Theor Ecol (2014) 7:115–125
DOI 10.1007/s12080-013-0204-6
ORIGINAL PAPER
The effect of predator limitation on the dynamics
of simple food chains
Christoph K. Schmitt · Stefan Schulz · Jonas Braun ·
Christian Guill · Barbara Drossel
Received: 14 December 2012 / Accepted: 21 October 2013 / Published online: 13 November 2013
© Springer Science+Business Media Dordrecht 2013
Abstract We investigate the influence of competition
between predators on the dynamics of bitrophic predator–
prey systems and of tritrophic food chains. Competition
between predators is implemented either as interference
competition, or as a density-dependent mortality rate.
With interference competition, the paradox of enrichment
is reduced or completely suppressed, but otherwise, the
dynamical behavior of the systems is not fundamentally
different from that of the Rosenzweig–MacArthur model,
which contains no predator competition and shows only
continuous transitions between fixed points or periodic
oscillations. In contrast, with density-dependent predator
mortality, the system shows a surprisingly rich dynamical
behavior. In particular, decreasing the density regulation of
the predator can induce catastrophic shifts from a stable
fixed point to a large oscillation where the predator chases
the prey through a cycle that brings both species close to
the threshold of extinction. Other catastrophic bifurcations,
such as subcritical Hopf bifurcations and saddle-node bifurcations of limit cycles, do also occur. In tritrophic food
chains, we find again that fixed points in the model with
predator interference become unstable only through Hopf
bifurcations, which can also be subcritical, in contrast to
the bitrophic situation. The model with a density limitation
shows again catastrophic destabilization of fixed points and
C. K. Schmitt · S. Schulz · J. Braun · B. Drossel ()
Physics Department, TU Darmstadt, Hochschulstraße 6,
64289 Darmstadt, Germany
e-mail: [email protected]
C. Guill
Institute for Biodiversity and Ecosystem Dynamics,
University of Amsterdam, P.O. Box 94248,
1090 GE Amsterdam, The Netherlands
various nonlocal bifurcations. In addition, chaos occurs for
both models in appropriate parameter ranges.
Keywords Population dynamics · Bifurcations · Food
chain · Regime shift
Introduction
Population dynamics that are predicted by mathematical
models can be very sensitive to the types of equations
used (Fussmann and Blasius 2005), implying that different biological situations can lead to very different population dynamics. In order to assess the relevance of the
results obtained with a model, it is important to understand the ecological meaning of the terms included and the
simplifying assumptions made in the model. While the oldest predator–prey model, the Lotka–Volterra model (Lotka
1925; Volterra 1926) which neglects predator saturation
effects, leads to marginally stable cycles, the Rosenzweig–
MacArthur model (Rosenzweig and MacArthur 1963),
which includes saturation effects, leads to either stable
fixed points or stable limit cycles, depending on the prey
carrying capacity (May 1972). The amplitude of the oscillation increases with increasing carrying capacity and the
period becomes slower, an effect that is called the paradox of enrichment (Rosenzweig 1971), which is, however,
usually not observed in nature (McCauley and Murdoch
1990), although Hopf bifurcations are observed (Fussmann
et al. 2000). Indeed, when prey refuges (via a Holling
type III functional response) or predator interference (via
a Beddington–De Angelis functional response (Beddington
1975; De Angelis et al. 1975)) are included in the model,
the Hopf bifurcation is shifted and oscillations are reduced,
and the paradox does not occur under certain parameter
116
combinations (Rall et al. 2008). Similar results are obtained
when other models for predator interference are used (Arditi
et al. 2004). When a ratio-dependent functional response
is used, the limit cycle collides eventually with the origin
in a so-called heteroclinic bifurcation, and both species go
extinct as the carrying capacity is increased beyond a critical value (Li and Kuang 2007). However, ratio-dependent
functional responses (Arditi and Ginzburg 1989) are not
considered to be realistic (Abrams 1997). A recent publication shows that in a stoichiometric predator–prey model
that includes limiting resources, an infinite-period bifurcation (saddle-node bifurcation on a limit cycle) occurs
(van Voorn et al. 2010). Such a bifurcation implies catastrophic behavior, because a stable fixed point can become
unstable by giving rise to large predator–prey oscillations.
A comprehensive overview over the possible dynamical
behaviors of predator–prey systems is given by Bazykin
(1998). Even the Bogdanov–Takens bifurcations and a variety of catastrophic transitions are possible, for instance,
when the predator density is limited such that it cannot
control the prey any more (see Chap. 3 in Bazykin 1998).
The range of possible dynamical behaviors becomes
considerably larger when tritrophic food chains are considered. In particular, in systems with three dynamical
equations, chaos can occur. Indeed, a tritrophic food chain
that is a generalization of the Rosenzweig–MacArthur
model was shown in 1991 to display a strange attractor that
is called “teacup attractor” because of its shape (Hastings
and Powell 1991), but chaotic behavior in food chains
was already observed much earlier (Hogeweg and Hesper
1978). A detailed analysis of this model and several citations of earlier investigations are given by Kusnetsov et al.
(2001), showing that the model displays Belyakov bifurcations, fold and period-doubling cycle bifurcations, and
the coexistence of stable limit cycles and strange attractors
with different geometries. Tritrophic food chain models can
display various global bifurcations (see van Voorn et al.
2010). Since in global bifurcations extended attractors
are destroyed or created, they have catastrophic effects if
they occur in natural systems. The dynamical trajectories
move to a different region in phase space when the system
undergoes a global bifurcation, leading to a regime shift.
In this paper, we focus on the effect of intraspecific competition of the predator on the dynamics of the system. The
Rosenzweig–MacArthur model uses a Holling type II functional response (Holling 1959), and the predator growth
rate depends only on prey density, not on predator density. However, empirical evidence indicates that predator
feeding rates depend also on predator density (Skalski and
Gilliam 2001), and for this reason, one of the model versions
that we will investigate uses the Beddington–De Angelis functional response (for a study on the effect of other
implementations of predator interference, see for instance
Theor Ecol (2014) 7:115–125
Bazykin et al. 1981). On the other hand, predator populations are limited in size also by factors independent of prey
availability, such as parasites or epidemics, nesting sites or
shelter, or a limiting resource. The effect of such factors can
be implemented as mortality rates that are proportional to
predator density, which are, for instance, discussed by Steele
and Henderson (1981), Bazykin (1998), and Gross et al.
(2005). One obvious effect of introducing terms that reduce
predator density is a reduction of oscillation amplitudes and
a shift of Hopf bifurcations to higher carrying capacities
or food intake rates. Apart from this stabilizing effect, we
will show below that the two mentioned ways of limiting
predator densities have fundamentally different effects on
the dynamical behavior of di- and tritrophic food chains. If
predator density is limited by a density-dependent mortality rate, catastrophic transitions can occur, where the system
moves from a stable fixed point to a cycle with a very large
amplitude, with predator and prey species going repeatedly
through periods of extremely low abundance, leading to a
large extinction risk due to random fluctuations. The bifurcations leading to this type of catastrophic behavior do not
occur with the Beddington–De Angelis functional response.
Model
We write predator–prey dynamics in the following general
form:
N
Ṅ = rN 1 −
− f (N, P ) · P
KN
Ṗ = λf (N, P ) · P − g(P ) · P
(1)
where N is the prey biomass density, and P is the predator biomass density. The parameter r is the maximum prey
growth rate, and KN is the carrying capacity of the prey. The
limiting factor for prey growth is usually the availability of
resources, which can be plant biomass if the prey is considered to be a herbivore, or chemical nutrients if the prey is
a plant species. Therefore, an increase of KN can generally
be interpreted as enrichment of the environment. The functional response f (N, P ) describes the feeding rate of the
predator and will be specified below. g(P ) is the mortality
rate of the predator. The ecological efficiency λ takes into
account that the predator can digest only part of the prey
biomass.
For a tritrophic food chain, the equations take the following form:
N
Ṅ = rN 1 −
− f1 (N, P1 ) · P1
KN
Ṗ1 = λf1 (N, P1 ) · P1 − f2 (P1 , P2 ) · P2 − g1 (P1 ) · P1
Ṗ2 = λf2 (P1 , P2 ) · P2 − g2 (P2 ) · P2
(2)
Theor Ecol (2014) 7:115–125
117
The widely used Rosenzweig–MacArthur predator–prey
model uses a constant mortality rate
(gi (Pi ) = di ) and
ai N
.
Holling type II functional response fi (N, P ) = 1+h
i ai N
This model takes into account that prey needs to be processed by the predator, giving rise to a “handling time”
hi . With an appropriate rescaling of the biomasses, we can
always achieve that ai hi = 1, which eliminates the parameter hi from the model. The parameter ai now has the
meaning of a maximum consumption rate (saturation value
of the functional response at infinite prey abundance). We
further rescale time such that r = 1. This means that the
maximum consumption rate ai and the mortality rate gi (Pi )
are expressed relative to the growth rate of the prey.
We will discuss two extensions of both bitrophic and
tritrophic food chains, which include competition between
predators. The first model with competition uses the
Beddington–De Angelis functional response (Beddington
1975; De Angelis et al. 1975):
fi (N, P ) =
ai N
1 + N + ci P
(3)
which can be motivated by the fact that predators lose time
fighting with each other when they meet, leading to a lower
consumption rate. The parameter ci quantifies the strength
of interference competition. A strict mechanistic derivation
based on predator time lost due to hiding prey has recently
been given by Geritz and Gyllenberg (2012).
The second model uses a mortality rate that depends on
predator density, which may be due to parasites or diseases
that propagate with a rate that depends on predator density
or to the fact that the environment can sustain only a certain
predator density because of limited availability of space or
resources other than the prey. In this case, parameters KP i
are introduced, which limit the predator populations. The
predator mortality has the form
gi (Pi ) = di +
λ ai
Pi ,
KP i
(4)
implying that the predator density cannot increase beyond
KP i (1 − di ).
Bitrophic predator–prey dynamics
Rosenzweig–MacArthur model
The Rosenzweig–MacArthur model, which contains no
direct predator competition, has been studied thoroughly
(Rosenzweig and MacArthur 1963; Rall et al. 2008). For
d
KN < λ a−d
, the predator becomes extinct since its growth
rate is negative even when the prey population is at the card
d
rying capacity. For λ a−d
< KN < 1 + 2 λ a−d
, the system
d
has a stable fixed point. For 1 + 2 λ a−d
< KN , the system has a stable limit cycle. The amplitude of the limit cycle
increases with increasing KN , making the species prone to
extinction by random fluctuations when KN is large. This is
the paradox of enrichment. For any set of parameters, there
is only one attractor towards which all trajectories tend that
start at values N > 0 and P > 0.
Model with interference competition
When a predator interference term is included in the functional response, one obtains the Beddington–De Angelis
function. The resulting model has been investigated thoroughly by Arditi et al. (2004), Rall et al. (2008), van Voorn
et al. (2008), and again a summary is given in the following.
The strength of the interference competition is controlled by
the parameter c (see Eq. 3), and in the limit case c = 0, the
Rosenzweig–MacArthur model is obtained. Figure 1 shows
the qualitatively different phase portraits, and Fig. 2 shows
a phase diagram (also known as two-parameter bifurcation
diagram or parametric portrait) for this system.
The extinction threshold of the predator is at the same
parameter value as in the Rosenzweig–MacArthur model,
independently of the competition strength c, because at vanishing predator density, competition is negligible. The phase
portraits shown for c/a = 0.25 are qualitatively similar to
those occurring for other ratios c/a < 1. For the case c = 0,
the predator isocline (i.e, the line Ṗ = 0) is vertical. The
line of the Hopf bifurcation can be calculated analytically
and is given by Rall et al. (2008):
Hopf
KN
=
(aλ − cdλ + d)2
.
(cλ − 1) acdλ2 + aλ2 (c − a) + d 2 (1 − cλ)
The Hopf bifurcation moves to larger values of KN when c
is larger.
For c > a, the prey isocline has a pole at N = KN (1 −
a/c) and is a decreasing function of the prey biomass N,
crossing the prey axis at KN . The fixed point is always stable in this case, and therefore, the paradox of enrichment
does not occur for high interference competition. The condition for this case can also be satisfied by reducing the
parameter a. This means that reducing the predator’s maximum consumption rate results in a stable fixed point (keep
in mind that a is expressed relative to the prey growth rate).
This can be understood as follows: Large c or small a means
that the influence of a predator on its prey is small because
predators kill few prey compared to the amount being produced. The prey can grow at low density even when the
predator biomass is very large. Prey dynamics is affected
by the presence of the predator sufficiently little, so that
the prey biomass always goes to a stable fixed point, as it
does in absence of the predator. In contrast, the periodic
118
Theor Ecol (2014) 7:115–125
predator biomass
KN =5
KN =1
6
Ṗ = 0
KN =10
5
Ṗ = 0
4
3
2
1
0
Ṗ = 0
Ṅ = 0
0
1
Ṅ = 0
Ṅ = 0
2
3 0
1
2
3
4
prey biomass
0
5
2
4
6
8
10
Fig. 1 Phase portraits for the bitrophic predator–prey model with
predator interference. The prey isocline (Ṅ = 0) and predator isocline
(Ṗ = 0) are also shown (solid lines). First three panels, isoclines and
typical trajectories for different enrichment levels (KN ) for the case
c < a. Rightmost panel, the case c > a for KN = 3. Parameter values are a = 1, d = 0.5, and λ = 0.85. The competition strength is
c = 0.25 in the first three graphs and c = 1.25 in the fourth one
behavior after the Hopf bifurcation indicates a strong negative response of the prey to changed predator densities:
Prey abundances increase only after the predator decreased
because of food deficit.
In the model with predator interference, more complex
situations such as catastrophic transitions from fixed points
to large-amplitude oscillations cannot occur. The nontrivial
fixed points of the prey biomass density are given by the
following:
positive population sizes, i.e., no tangent (saddle-node)
bifurcations can occur.
∗
N1,2
KN
(λ(c − a) + d) ±
=
2λc
KN
2λc
2
(λ(c − a) + d)2 +
Kd
.
λc
Next, we consider the case where predator competition is
due to density-dependent mortality, implemented via the
quadratic term (4). The functional response is a Holling
type II function. Investigations of the bifurcations occurring for such equations for the case d/r = 1 are given by
Kusnetsov (2004, p. 328ff) and by Bazykin (1998). In this
model, the prey isocline is the same as in the Rosenzweig–
MacArthur model, but the predator isocline now takes the
form IP (N ) = N KPN(λλ aa−d) . It is no more linear but bends
to the right and approaches a constant value for large N (see
Fig. 3), reflecting the imposed restriction on the sustainable
predator biomass, in the same way as the logistic growth
restricts the prey biomass.
predator biomass
Because all parameters have positive values, there can be at
most one fixed point where both prey and predator biomass
density are positive. Further analysis of the isoclines reveals
that in the case with predator interference, the prey and
predator isoclines can never be tangent to each other for
Model with density-dependent predator mortality
Fig. 2 Phase diagram for the bitrophic predator–prey model with
predator interference. The dotted gray line marks the transcritical
bifurcation, the solid black line, the Hopf bifurcation. A stable limit
cycle exists below the Hopf bifurcation line. The parameter values are
as in Fig. 1
16
14
12
10
8
6
4
2
0
Q R
KN =60
Ṅ = 0
S
Ṗ = 0
0
10
20 30 40
prey biomass
50
60
Fig. 3 Phase portrait of the bitrophic predator–prey system with
density-dependent predator mortality beyond the infinite-period bifurcation, for KP = 40 and KN = 60. A saddle-node bifurcation has
created two fixed points Q and R on the original limit cycle
Theor Ecol (2014) 7:115–125
For low KN , the system behaves like the one with interference competition. First, a stable positive fixed point
emerges, which becomes unstable at a Hopf bifurcation,
spawning a limit cycle. With high competition (small KP ),
the Hopf bifurcation does not occur.
However, unlike in the system with interference competition, the limit cycle can disappear as KN is increased
further, resulting again in a stable fixed point as the attractor of the system. This can happen in several ways, which
can be obtained by increasing KN for different values of KP
(see also the phase diagram in Fig. 4):
1. Another supercritical Hopf bifurcation (for instance for
KP = 16). The limit cycle shrinks and vanishes, turning
the initial fixed point stable again.
2. A variant of the first case occurs for values KP 17.5.
Here, the stable limit cycle does not shrink to zero
with increasing KN , but collides with an unstable limit
cycle that has emerged via a subcritical Hopf bifurcation of the fixed point, which has become stable again.
Since the subcritical Hopf bifurcation and the subsequent saddle-node bifurcation of limit cycles occur very
close to each other and to the fixed point, this scenario
is hardly distinguishable from the first one, even though
it represents in principle a catastrophic scenario.
3. A saddle-node bifurcation (of fixed points) happens on
the limit cycle. This scenario occurs for KP > 20
and is called an infinite-period bifurcation. The period
of the oscillation increases, and eventually, the limit
cycle turns into a heteroclinic orbit between the newly
spawned saddle and node. In this case, the limit cycle
Fig. 4 Phase diagram of the bitrophic predator–prey system with
density-dependent predator mortality in the KP –KN plane. The second diagram is a zoom into the first one. Lines indicate bifurcations
of codimension 1. Solid black, Hopf bifurcations; dashed gray, saddlenode bifurcations; dashed black, saddle-node bifurcation of limit
cycles; solid gray, homoclinic bifurcations. The marked points indicate
119
also has a large amplitude when it disappears. Furthermore, small deviations from the new fixed point can
cause the system to go through a large orbit before
returning to the fixed point. Like the original limit
cycle, this takes the biomass densities close to 0, leaving
the species vulnerable to extinction.
4. In a small parameter region (KP = 18.5 to 20) around
the end of the Hopf bifurcation line in a Bogdanov–
Takens bifurcation, many other combinations of bifurcations occur, which we do not discuss further due
to their small ecological relevance. An equivalent system is analyzed in great detail in Chap. 3.5.2 by
Bazykin (1998), and the phase diagram is also shown by
Kusnetsov (2004, p. 330). Because our value d/r = 0.5
is different from their choice d/r = 1, in our case,
the two saddle-node lines do not rejoin, and there is
no second Hopf bifurcation line; however, the rest of
our phase diagram is equivalent to the ones shown
by Bazykin (1998) and Kusnetsov (2004). Our phase
diagram was obtained numerically using the AUTO
software package (Doedel et al. 2007).
Tritrophic food chain
Rosenzweig–MacArthur model
In the tritrophic food chain with ci = 0 and g(Pi ) = di Pi ,
period-doubling cascades and chaos are observed, as analyzed in detail by Kusnetsov et al. (2001) and shown in the
bifurcation diagram Fig. 5.
bifurcations of codimension two. GH, generalized Hopf bifurcation
(the Hopf bifurcation changes from supercritical to subcritical, spawning a saddle-node bifurcation of limit cycles); CP, Cusp bifurcation
(two saddle-node bifurcations meet); BT, Bogdanov–Takens bifurcation (a Hopf bifurcation line and a homoclinic bifurcation line end on
a saddle-node bifurcation line)
120
Theor Ecol (2014) 7:115–125
7
biomass (a.u.)
6
second predator
5
4
3
2
1
0
0.5
1
1.5
2
3
2.5
KN
3.5
4
4.5
5
Fig. 5 Bifurcation diagram for the tritrophic food chain without
predator competition (Rosenzweig–MacArthur model). The parameter values are a1 = 0.8, d1 = 0.2, a2 = 0.133, d2 = 0.033, and
λ = 0.85. Only the fixed point values and local extrema of the top
predator biomass are shown
Just as in the two-species model, the predators can only
exist above a minimum value of KN , which is higher for the
top predator. With increasing nutrient level, the global fixed
point goes through a Hopf bifurcation. Again, the paradox of enrichment can be found. With further enrichment,
the system goes through a series of period-doubling bifurcations towards a chaotic behavior. Further increasing the
enrichment leads to a series of period-halving bifurcations
that end in a stable limit cycle. This cycle’s minimum and
maximum both decrease with increasing KN .
There is a second effect that does not occur in twodimensional predator–prey systems: The basin of attraction
of the attractor does not cover all of phase space, because
the attractor of the two-dimensional system can also be an
attractor in the three-dimensional one if extinction of the
second predator occurs for some of the starting values. Furthermore, as shown by Kusnetsov et al. (2001), chaotic and
periodic attractors can coexist.
For a detailed analysis of this model, including phase
diagrams, we refer again to Kusnetsov et al. (2001).
the isoclines diverges, the system can show periodic oscillations, but no chaos. When c1 > a1 (i.e., the prey isocline
diverges), the prey population is large and fluctuates only
little, while the two predators have a periodic oscillation if
c2 is small enough. When the second isocline diverges, i.e.,
2
when c2 > λ aa1 −d
, the second predator is strongly limited
1
by competition and has no large effect on the first predator. The system behaves similarly to the two-species model,
with the phase diagram being qualitatively the same as in
Fig. 2. However, we found that the Hopf bifurcation turns
subcritical when c2 is lowered, giving rise to a saddle-node
bifurcation of limit cycles and to the coexistence of a stable fixed point with a stable limit cycle. When changing the
parameter KN , a catastrophic transition from a fixed point
to a limit cycle and back can occur.
The most complex situation arises in the intermediate parameter range, when c2 is around 0.2. The phase
diagram is shown in Fig. 6. In contrast to the Rosenzweig–
MacArthur model, we find subcritical Hopf bifurcations
and saddle-node bifurcations of limit cycles, in addition
to the period-doubling and period-halving bifurcations that
surround the chaotic region.
As in the two-species system, in our numerical simulations, the fixed point always became unstable in a Hopf
bifurcation; saddle-node bifurcations of fixed points were
never observed. However, in contrast to the two-species system, the Hopf bifurcation can become subcritical, which
Model with interference competition
The strength of interference competition of the two predators is given by the parameters c1 and c2 in the functional responses. For small interference competition, the
dynamical behavior is similar to the tritrophic Rosenzweig–
MacArthur model (Fig. 5). The main change with respect
to Fig. 5 is that the bifurcation diagram looks stretched in
the KN direction. In the opposite, limit of strong interfer2
ence competition, when c1 > a1 and c2 > λ a1a−d
, i.e.,
1
when the isoclines of the prey and the first predator diverge,
we find that the fixed point remains stable with increasing
enrichment. The stabilizing effect that has been observed in
the two-dimensional case is thus found again. The stability can be shown analytically if c1 > 1. When only one of
Fig. 6 Phase diagram of the three-species system with interference
competition, with the top predator’s competition parameter being c2 =
0.2. The solid black line indicates a Hopf bifurcation, the dotted black
lines are period-doubling bifurcations, and the dashed black lines
are saddle-node bifurcations of limit cycles. On the right, a perioddoubling and saddle-node bifurcation of limit cycles are so close to
each other that they can hardly be distinguished in the figure and
appear as a dash-dotted line. To the right of these bifurcations, the
system has again a stable limit cycle
Theor Ecol (2014) 7:115–125
means that a catastrophic shift from a fixed point to a
finite-amplitude oscillation can occur. Conversely, a finiteamplitude oscillation can become unstable due to a saddlenode bifurcation of limit cycles, and the system goes to
a fixed point. Compared to the catastrophic transitions
observed in the situation with density-dependent predator
mortality, the fixed point lies within the limit cycle and
not outside. In addition, nonlocal catastrophic transitions
between limit cycles and chaos occur. In contrast to the
Rosenzweig–MacArthur model, where the limit cycles are
interwoven with the chaotic attractors (Kusnetsov et al.
2001), they now occur also outside of the chaotic attractors.
Model with density-dependent predator mortality
Finally, we study the model with density-dependent predator mortality. Just as for the case of interference competition,
high competition can prevent chaotic behavior and (if even
higher) limit cycles altogether. In general, the dynamical
behavior of this model is very rich. Due to the complexity of the behavior and to the high dimension of parameter
space, we limit ourselves to giving an impression of the
interesting phenomena observed in this model, without performing a complete and thorough analysis. For this reason,
all parameters except for the carrying capacities KN , KP 1 ,
and KP 2 are fixed at a1 = 0.8, d1 = 0.2, a2 = 0.133,
d2 = 0.033, and λ = 0.85. We vary KN and KP 1 for values
of KP 2 = 10, 18 and 40. The cases of high and low KP 2 are
fairly well represented by the chosen values 40 and 10, but
the intermediate case is just one of many different stages of
transition between the two limit cases.
High values of KP 2
For high values of KP 2 , the dynamical behavior resembles
to some extent the one for low c2 in the case of interference competition. Figure 7 shows the phase diagram, a
bifurcation diagram, and two attractors. We observe a Hopf
bifurcation and a period-doubling cascade to chaos and
back, but no saddle-node bifurcations. The chaotic region
occurs for large values of both predator carrying capacities and low carrying capacity of the resource, KN , where
the density limitation of the two predators has only a small
effect on the dynamics. In contrast to Fig. 5, the chaotic
region in the bifurcation diagram is much more narrow,
and the chaotic attractor looks somewhat different from the
well-known teacup attractor. Beyond the chaotic region, the
limit cycle comprises an oscillation of the P1 –N pair as well
as one of the P2 –P1 pair, as shown in the bottom right graph
in Fig. 7. For other parameter values, we observe mainly
oscillations of one of the two pairs: For small KN , the oscillation is primarily driven by the interaction between the
121
resource N and its predator P1 , and the Hopf bifurcation
line is almost parallel to the KP 1 axis in the phase diagram
of Fig. 7. For large KN with small KP 1 , the resource density stays close to its upper limit KN , and P1 has plenty of
food and acts like a resource for P2 . The oscillation occurs
now between P1 and P2 , and the Hopf bifurcation line is
almost parallel to the KN -axis for low values of KP 1 and
high values of KN .
Small values of KP 2
Figure 8 shows the phase diagram for KP 2 = 10. The phase
diagram resembles the two-species system with densitydependent predator mortality, because the density limit
KP 2 = 10 of the top predator is small, so that this predator has only a small effect on the two other species. With
increasing KN , the system undergoes first a Hopf bifurcation, and later, the limit cycle is again replaced with a
fixed point. As in the two-species system, the latter transition is not smooth, but catastrophic. The left graph in Fig. 8
shows the Hopf and saddle-node bifurcation lines, which
resemble those of the two-species model. In contrast to the
two-species system, there is no infinite-period bifurcation
for this parameter combination. Instead, there is a parameter region where the two different attractors (limit cycle
and fixed point) coexist. The coexistence region is limited
by a saddle-node bifurcation on one end and a homoclinic
bifurcation at the other end. The homoclinic line behaves
very differently from the two-species model, initially staying close to the bottom saddle-node line and running parallel
to the upper saddle-node line for large values of KN . As
in the two-species case, the Hopf bifurcation changes from
super- to subcritical, spawning a saddle-node bifurcation of
limit cycles. But instead of staying close to the Hopf bifurcation, the lines separate. The saddle-node bifurcation of
limit cycles goes up and left, staying above the Hopf line,
then joins another of its kind in a cusp shape. The other line
crosses below the Hopf line and then runs parallel to it. After
crossing the bottom one of the regular saddle-node lines, it
runs very close to the homoclinic bifurcation, parallel to the
upper saddle-node line.
The zoom on the right side in Fig. 8 further reveals that
the ecologically less relevant details of the phase diagram
are also different from the two-species case. For instance,
the system shows a zero-Hopf bifurcation (the Hopf line
becomes tangent to a saddle-node line), and the Bogdanov–
Takens point, where the Hopf bifurcation line ends, is now
on the lower branch of the saddle-node bifurcations.
Intermediate values of KP 2
The complexity of the system can be appreciated when
considering the Hopf bifurcation line at intermediate values
122
Theor Ecol (2014) 7:115–125
top
predator
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
4
3
0
1
2
prey
3
1.5
3
5
60
0.5
.
2
3.5
2.5
1
Fig. 7 Tritrophic food chain with density-dependent predator mortality for high carrying capacity of the top predator KP 2 = 40. Top
left, phase diagram. The lines are from left to right: dotted gray,
transcritical bifurcation; solid black, Hopf bifurcation; several dotted
black, period doubling bifurcations. Top right, bifurcation diagram
for KP 1 = 17.5; bottom left, chaotic attractor for KN = 5.7 and
KP 1 = 17.5. Bottom right, limit cycle that alternates between an oscillation of the resource and intermediate predator and an oscillation of
the intermediate predator and top predator. Parameters are KN = 30
and KP 1 = 17.5
Fig. 8 Phase diagram for the tritrophic food chain with densitydependent predator mortality, for KP 2 = 10. The second graph shows
a zoom into the region close to the cusp bifurcation (CP), where the
two dashed gray saddle-node bifurcation lines end. The ZH marks
a zero-Hopf bifurcation (where the solid black Hopf bifurcation line
becomes tangent to a saddle-node bifurcation line). BT marks the
Bogdanov–Takens point, from which a solid gray homoclinic bifurcation line spawns. As can be seen in the first picture, it approaches and
eventually runs parallel to the dashed black saddle-node of limit cycle
line spawned at a generalized Hopf bifurcation GH
Theor Ecol (2014) 7:115–125
of KP 2 in the KN –KP 1 plane, as shown in Fig. 9. The
upper branch of the Hopf bifurcation line, together with
the saddle-node lines, resembles the picture obtained for
smaller KP 2 . Here, the enrichment KN and the carrying
capacity for the first predator, KP 1 , are large enough, so
that the population size of the second predator can increase
to values where the density limitation is strongly felt. On
the other hand, when the enrichment KN is small (left
vertical branch of the Hopf bifurcation line in Fig. 9), the
predators are limited by food. So the density limitations of
the predators are not felt, and a supercritical Hopf bifurcation followed by chaos occurs with increasing KN , as in the
Rosenzweig–MacArthur model and at high values of KP 2 .
The lower horizontal part of the Hopf bifurcation line can
be understood in the same way as in the case of high KP 2 :
The first predator eats almost at the maximum rate, with
the resource being close to its carrying capacity. The first
and second predators together behave like the two-species
system, with a Hopf bifurcation occurring as KP 1 increases.
Between the two main branches, the Hopf bifurcation
line has a narrow and high peak, where it changes from
super- to subcritical. Note that the limit cycles spawned
when crossing this peak usually correspond to each other
and do not interact with the one spawned when crossing the
left branch of the Hopf line. This can be seen in Fig. 10.
Between this peak and the left Hopf branch, there is a complex chaotic region, which, in addition to period doublings,
123
Fig. 10 Bifurcation diagram for the tritrophic food chain with densitydependent predator mortality, with KP 1 = 30 and KP 2 = 18. Shown
are the fixed point values (black) of the intermediate predator biomass
and maxima and minima for limit cycles (gray). Solid lines indicate
stable fixed points or limit cycles, while dashed lines, unstable ones.
Around KN = 7, two Hopf bifurcations (H) occur in quick succession.
This stabilizes an otherwise unstable fixed point in a small parameter
interval. A letter D indicates a period-doubling bifurcation. Between
these bifurcations, a period-doubling cascade occurs, leading to chaos
(not shown). The letter L marks a saddle-node bifurcation of limit
cycles
also includes various saddle-node bifurcations of the limit
cycles both from the left branch and from the peak.
As previously mentioned, many other transitional states
between the case of high KP 2 and low KP 2 exist, some
including two separate Hopf lines and three Bogdanov–
Takens bifurcations. There is much dynamical complexity
left to explore; however, we think that the cases that we discussed give a good impression of the important features that
are relevant for understanding and explaining ecological
phenomena in this type of system.
Finally, we mention that the system can also show a
trans-critical bifurcation of a limit cycle, where the amplitude of the oscillation of the top predator and its population
size decrease to zero, and the top predator cannot survive
beyond the bifurcation.
Conclusions
Fig. 9 Phase diagram for the tritrophic food chain with densitydependent predator mortality, for KP 2 = 18. The solid black Hopf
line now takes a turn, eventually changing from super- to subcritical
at the topmost GH point (the dashed black saddle-node bifurcations
of limit cycles of the other four GH points run so close to the Hopf
line that the net effect is negligible). In the chaotic region between the
peak of the Hopf line and the leftmost part of it, there are many saddlenode bifurcations of limit cycles (dashed black) and period doublings
(dotted black). The right part of the phase diagram is very similar to
the KP 2 = 10 case
We investigated the influence of competition between
predators on the dynamics of di- and tritrophic food chains.
Competition between predators was taken into account
either as predator interference or as a density-dependent
mortality rate. These two cases correspond to situations
where population growth is either limited by competition
for food or by other factors such as parasites or epidemics,
124
nesting sites, or shelter. We found that these two cases lead
to very different types of dynamical behavior.
For ditrophic systems, part of the investigation could
be done by analytical means; otherwise, we used computer simulations and the continuation software “AUTO”
(Doedel et al. 2007). We mention again that a full analysis of the dynamics of these models is beyond the
scope of this paper, and that the included bifurcation diagrams should not be considered to be complete. Additional
bifurcations likely exist, and changes in parameters that
were fixed may strongly affect the dynamics. The results
were compared to each other and to the Rosenzweig–
MacArthur model, which does not include (direct) predator
competition.
Below a critical value of KN (the enrichment), the top
predator dies out. Competition between predators does not
have any influence if the predator biomasses are small. All
models show the Hopf bifurcations with increasing KN ,
resulting in a stable limit cycle afterward. In the threedimensional case, a further increase of KN typically leads
to a chaotic attractor if intraspecific competition of neither
predator is too large, and these models also show the coexistence of fixed points and limit cycles, and of limit cycles and
chaos. In all models, the paradox of enrichment is found. In
the two-dimensional as well as in three-dimensional models,
this “paradox” can be suppressed with higher competition,
and then the fixed point remains stable for all KN .
There is a fundamental difference between the dynamics
of systems with interference competition and systems with
a density-dependent predator mortality. While for interference competition fixed points always become unstable
via Hopf bifurcations, the fixed points of models with
density-dependent predator mortality also undergo saddlenode bifurcations. Even though the Hopf bifurcation in the
systems with interference competition can become subcritical in the three-dimensional model, the limit cycle to which
the dynamics jump after the bifurcation encloses the fixed
point. When the system with density-dependent predator
mortality goes trough a saddle-node bifurcation, the dynamics can go to a limit cycle that lies outside the previous
fixed point and has a large-amplitude oscillation where the
population densities can come close to the extinction threshold. In contrast to the three-dimensional system, the twodimensional system with interference competition shows no
catastrophic transitions at all, but only a supercritical Hopf
bifurcation. This is very different from the two-dimensional
system with density-dependent predator mortality, which
shows several types of catastrophic transitions, which are
due to saddle-node bifurcations of fixed points and limit
cycles, infinite-period bifurcations, and transcritical bifurcations. The most prominent catastrophic bifurcation in this
system is an infinite-period bifurcation. Before the bifurcation, at the stable fixed point, the predator density is limited
Theor Ecol (2014) 7:115–125
by factors other than prey. At the infinite-period bifurcation, the density limitation is no longer strong enough
to prevent the predator from chasing the prey through a
predator–prey oscillation cycle. In the tritrophic food chain,
this scenario is slightly modified, as the saddle-node bifurcation of fixed points does not occur on the limit cycle, but
in its neighborhood.
Catastrophic scenarios such as those described in the
models with density-dependent predator mortality can occur
if the predator is released from control by a parasite or parasitoid. For example, a strong reduction or even extinction of
parasitoids following inappropriate use of insecticides has
been suggested to initiate pest outbreaks in tropical plantation crops (Godfray and Hassel 1989). The catastrophic
transition could, in principle, also be caused by conservation
measures. If the predator density is limited by a lack of safe
nesting sites or shelter instead of the availability of food,
increasing the overall habitat quality could also increase the
carrying capacity of the predator up to the point where the
prey population suddenly collapses. Similarly, a reduction
of the extensive use of agricultural fertilizers could drive
the catastrophe by derichment of the environment, i.e., by
decreasing the basal species’ carrying capacity.
The second important conclusion from our work is that
both types of competition alleviate the paradox of enrichment by moving the Hopf bifurcation to higher values of
the carrying capacity of the resource. However, for the
tritrophic food chain, this can simultaneously lead to a catastrophic version of the paradox, because the Hopf bifurcation
may become subcritical. Even though the oscillation is prevented at first as the enrichment increases, the onset of the
oscillation can occur in the form of a sudden jump to a
large-amplitude oscillation.
Acknowledgments This project was supported by the German
Research Foundation under contract number Dr300/10-1. We thank
Wolfram Just for useful discussions and the referees for useful comments.
References
Abrams PA (1997) Anomalous predictions of ratio-dependent models
of predation. Oikos 80:163–171
Arditi R, Ginzburg LR (1989) Coupling in predator-prey dynamics:
ratio dependence. J Theor Biol 139:311–326
Arditi R, Callois J-M, Tyutyunov Y, Jost C (2004) Does mutual interference always stabilize predator-prey dynamics? A comparison of
models. C R Biolog 327:1037–1057
Bazykin A (1998) Nonlinear dynamics of interacting populations.
World Scientific, River Edge
Bazykin AD, Berezovskaya FS, Denisov GA, Kusnetsov YuA (1981)
the influence of predator saturation effects and competition among
predators on predator-prey system dynamics. Ecol Model 14:39–
57
Theor Ecol (2014) 7:115–125
Beddington JR (1975) Mutual interference between parasites or predators and its effect on searching efficiency. J Anim Ecol 44:331–
340
De Angelis DL, Goldstein RA, O’Neill RV (1975) A model for trophic
interaction. Ecol 56(4):881–892
Doedel et al. (2007) AUTO—software for continuation and bifurcation problems in ordinary differential equations http://indy.cs.
concordia.ca/auto/. Accessed 5 July 2013
Fussmann GF, Blasius B (2005) Community response to enrichment is highly sensitive to model structure. Biol Lett 1:9–
12
Fussmann GF, Ellner SP, Shertzer KW, Hairston NG Jr (2000) Crossing the Hopf bifurcation in a live predator-prey system. Sci
290:1358-1360
Geritz S, Gyllenberg M (2012) A mechanistic derivation of
the DeAngelis–Beddington functional response. J Theor Biol
314:106–108
Godfray HCJ, Hassel MP (1989) Discrete and continuous insect
populations in tropical environments. J Anim Ecol 58:153–
174
Gross T, Ebenhöh W, Feudel U (2005) Long food chains are in general
chaotic. Oikos 109:135–144
Hastings A, Powell T (1991) Chaos in a three-species food chain. Ecol
72(3):896–903
Hogeweg P, Hesper B (1978) Interactive instruction on population
interactions. Comp Biol Med 8:319–327
Holling CS (1959) Some characteristics of simple types of predation
and parasitism. Can Entomol 91(7):385–398
Kusnetsov YuA (2004) Elements of applied bifurcation theory, 3rd
edn. Springer, New York
Kusnetsov YuA, de Feo O, Rinaldi S (2001) Belyakov homoclinic
bifurcation in a tritrophic food chain model. SIAM J Appl Math
62:462–487
125
Li B, Kuang Y (2007) Heteroclinic bifurcation in the Michaelis–
Menton-type ratio-dependent predator-prey system. SIAM J Appl
Math 67:1453–1464
Lotka AJ (1925) Elements of physical biology. Williams & Wilkins
company, Baltimore
May RM (1972) Limit cycles in predator-prey communities. Science
177:900–902
McCauley E, Murdoch WW (1990) Predator-prey dynamics in environments rich and poor in nutrients. Nature 343:455–457
Rall BC, Guill C, Brose U (2008) Food-web connectance and predator
interference dampen the paradox of enrichment. Oikos 117:202–
213
Rosenzweig ML (1971) Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science
171(3969):385–387
Rosenzweig ML, MacArthur RH (1963) Graphical representation and
stability conditions of predator-prey interactions. Am Nat 97:209–
225
Skalski GT, Gilliam JF (2001) Functional responses with predator
interference: viable alternatives to the Holling type II model.
Ecology 82:3083–3092
Steele JH, Henderson EW (1981) A simple plankton model. Am Nat
117:676–691
van Voorn GAK, Stiefs D, Gross T, Kooi BW, Feudel U, Kooijman
SA (2008) Stabilization due to predator interference: comparison of different analysis approaches. Math Biosci Eng 5:567–
83
van Voorn GAK, Kooi BW, Boer MP (2010) Ecological consequences
of global bifurcations in some food chain models. Math Biosci
226:120–133
Volterra V (1926) Variations and fluctuations of the number of individuals in animal species living together. ICES J Mar Sci 3:3–51.
Reprinted and translated into English (1928)