Ecological Complexity 8 (2011) 92–97
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Ecological Complexity
journal homepage: www.elsevier.com/locate/ecocom
Intransitive loops in ecosystem models: From stable foci to heteroclinic cycles
John Vandermeer *
Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor, MI 48109, United States
A R T I C L E I N F O
A B S T R A C T
Article history:
Received 24 July 2010
Received in revised form 11 August 2010
Accepted 13 August 2010
Available online 18 September 2010
Several forms of intransitive loops are investigated with regard to their stabilizing properties in simple
dynamic models. The intransitive loops are stabilizing when there is an odd number of species in the
system. Furthermore, an intransitive loop can stabilize a chain of negative interactions that would
otherwise be unstable.
ß 2010 Published by Elsevier B.V.
Keywords:
Competition
Intransitive loops
Heteroclinic cycles
1. Introduction
Standard ecological theory treats competitive communities as if
they exist along a gradient from those in which competition
coefficients are small (presumably because niches do not overlap too
much) to those in which competition coefficients are large. The first,
dominance-controlled communities (Yodzis, 1978), are perhaps the
most common in nature, if indeed Gause’s principle is approximately
true. The second, founder-controlled communities, are perhaps less
likely in that standard ecological theory predicts their downfall (i.e.,
all but one species are doomed to competitive displacement with the
standard model). In this communication I explore the consequences
of various structures that defy the basic conclusion of Gause’s
principle, allowing many species to coexist even though their
competition coefficients may be extremely large. The fundamental
idea stems from the notion of an intransitive loop.
May and Leonard (1975) first noted the unusual state of a
competitive arrangement that mimicked the classical children’s
game of rock, scissors, paper (the rock dulls the scissors, the
scissors cuts the paper, the paper covers the rock). Using the simple
Lotka–Volterra form,
dx1
¼ x1 ð1 x1 a12 x2 a13 x3 Þ
dt
(1a)
dx2
¼ x2 ð1 x2 a21 x1 a23 x3 Þ
dt
(1b)
dx3
¼ x3 ð1 x3 a32 x2 a31 x1 Þ
dt
(1c)
* Corresponding author.
E-mail address: [email protected].
1476-945X/$ – see front matter ß 2010 Published by Elsevier B.V.
doi:10.1016/j.ecocom.2010.08.001
they effectively noted that if a12 a13, a23 a21, and a31 a32,
this would represent a case of intransitive competition (2 beats 1, 3
beats 2, but 1 beats 3). If the inequalities are sufficiently large, a
heteroclinic cycle would arise, in which the carrying capacity of
each species would be alternatively approached as the zero values
for the other species are approached. The underlying dynamics are
well-known mathematically (see May and Leonard, 1975) and can
be described by simple qualitative graphs. There exists a surface
intersecting the three axis at (0,0,1), (0,1,0) and (1,0,0) and any
trajectory rapidly moves to this surface (the surface itself is a stable
manifold). Once arriving at that surface the system oscillates with
each species alternatively taking the dominant role. If the above
inequalities are not very large, the system exhibits relaxed
oscillations on the manifold (see Fig. 1a). But as the inequalities
increase, a bifurcation point is reached in which the stable focal
point on the manifold reverses stability and the system oscillates in
an unstable fashion, continually approaching more closely the
points intersected by the stable manifold (Fig. 1b). Note that the
manifold itself is still stable, but the non-zero equilibrium points
have become unstable. I will refer to this type of bifurcation as a
‘‘heteroclinic bifurcation’’ (Vandermeer, 2006), in that the three
equilibria on the axes (the carrying capacities, are saddles, and
trajectories approach them along one axis, but are repelled along
another axis). Metaphorically, the surface to which all trajectories
are attracted (the manifold) is itself with an internal gravitation,
concave when equalities are small, such that oscillations tend to
converge on its center, but reversing to become convex at the
bifurcation point, such that the ‘‘high point’’ on the surface tends to
repel the oscillations (compare Fig. 1a and b).
Several important modifications of this basic bifurcation
framework enter as we increase the dimensionality of the system.
First, it is not possible to construct a stable or heteroclinic system
with an even number of species. Second, with more than three
[(Fig._1)TD$IG]
J. Vandermeer / Ecological Complexity 8 (2011) 92–97
93
Fig. 1. The manifold on which an intransitive change is manifest.
species, the Hopf bifurcation (point at which the manifold surface
changes from concave to convex) results in a limit cycle and then,
with further increase in the size of the inequalities, a heteroclinic
cycle emerges, focused on the carrying capacities. It appears to be
the case that all cases of an odd number of species follow this basic
pattern, whereas all cases of even number of species result in
extinction of half of the components, leaving the other half living
independently at their carrying capacities.
2. Multidimensional niches and intransitive loops
The metaphor of normal-like distributions along an environmental gradient representing competition continues to motivate
undergraduate ecology students. If overlap of these distributions is
taken as proportional to competition intensity, it is evident that
intransitive structures are not possible. However, using the
elementary idea of niche overlap coupled with Hutchinson’s
notion of the niche as hypervolume, intransitive loops can emerge
quite simply.
Formally, if E represents an environmental gradient and W(E)
represents some estimate of fitness at E, then the competition
coefficient can be thought of as,
cðW i \ W j Þ
0 W i ðEÞ dE
ai j ¼ R 1
(1)
where c is a scaling factor that indicates the intensity of
competition in areas of the environmental gradient with overlap.
It is evident from (1) that it is not possible to obtain the following
simultaneous relations,
a21 > a12
(2a)
a32 > a23
(2b)
a13 > a31
(2c)
which define an intransitive loop. Since Wi \ Wj = Wj \ Wi, it
follows from (2a) that
Z
1
W 1 ðEÞ dE >
0
Z
1
W 2 ðEÞ dE
0
and from (2b) that
Z
1
W 2 ðEÞ dE >
0
Z
0
1
W 3 ðEÞ dE
whence the requirement from (2c) that,
Z
1
W 3 ðEÞ dE >
0
Z
1
W 1 ðEÞ dE
0
cannot be true. Thus, with the formulation 1 all competitive
interactions are transitive. It follows from this that permanent
communities with strong competition (founder-controlled communities) are not possible, that the interactions will culminate in
the complete takeover by species i, if aji > aij for all values of j.
However, if there are two niche axes, E1 and E2, with fitness
functions 1 Wi ðE1 Þ and 2 Wi ðE2 Þ for species i, then a simple extension
of 1 is
ai j ¼
c
2
"
1 Wi ðE1 Þ \ 1 W j ðE1 Þ
R1
0 1 Wi ðE1 Þ dE
#
þ
2 Wi ðE2 Þ \ 2 W j ðE2 Þ
R1
0 2 Wi ðE2 Þ dE
whence condition 2 is easily satisfied, a graphical example of which
is shown in Fig. 2.
3. Stabilizing a large competitive chain with a small
intransitive loop
Extremely simplified conceptual models of resource competition are frequently employed to investigate generalized consequences of particular competitive arrangements. One of the
most popular, as noted above, is envisioning a series of species
along a single resource gradient in which competition results in
some sort of species packing rule. Thus, for example, beginning
with the arrangement in Fig. 3a, if niche overlap (the intersections
of the various curves) results in sufficiently high competition
coefficients, the species become packed as in Fig. 3b. This is
sometimes referred to as the ‘‘limiting similarity’’ of species on a
resource gradient.
However, a rather large number of species can be held together
even if undergoing extremely high levels of competition if an
intransitive loop is involved somewhere within the overall
structure (which, as shown above, is possible if there is more
than one niche dimension). To see the basic principle involved, we
begin with a simple specific form as follows:
dx1
¼ x1 ð1 x1 ax2 Þ
dt
(3a)
[(Fig._2)TD$IG]
J. Vandermeer / Ecological Complexity 8 (2011) 92–97
94
Fig. 2. Intransitive loop in a two dimensional niche space. Species 1 and 2 interact in niche axis 1 with species 1 dominant (since its overall niche occupancy in this dimension
is larger than species 2). Species 2 and 3 interact in niche axis 2, with species 2 dominant. Species 1 and 3 interact in niche axis 1, with species 3 dominant.
dx2
¼ x2 ð1 x2 ax3 Þ
dt
(3b)
dx3
¼ x3 ð1 x3 ax1 Þ
dt
(3c)
dxi
¼ xi ð1 xi axi1 Þ
dt
for i ¼ 4 . . . s
(3d)
Setting a = 1.1 (ensuring complete dominance by species i over
species i 1 for all species indices >3), it is evident that an
arbitrarily large value of s can be sustained. If any of the three
species in the intransitive loop are eliminated (i.e., i = 1, 2, or 3) the
system decomposes to a system of s/2 species, with every other
species extinct (as in Fig. 3b). Thus, the competitive sorting process
[(Fig._3)TD$IG]
Fig. 3. (a) Representation of the niches of seven species on a single niche axis. (b)
Resolution of the strong competition represented in (a). Conceptually the
arrangement in (b) is maximal species packing, with overlaps between adjacent
niche occupancies small.
that leads from a situation like Fig. 3a to the species packing and
limiting similarity of Fig. 3b, is effectively cancelled by the
introduction of the intransitive loop.
Consider the simple example of Eqs. (3) with s = 6, as
diagrammed in Fig. 4a. If a is less than the Hopf bifurcation point
for the subsystem {1,2,3}, the system as a whole, with all six
species, is maintained at a focal point equilibrium (Fig. 4b). On the
other side of the Hopf bifurcation the system oscillates with each of
the three transitive species perfectly out of phase with one another
(Fig. 4c). When in the heteroclinic orbit phase it is intuitively
obvious what is happening. If species 3 is at a low point in its cycle,
species 4 is able to increase, thus putting pressure on species 5,
which consequently is less able to put competitive pressure on
species 6. However, when species 3 increases in its oscillatory cycle
it puts competitive pressure on species 4, which releases the
pressure on species 5, which can then put more competitive
pressure on species 6. Basically the same dynamics is operative to
the left of the Hopf bifurcation (Fig. 4b), but it is less intuitively
obvious since the system as a whole approaches, in the limit, a nonoscillatory situation.
It is evident that the s in Eq. (3d) can be arbitrarily large. That is,
the species in a transitive nonreciprocal chain (as in Fig. 3a, but
with competition going in one direction, and connected to an
intransitive loop) can be maintained in an arbitrarily large
assemblage even with strong competition coefficients, as long as
the system is not to the right of the Hopf bifurcation, at which point
the heteroclinic oscillations eventually drive one of the species
close enough to zero to conclude that a local extinction has
occurred (formally, the actual arrival at zero population density is
only in the limit).
It is also evident that the intransitive loop can be much larger
and give precisely the same results, but only if there is an odd
[(Fig._4)TD$IG]
J. Vandermeer / Ecological Complexity 8 (2011) 92–97
95
Fig. 4. Illustration of the stabilizing potential of an intransitive loop. (a) basic structure of system (equation system 2). (b) Time series of all six species prior to Hopf bifurcation.
(c) Time series of all six species after the Hopf bifurcation.
number of species in the loop. However, in three dimensions the
Hopf bifurcation leads immediately to a heteroclinic cycle, whereas
in all odd dimensions greater than three, the Hopf bifurcation leads
first to a stable limit cycle, which, with increasing competition
coefficients, eventually leads to a heteroclinic cycle. A complete
bifurcation diagram for a 5D intransitive loop is shown in Fig. 5.
coefficient. Thus the equations,
dx1
¼ x1 ð1 x1 ax2 Þ
dt
(4a)
dx2
¼ x2 ð1 x2 ax3 Þ
dt
(4b)
dx3
¼ x3 ð1 x3 ax1 bx4 Þ
dt
(4c)
dx4
¼ x4 ð1 x4 ax6 bx3 Þ
dt
(4d)
4. Coupling intransitive loops
Since intransitive loops are inherently oscillatory, it makes
sense to treat two or more of them as coupled oscillators. That is, a
competitive community can be constructed in which two (or more)
[(Fig._5)TD$IG]3D loops are coupled with a simple extension of the competition
Fig. 5. Bifurcation diagram for the basic 5 species intransitive loop. Two bifurcation diagrams are superimposed here, the first on an arithmetic scale (the left y axis) showing
local maxima and local minima, the second on a logarithmic scale (the right y axis), showing only the local minimum. The small curved arrows indicate which bifurcation
diagram is associated with which axis.
[(Fig._6)TD$IG]
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J. Vandermeer / Ecological Complexity 8 (2011) 92–97
Fig. 6. Limit cycle resulting from weak coupling of two three dimensional intransitive loops. Note the ‘‘ghost’’ of the two original focal point attractors in the cycle.
dx5
¼ x5 ð1 x5 ax4 Þ
dt
(4e)
dx6
¼ x6 ð1 x6 ax5 Þ
dt
(4f)
represent a coupling of two 3D intransitive loops through
competition between species 3 and 4. As with other coupled
oscillators, this coupling of species 3 and 4 brings the two
oscillating systems into coordination with one another. In this
particular form, for almost any strength of the coupling, b, the
system moves into a phase locked state with the pairs {1,5}, {2,6}
and
[(Fig._7)TD$IG] {3,4} in complete reverse phase with one another. Given the
almost perfect out-of-phase locking of the system, its overall
behavior can be examined more easily with the following
transformations,
D1;5 ¼ x1 x5
D2;6 ¼ x2 x6
With a set such that the uncoupled system is in its stable
oscillatory mode, increasing b causes the system to move rapidly
to a limit cycle. The details of that cycle reflect the original two
Fig. 7. The heteroclinic cycle resulting from an increase in the strength of the coupled nontransitive loop. (a) General structure of the coupled system. (b) Phase plane in the
delta space (see text) for two of the three transformed variables. (c) 3D phase plane in the delta space. (d) Time series illustrating the increasing time spent near the zero and
1.0 points for each of the delta variables (note the increasing time spent near one of the equilibrium points, a characteristic of heteroclinic cycles).
J. Vandermeer / Ecological Complexity 8 (2011) 92–97
focal point attractors (Fig. 6). Increasing the coupling (increasing
the value of b in system 4), the system retains the overall structure
of the original coupled cycle (Fig. 6) but rapidly moves toward a
heteroclinic cycle (Fig. 7).
In coupling larger intransitive loops, similar qualitative
behavior is observed, namely, at low levels of coupling and either
focal point attractors or simple limit cycles in the original loops, the
coupled system synchronizes. As the coupling increases, the
oscillations become more complicated and frequently become
chaotic. Finally, with yet further increases in coupling intensity
there is a heteroclinic bifurcation and some component(s) of the
system are effectively lost (approach the value of zero in the limit).
5. Conclusions
For any ecosystem composed of strongly competitive species, as
that ecosystem becomes large, it is likely that intransitive loops
will exist. Here the dynamic nature of such loops is discussed.
Somewhat surprisingly, an intransitive loop can act as a stabilizing
force, in that other species may be influenced by it in a way that
prevents what would otherwise be a competitive exclusion. This
may result in a competitive chain that is almost completely
transitive, yet stabilized by a single intransitive loop. Thus, the
classical notion that species arranged along a single environmental
gradient will sort out such that overall competition is somehow
minimized (the species packing effect) may be dramatically altered
if an intransitive loop is involved.
However, not all intransitive loops generate this stabilizing
effect. Loops containing an even number of species are inherently
unstable, with half of the species surviving by virtue of the fact that
the other half disappear. Thus, only loops with an odd number of
97
species have this effect. Furthermore, a generalized pattern is
observed as the competition coefficients connecting the species in
the loop become large. As graphically illustrated in Fig. 1, the
system moves from a stable oscillation to a heteroclinic cycle. More
detailed examination (Fig. 5) shows a pattern of a stable focal point
passing through a Hopf bifurcation leading to a limit cycle and
eventually to a heteroclinic cycle.
Elementary considerations of the relationship between competition coefficients and the ecological niche reveal that intransitive loops cannot exist on a single niche dimension, but can easily
be envisioned on two or more niche dimensions.
Since intransitive loops are likely to emerge in large ecosystems
and since they are fundamentally oscillatory, the nature of the
oscillations under coupling is of interest. Generally, coupling
intransitive loops generates complicated cycles, frequently chaotic. However, a generalization seems to emerge in that coupling
intransitive loops (with odd numbers of species) seems to follow a
similar bifurcation pattern as is observed when increasing the
competition coefficient in a simple odd species loop (Fig. 5). A
stable focus yields a limit cycle which goes through cascading
bifurcations to chaos and eventually leads to a heteroclinic cycle,
sometimes a very complicated one (Fig. 7).
References
May, R.M., Leonard, W.J., 1975. Nonlinear aspects of competition between three
species. SIAM J. Appl. Math. 29, 243–253.
Vandermeer, J., 2006. Cascading thresholds to heteroclinicity in an ecosystem
model. Int. J. Ecodyn. 1, 373–379.
Yodzis, P., 1978. Competition for Space and the Structure of Ecological Communities. Springer-Verlag, Berlin.