2002,
Lewis-epstein and Hadany 2020, Liao et al. 2019) and computer simulated
populations (Reichenbach et al. 2007). Interestingly, RPS dynamics go
beyond simply explaining coexistence but can accommodate other processes
such as dispersal (Reichenbach et al. 2007, Kerr et al. 2002) and
cooperation (Lewis-epstein and Hadany 2020). Additionally, spatially, or
temporally heterogenous habitats in populations with low mobility are
known to encourage RPS dynamics and polymorphism (Reichenbach et al.
2007), which may set the stage for new species to arise via assortative
mating or allopatric speciation (Gray and McKinnon 2006). It’s clear
that the flexibility and explanatory power of the RPS model makes it a
useful tool for understanding evolutionary processes in real systems.
However, the ubiquity of these dynamics within populations remains in
question as many studies investigating RPS dynamics have been conducted
in experimental laboratory conditions. Nonetheless, work with natural
populations (Sinervo and Lively 1996, San-Jose et al. 2014) and the
roles that similar selective processes (such as NFDS) play in natural
systems, indicate that RPS dynamics may be more prevalent than
previously thought.
Clearly RPS dynamics are a powerful tool in explaining evolutionary
phenomena such as genetic diversity, polymorphism and even speciation.
Even over short time periods, RPS dynamics can influence the genetic
variation in a population, leading to diverse evolutionary outcomes.
However, the same non-hierarchical interaction model can also be adapted
to explain ecological patterns.
ECOLOGICAL DYNAMICS: INTRANSITIVE COMPETITION
Many of the ecological theories that attempt to explain coexistence such
as recourse partitioning/niche differentiation (Tilman 1982) or models
invoking invasibility (Lotka-Volterra models; Lotka 1925, Volterra 1926)
approach coexistence in a pairwise fashion and begin to struggle with
increasing levels of community complexity. Real systems contain many
more species than there are limiting resources and interactions occur
between multitudes of species simultaneously rather than just two at a
time. The key limitation of these models is that they regard competition
as purely hierarchical and binary, such that in more complex systems one
superior competitor should displace all others (or perhaps one
competitor for each limiting resource). This is unlikely to be
realistic, as many species exist that are competitively similar and even
species that are competitively distinct have been shown to coexist
(Maynard et al. 2017). Therefore, it is possible – and perhaps even
likely – that competition is more often intransitive in nature, whereby
there is no single best competitor, but rather competitive outcomes are
determined by environmental variables and community/population
compositions (Soliveres et al. 2018, Soliveres et al. 2015, Gallien
2017). Like the RPS dynamics described above, intransitive relationships
result in an oscillating or dynamic equilibrium.
Multiple factors influence the stability of an intransitive system. Most
basically, system stability increases with increasing intransitivity
(Laird and Schamp 2006) such that a purely hierarchical or transitive
competitive system results in the competitive exclusion of all but one
competitor. A purely intransitive system – in which all competitors are
part of an intransitive loop and coexistence is impossible in the
absence of the loop – is particularly powerful in allowing for
coexistence. In fact, models of such systems show that coexistence is
possible even in the absence of any niche differences (Allesina and
Levine 2011, Laird and Schamp 2006). However, pure intransitivity is
also unlikely. Complete intransitivity requires an unlikely scenario in
which all species are present at the moment of community establishment
(Gallien et al. 2017). Natural systems most likely exist along a
gradient from purely transitive to purely intransitive competition
(Laird and Schamp 2006, Soliveres and Allan 2018, Gallien et al. 2017).
Intransitive competition also allows for the co-existence of not just
three species (as in the classic RPS scenario), but a theoretically
limitless number of competitors where each beats the following and is
beaten by the previous (Figure 2). Therefore, the length of the
intransitive loop also stabilizes a system, with longer loops exhibiting
more stable coexistence, by reducing the impact that any single species
has on the stability of the loop (Gallien et al. 2017). As the number of
species increases, so too does the number of possible competitive
interactions and intransitive loops. The existence and/or outcomes of
these interactions can also influence the stability of the system
(Allesina and Levine 2011, Vandermeer 2011, Gallien et al. 2017, Gallien
2017). As interactions increase within the intransitive loop nested
loops are formed which further stabilize the system (Figure 2b), similar
to the effects seen in food webs (Stouffer and Bascompte 2011) and
plant-animal mutualistic networks (Bascompte et al. 2003). Gallien et
al. (2017) note that the stabilizing effects of nested loops becomes
increasingly harder to quantify as they increase in length and
complexity. This may be because they depend on the length of the inner
loop(s) and the fitness differences between loop members. However,
evidence across five major taxonomic groups indicate that intransitivity
in natural systems is commonly nested, with interactions occurring more
frequently between species with similar competitive abilities (Soliveres
et al. 2018). There is also evidence to suggest that intransitive loops
in highly diverse communities not only enable coexistence but promote
additional diversity (Maynard et al. 2017). Nonetheless, stability of
the loops is contingent on an odd number of species. An even numbered
loop leads to the benefit of half the species to the exclusion of the
other half (Vandermeer 2011), as is the case in a two species system
where one outcompetes the other.
Intransitive competitive networks quickly become highly complex. A seven
species network with all species interacting with each other (Figure 2b)
already contains 49 possible competitive interactions in which the
outcomes of any given interaction may be influenced by the identity of
the species, environmental variables, the nestedness of the loops and
the outcomes of other interactions within the network (Allesina and
Levine 2011). This complexity makes studying intransitivity in natural
systems difficult. Empirical support for intransitivity’s role in
maintaining coexistence and shaping community structure remains
relatively scare. Additionally, the research that does exist is
complicated by the lack of a commonly accepted index or methodology to
measure intransitivity (Table 1: See Laird and Schamp 2018 for a
review). The methods chosen to measure intransitivity make different
assumptions about the nature of the competitive system and may quickly
increase the complexity and the feasibility of studying natural systems.
For example, a common method of measuring intransitivity is via
competitive reversals. Competitive reversals occur when a species that
would be “lower” in a purely hierarchical system beats a species that
would be “higher”, thereby creating a loop. However, measuring the
number of competitive reversals quickly becomes computationally
intensive. For an n-species system, the number of competitive reversals
needed to reach a possible pure hierarchy must be calculated for all n!
possible hierarchies. This means the 7 species system mentioned above
involves calculating the number of competitive reversals across 5040
possible hierarchies, but an 11 species system requires 11! = 39,916,800
(Laird and Schamp 2018, Slater 1961).
Despite the challenges, intransitive competition has emerged as a
popular avenue of research for explaining ecological phenomena such as
coexistence and community structure. Work with plants, coral and fungi
suggests that intransitive competition is the rule rather than the
exception (Buss and Jackson 1975, Soliveres et al. 2018, Soliveres et
al. 2015, Browne and Karubian 2016). An analysis by Soliveres et al.
(2018) of hundreds of plant communities across Europe showed that
intransitivity was present in almost 70% of dryland and over 80% of
grassland plant communities. Furthermore, a review by Gallien (2017)
found evidence of intransitivity in a diverse set of taxa including
aquatic invertebrates, microorganisms, ants and lizards. Studies also
show that intransitivity can work in tandem with other diversity
promoting mechanisms such as habitat spatial heterogeneity or species
mobility (Soliveres et al. 2018, Allesina and Levine 2018, Reichenbach
et al. 2007, Levine et al. 2017, Soliveres and Allan 2017).
While intransitivity has the potential to explain both coexistence and
community structure, fundamental weaknesses in the current body of
research makes interpreting its effects on natural systems difficult.
For example, questions remain about how to accommodate other theories of
coexistence alongside intransitive competition. However, research
indicates that intransitive competition alone is unlikely enough for
stable coexistence (Vandermeer 2011, Levine et al. 2017, Gallien et al.
2017, Soliveres and Allan 2017) indicating that pairwise mechanisms of
coexistence and intransitive competition may work together to maintain
diverse communities in nature. Another problem is the lack of a
universally accepted index or methodology for measuring intransitivity
(Table 1), and therefore the strength (along the gradient from a pure
hierarchy to pure intransitivity) and even the presence of
intransitivity in most communities remains unknown or underestimated.
Finally, natural communities are incredibly complex and contain dozens
if not hundreds of species. This complexity leaves open questions as to
how intransitive competition patterns function across different spatial
(Reichenbach et al. 2007, Levine et al. 2017) and temporal (Gallien et
al. 2017, Laird and Schamp 2018, Le Gac et al. 2012) scales, trophic
levels (Levine et al. 2017, Cameron et al. 2009, Soliveres and Allan
2018) and functional groups (Gallien 2017, Kassen 2002) In addition to
these ecologically oriented questions there is also a need to
investigate how these intransitive ecological interactions may be
altering evolutionary outcomes. Despite these uncertainties, the
potential explanatory power, and the existing evidence for the
prevalence of intransitivity provides a case for these dynamics playing
a pivotal role in co-existence theory.
ECO-EVOLUTIONARY APPLICATIONS: BRIDGING THE DIVIDE
RPS dynamics and intransitivity represent the same underlying logic,
that interactions within a system are conditional on each other and thus
outcomes may differ from those observed in pair-wise interactions.
Therefore, these theories represent a unique theoretical tool for
exploring eco-evolutionary processes as the same mechanistic approach
can be used to explain competition, selection, and coexistence on
multiple levels of organization. With research interest in both
intransitive interactions and eco-evolutionary theory growing, we
propose that intransitivity can be used as a framework to continue to
explore feedbacks between ecology and evolution.
Some previous research has also begun to provide direct or indirect
evidence for the interactive effects of intransitivity in
eco-evolutionary processes (Cameron et al. 2009, Huang et al. 2017,
Koskella and Lively 2009). While Cameron et al. (2009) do not explicitly
show evolution happening over the course of their experiment, they do
show that intransitivity along with certain environmental conditions can
stabilize a hemi-parasite, host, and resistant species system. This
RPS-enabled coexistence could lead to coevolution, as other
host-parasite systems have previously shown (Koskella and Lively 2009,
Brown and Tellier 2011, Ebert 2008). While co-evolution in this specific
instance is only speculative, it points to the need to move towards a
broader eco-evolutionary understanding of natural systems, as
ecologically orientated studies often fail to track the genetic changes
needed to identify evolution in action.
The flexibility of intransitivity and RPS dynamics to work within and
between species, as well as to emerge in a variety of scenarios further
showcase the potential that intransitivity has in connecting ecology and
evolution. Research has shown that intransitivity is found across the
living world in a variety of taxa and in a diverse set of circumstances
(Soliveres et al. 2018, Gallien 2017). Many of the scenarios which
commonly or theoretically produce intransitivity (Figure 3) are
ecological in nature and allow for stable coexistence and long-term
interaction between species (or genetic variants within a species). In
this way, intransitivity sets the stage for either co-evolution or
unidirectional evolution of one species (or morph) in response to
selective pressures created by other members of the intransitive loop.
These evolutionary changes can in turn change the ecological
interactions or the population dynamics of phenotypes within the system.
The ubiquity of these patterns, their potential as powerful stabilizing
forces and their flexibility to work across both ecology and evolution
lead us to believe that intransitivity functions across complete
eco-evolutionary feedback loops. Establishing the presence and
mechanisms of such feedbacks remain a major goal within eco-evolutionary
research (Fussman et al. 2007, Pelletier et al. 2009, Hendry 2019) and
we believe that intransitivity represents an ideal framework for these
investigations. Yet, no studies have provided direct empirical support
for this hypothesis.
To test the potential for such eco-evolutionary intransitivity
feedbacks, we have adapted a set of simulations to examine whether
intransitivity at the population-level leads to greater stability
between species at the community level. Using the model of Maynard et
al. (2019) we explored the effects of within-species intransitive
relationships on community-level intransitivity by simulating the
dynamics of a zero-sum competitive communities with phenotypic
variation. This model is a generalization of the replicator-mutator
equation, allowing for competitive interactions among phenotypes nested
within species (Hofbauer 1985). The dynamics are described by two key
parameters: phenotypic similarity (𝜏), which ranges from 0 to 1,
quantifying the average correlation between phenotypes’ competitive
abilities within a species; and phenotypic memory (ρ) which ranges
from \(\frac{1}{m}\) to 1 (with m being then number of
phenotypes within a species), capturing the probability that the
offspring of an individual have the same phenotype as the parent. Using
this model, we randomly generated 15 million interaction matrices across
a gradient of ρ and 𝜏 values, with the number of species fixed
at n =5 and the number of phenotypes fixed at m= 3 per
species (see Maynard et al. 2019 for details). For each random set of
matrices, we implemented two different scenarios: (1) where all of the
3x3 intraspecific sub-matrices of H are perfectly intransitive
rock-paper-scissor matrices, and (2) where all of the 3x3 intraspecific
sub-matrices are perfectly hierarchical. We then integrated the dynamics
of the community, under each of the two scenarios, until they reached
equilibrium. To quantify the effect of intraspecific intransitivity on
interspecific intransitivity, we calculated the species-level
intransitivity of the initial and final communities using Kendall and
Babington Smith’s d (Kendall and Babington Smith 1940, Laird and Schamp
2018), and then calculated the relative difference between the two
scenarios (perfectly intransitive vs. perfectly transitive intraspecific
relationships). The resulting difference gives the net effect of
intraspecific intransitivity on interspecific intransitivity across a
gradient of ρ and 𝜏 values (Fig 4).
We found clear evidence for a positive correlation between the
intransitivity of phenotypes within a population, and the resulting
intransitivity between species within a community. In all cases,
intransitive (rock-paper-scissors) relationships among the three
phenotypes led to an average increase in community-level intransitivity
between species (3.6% ± 0.009% across all combinations, p< 0.001), demonstrating clear positive feedbacks between
intra- and inter-specific intransitivity. This effect was largest under
high values of ρ and low values of 𝜏 (low similarity but high memory),
exhibiting upwards of 10% increase in community-level intransitivity
due to intraspecific intransitivity. These results demonstrate that this
benefit is strongest when phenotypes have unique interactions with other
species and that there is a strong benefit to being more competitive.
Interestingly, this is the same general region where Maynard et al.
(2019) observed that phenotypic variation provided the strongest
stabilizing force on the dynamics of the system, suggesting that the
intransitivity and stability of the system may be partially linked. This
relationship only declines at the extreme, as ρ→1 and 𝜏→0, where
communities have high dynamical stability to begin with (Maynard et al.
2019), suggesting that differences in intraspecific interactions have
little effect when the dynamics are already globally stable. At the
other extreme, intraspecific intransitivity had a negligible effect when
there were no competitive differences among phenotypes to begin with (𝜏
=1, Fig. 4, top black line), highlighting that intraspecific dynamics
have no effect on community-level dynamics when there is no variation in
fitness among phenotypes (i.e., phenotypes differ in name only).
Collectively, these simulation results show proof-of-concept of
feedbacks between intra- and interspecific intransitivity, while
demonstrating that the same conditions which promote stability and
robustness in these communities are the same conditions whereby
intraspecific intransitivity promotes interspecific intransitivity.
Given the potential for feedbacks between intransitivity within and
across species, these dynamics provide a conceptual framework for
linking ecological and evolutionary dynamics. As the emerging fields of
RPS and intransitive dynamics continue to grow, we highlight the need
for future empirical research to focus on evaluating the presence of
such feedback mechanisms within different ecological contexts. The
possibility of these frameworks to help us find tangible linkages
between the mechanisms driving diversity within and across species
represents a valuable research avenue for developing a more holistic
understanding of stability within ecological systems.
DISCUSSION
We propose that intransitivity represents a unifying mechanism to
advance eco-evolutionary theory. This work does not represent a
comprehensive evaluation of all intransitivity research, but rather aims
to collect the conceptual building blocks needed to use intransitivity
as a framework for investigating feedbacks between ecology and
evolution. While our results support the feasibility of intransitive
eco-evolutionary feedback loops, more research will be needed to
determine if these phenomena are at play in real systems. Empirical
studies showing RPS dynamics and intransitive interactions at work
remain relatively rare, especially in larger and more complex systems
like metapopulations, communities and ecosystems. However, recent
research has increasingly recognized the potential that RPS dynamics and
intransitive competition have in explaining foundational questions of
both ecology and evolution. Even without realizing it, parallels between
the two disciplines via these mechanisms have already been drawn. In
talking about intransitive competition and the ecological dynamics of
coexistence in complex communities, Levine et al (2017) cite RPS
dynamics within a single species (such as the example inU.stansburiana ) as the best empirical evidence for these dynamics
at work in natural systems. Furthermore, even the evolutionary
orientated studies (Sinervo and Lively 1996, San-Jose et al. 2014, Kerr
et al. 2002) are centered around the ecological processes of
competition, reproductive strategies, and behavior. Incorporating both
disciplines into future intransitivity studies can only serve to improve
the strength of research and theory.
While there is much more work on eco-evolutionary theory than presented
here, there remains a lack of research that empirically establishes full
feedbacks between ecology and evolution. Hendry (2019) cites the need to
conduct more empirical experiments in non-laboratory settings to show
how contemporary evolution may influence the ecological dynamics at play
in natural populations, communities, and ecosystems. We suggest that
increasingly sophisticated mathematical models to measure intransitivity
and improved genetic tools to track evolutionary changes will show that
intransitivity works simultaneously across both ecology and evolution to
shape and stabilize natural systems. Nevertheless, finding empirical
support for such feedbacks remains a considerable challenge. The lack of
long-term empirical studies investigating both RPS enabled polymorphisms
and intransitive competition, makes determining the conditions that
permit these patterns hard to determine. These mechanisms seem to
operate on longer temporal scales than those of the pair-wise
interactions that are often investigated. However, whether these
dynamics can assemble quickly and spontaneously or whether they require
long periods of co-existence and adaptation remains unclear.
Additionally, the circumstances under which intransitivity or
RPS-dynamics begin to breakdown are also unknown as even unstable
even-numbered intransitive loops can decay into stabilized odd-numbered
loops (Levine et al 2017). Research into these boundary conditions and
the maintenance of intransitive loops across larger spatial and temporal
scales are necessary to solidify their role in both ecological and
evolutionary processes.
Even studies that do attempt to measure intransitivity in natural
systems may underestimate its effects given that many of the commonly
used indices rely on pair-wise measures of competitive outcomes which
may incompletely capture the complexity of these inherently multispecies
interactions (Table 1). Clearly, there is a need for an improved and
standardized measure of intransitivity which can be used in both
ecological and evolutionary contexts. While our model relied on Kendall
and Babington Smith’s d (Kendall and Babington Smith 1940, Laird and
Schamp 2018), which does measure the number of three-species
intransitive triads, this measure may quickly become too computationally
intensive in larger systems.
CONCLUSIONS
Overall, a growing body of research highlights the tangible linkages
between the intransitive dynamics of phenotypes within populations as
well as between species within a community, and how these can give rise
to the stable maintenance of diversity across scales. The potential for
feedback mechanisms to operate between these scales means that
intransitivity may present a valuable framework for enhancing our
understanding of eco-evolutionary dynamics. Our simulations provide some
initial support for the potential existence of such feedbacks, but we
propose that future empirical research should strive to test the
strength of such linkages within natural systems. Given the prevalence
of intransitivity, and the importance of such stabilizing mechanisms
across evolutionary and ecological scales, this research has the
potential to provide insights into some of the fundamental questions in
ecology and evolution. Questions like; How can we explain genetic
variation within populations? Or why is there such an abundance of
biodiversity on our planet? We propose that intransitivity can also
provide a conceptual framework for explaining the complexity of natural
systems and shifting our understanding towards more holistic
interpretations. In so doing, it might allow us to better understand the
natural world, giving us insights into the connections that are
fundamental in shaping natural systems.
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