# Chaos in Ecology: The Topological Entropy of a Tritrophic Food Chain Model

Authors: Jorge Duarte; Cristina Januário; Nuno Martins
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

An ecosystem is a web of complex interactions among species.
With the purpose of understanding this complexity, it is necessary to study basic food chain dynamics with preys, predators and superpredators interactions. Although there is an elegant interpretation of ecological models in terms of chaos theory, the complex behavior of chaotic food chain systems is not completely understood. In the present work we study a specific food chain model from the literature. Using results from symbolic dynamics,
we characterize the topological entropy of a family of logistic-like Poincaré return maps that replicates salient aspects of the dynamics of the model. The analysis of the variation of this numerical invariant, in some realistic system
parameter region, allows us to quantify and to distinguish different chaotic regimes. This work is still another illustration of the role that the theory of dynamical systems can play in the study of chaotic dynamics in life sciences.


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## Body

## 1. Motivation and Preliminaries

The  detailed
examination  of food   chain dynamics is crucial to the study of important
ecological systems. Basic food chains can be thought as fundamental “building
blocks” of an ecosystem. The chaos theory, which has always intertwined
with complex population dynamics since its inception,
is accepted as an important part of a paradigm by which ecological complexity
can be understood.A compelling reason to studyecochaos, beside
the challenging mathematical problems that occur in this context, is due to a
recent discovery (which may have important management implications) that the
average superpredator biomass in various tritrophic food chain models is
maximum at the onset of chaos (see [1, 2],
and references therein).The main dynamical features of the well-known
Lotka-Volterra models for two species interactions, which are spatially
homogeneous and time autonomous, are steady states and limiting cycles.
However, models for tritrophic food chains are extremely rich in complex dynamics.
The strong nonlinearity and high dimensionality of the phase  space are a significant obstacles in
understanding the qualitative behavior of such systems.The dynamical principles and mechanisms underlying the
chaotic food chain behavior can be analyzed with comprehensive studies of
low-dimensional systems, which emerge from the food chain models. Indeed, we
can gain some important qualitative insights by studying representative return
maps, considered as one-dimensional interval maps.The aim of this work is to provide a contribution for
the detailed analysis of the chaotic behavior of the time-diversified
Rosenzweig-MacArthur tritrophic model, which is cast as a singularly perturbed
system. More precisely, using results of symbolic dynamics theory, we compute
the topological entropy of a family of logistic-like Poincaré return maps (with
the shape of a unimodal map) presented in [4], which incorporates fundamental dynamical properties
of the three-dimensional attractor. This measure of the amount of chaos in a
dynamical system is the most important numerical invariant related to the orbit
growth and its variation with particular parameters gives us a finer
distinction between states of complexity.For the sake of clarity, the next paragraph begins
with an overview of the food web model considered (the reader is referred to
[4]).


## 2. Description of the Model

The
Rosenzweig-MacArthur model for tritrophic food chains is given by the following
differential system:(2.1)X.=X(r−rXK−p1YH1+X),Y.=Y(c1XH1+X−d1−p2ZH2+Y),Z.=Z(c2YH2+Y−d2).It is composed of a logistic prey X, a predator Y, and a top predator Z. The model assumes predator/top
predator responses that reflect the more realistic situations in the
saturation capture rate of prey by predator/predator by top predator.In order to simplify the mathematical analysis, it is
appropriate to nondimensionalize the previous equations (2.1) so that the scaled system contains a minimal
number of parameters. We use the same scaling of variables and parameters as in
[4],(2.2)t→c1T,x→1KX,y→p1rKY,z→p2p1c1rKZ,ζ=c1r,ϵ=c2c1,β1=H1K,β2=H2Y0,withY0=rKp1,δ1=d1c1,δ2=d2c2. This procedure leads to the following dimensionless
form (2.3)ζx.=x(1−x−yβ1+x),y.=y(xβ1+x−δ1−zβ2+y),z.=ϵz(yβ2+y−δ2).In
the context of ecology, the parameters have the following definitions:(i)
ζ=c1/r,
where c1 is the maximum per capita growth rate of the
predator and r is the maximum per capita growth rate for the
prey;(ii)
ϵ=c2/c1,
where c2 is the maximum per capita growth rate of the
top predator and c1 is the maximum per capita growth rate for the
predator;(iii)
β1 is the dimensionless semisaturation constant
measured against prey's carrying capacity;(iv)
β2 is the dimensionless semisaturation constant
for the predator, measured against its predation capacity;(v)
δ1=d1/c1,
where d1 is the per capita natural death rate of the
predator and c1 is the maximum per capita growth rate for the
predator;(vi)
δ2=d2/c2,
where d2 is the per capita natural death rate of the
top predator and c2 is the maximum per capita growth rate for the
top predator.As in [4], we assume throughout
that(2.4)0<β1<1,0<β2<1,which can be interpreted to mean
that both the predator and the superpredator are good hunters. We also assume
throughout that(2.5)0<δ1=d1c1<1,0<δ2=d2c2<1.This is in fact a default
assumption because the condition of either δi>1  (i=1,2) would lead to
the collapse of the tritrophic food chain. More specifically,
with d1>c1,
the predator dies out faster than it can reproduce even at its maximum
reproduction rate. With d2>c2,
the top predator must die out by the same reasoning. In both cases, we would
have a trivial tritrophic food chain, whose dynamics is entirely understood.Under the “trophic time diversification
hypothesis,” which states thatthe maximum per capita growth rate
decreases from bottom to top along the food chain, namely,(2.6)r>c1>c2>0or equivalently(2.7)0<ζ≪1,0<ϵ<1.Equation (2.3) become a singularly perturbed system of
three-time  scales. The rates of change for the prey, predator, and top predator
are fast, intermediate, and slow, respectively. From the ecological point of
view, it means that the prey reproduces faster than the predator which in turn
reproduces faster than the top predator.


## 3. Unimodal Return Maps. Symbolic Dynamics, Topological Entropy, and Chaos

As we mentioned before, this paper aims to address a
study of the topological entropy of a family of logistic-like Poincaré return
maps associated to the model (2.3). The existence of these maps, for some
system realistic parameter region (see region Ω in Figure 8 represented below), was
demonstrated in [4]
using a geometric method of singular perturbations, which have proved to be
extremely effective for ecological models (see [4–7]).For numerical investigation, we will use
throughout(3.1)ζ=0.1,β1=0.3,β2=0.1,δ1=0.2,and consider ϵ and δ2 as control parameters. As we saw earlier, the
parameter ϵ is the reproduction rate ratio of the top
predator over the predator and the parameter δ2 is the ratio of the per capita natural death
rate of the top predator over its reproduction rate.


## 3.1. Return Maps

Using numerical integration of the system (2.3), we can gain some insights about the
geometry of the trajectories in the long run. After an initial transient, a
structure emerges when the solution (x(t),y(t),z(t)) is visualized as a trajectory in
three-dimensional space (see Figure 1). The projection of the three-dimensional
trajectory onto a two-dimensional plane is exhibited in Figure 2.Figure 1
Solution visualized as a trajectory in the
three-dimensional space forϵ=0.4 and δ2=0.65.Figure 2
Projection of the three-dimensional trajectory onto
thezy-plane for ϵ=0.4 and δ2=0.65.With the purpose of understanding the main features of
the three-dimensional flow, we can use the Poincaré map technique to reduce the
dimensionality of the phase space and so make the analysis simpler. Now we
briefly describe the construction of a Poincaré map.Consider ann-dimensional system dx/dt=f(x).
Let P be an (n−1)-dimensional surface, called a Poincaré
section. P is required to be transverse to the flow. The
Poincaré map F is a map from P to itself, obtained by following trajectories
from one intersection with P to the next. If xn∈P denotes the nth intersection, then the
Poincaré map is defined by xn+1=F(xn).
In our particular case, we have a system with three dynamical variables and we
consider a Poincaré plane of the form y=k(k∈ℝ),
namely, Γ1:y=0.2 (see Figure 2). After allowing the initials to
decay, we record the successive intersections of the trajectory with the plane,
which are specified by two coordinates xn and zn.
The logistic-like iterated map of Figure 3 consists of pairs (zn,zn+1),
obtained from the successive second coordinates of the points defined by the
Poincaré map. The obtained iterated map dynamically behaves like a unimodal map
(family of continuous maps on the interval with two monotonic subintervals and
one turning point).Figure 3
The iterated map forϵ=0.4 and δ2=0.65.In order to see  the long term behavior for different
values of the parameters at once, we plot typical bifurcation diagrams. The
dynamics bifurcate into stable equilibria with the
increase in the parametersϵ and δ2 (see Figures 4 and 5).Figure 4
Bifurcation diagram forzn as a function of ε,
with δ2=0.662 and ϵ∈[0.36,1.0[.Figure 5
Bifurcation diagram forzn as a function of δ2,
with ϵ=0.5 and δ2∈[0.585,0.686[.At this point, we are in a
position to devote our attention to the study of the topological entropy of the
logistic-like return maps using results of symbolic dynamics theory.


## 3.2. Symbolic Dynamics, Topological Entropy, and Chaos

In this paragraph, we describe techniques of symbolic
dynamics, in particular some results concerning to Markov partitions associated
to unimodal maps. For more details see [8–10].A unimodal mapf on the interval I=[a,b] is a 2-piecewise monotone map with one critical
point c.
Thus I is subdivided into the following
sets:(3.2)IL=[a,c[,IC={c},IR=]c,b],in such way that the restriction
of f to interval IL is strictly increasing and the restriction of f to interval IR is decreasing (see Figure 3). Each
of such maximal intervals on which the function f is monotone is called a lap of f,
and the number ℓ=ℓ(f) of distinct laps is called the lap number of f.
Beginning with the critical point of f, c (relative extremum), we obtain the
orbit(3.3)O(c)={xi:xi=fi(c),i∈ℕ}.With  the purpose of studying the
topological properties, we associate to the orbit O(c) a sequence of symbols, itinerary (i(x))j=S=S1S2…Sj…,
where Sj∈𝒜={L,C,R} and(3.4)Sj=L,iffj(x)<c,Sj=C,iffj(x)=c,Sj=R,iffj(x)>c.The turning point c plays an important role. The dynamics of the
interval is characterized by the symbolic sequence associated to the critical
point orbit. When O(c) is a k-periodic orbit, we obtain a sequence of
symbols that can be characterized by a block of length k,
the kneading sequence S(k)=S1S2…Sk−1C.We introduce, in the set of symbols, an order relationL<C<R.
The order of the symbols is extended to the symbolic sequences. Thus, for two
of such sequences P and Q in 𝒜ℕ,
let i be such that Pi≠Qi and Pj=Qj for j<i.
Considering the R-parity of a sequence, meaning odd or even
number of occurrence of a symbol R in the sequence, if the R-parity of the block P1…Pi−1=Q1…Qi−1 is even, we say that P<Q, if Pi<Qi.
And if the R-parity of the same block is odd, we say that P<Q, if Pi>Qi.
If no such index i exists, then P=Q.The ordered sequence of elementsxi of O(c) determines a partition 𝒫(k−1) of the interval I=[f2(c),f(c)]=[x2,x1] into a finite number of subintervals labeled
by I1,I2,…,Ik−1.  To this partition, we associate a (k−1)×(k−1) transition matrix M=[aij] with(3.5)aij={1ifIj⊂f(Ii),0ifIj⊈f(Ii).Now we consider the topological entropy. As we pointed
out before, this important numerical invariant is related to the orbit growth
and allows us to quantify the complexity of the phenomenon. It represents the
exponential growth rate for the number of orbit segments distinguishable with
arbitrarily fine but finite precision. In a sense, the topological entropy describes
in a suggestive way the total exponential complexity of the orbit structure
with a single number.A definition of chaos in the context of
one-dimensional dynamical systems states that a dynamical system is called
chaotic if its topological entropy is positive. Thus, the topological entropy
can be computed to express whether a map has chaotic behavior, as we can see in
[11, 12]. In these references,
Glasner and Weiss, in a discussion of Devaney's definition (of chaos), proposed
positive entropy as a strong property for the characterization of complex
dynamical systems, more precisely, as the essential criterium of chaos.
Important results were constructed using this property (see [13, 14]).The topological entropy of a unimodal interval mapf,
denoted by htop(f),
is given by(3.6)htop(f)=logλmax(M(f))=logs(f),where λmax(M(f)) is the spectral radius of the transition
matrix M(f) and s(f) is the growth rate,(3.7)s(f)=limk→∞ℓ(fk)k,of the lap number of fk (kth-iterate of f) (see [10, 15, 16]). In summary, for each value of the parameter, the
computation begins with the symbolic codification of the critical point orbits
which determines a Markov partition of the interval. Then, we compute the
transition matrix induced by the interval map on the Markov partition. Finally,
the topological entropy is given by the logarithm of the highest eigenvalue of
this transition matrix.In order to illustrate the outlined formalism about
the computation of the topological entropy, we discuss the following
example.Example 3.1.
Let us consider the map of Figure3. The orbit of the turning point defines the period-6 kneading sequence (RLLLLC)∞. Putting the orbital points in order we
obtain(3.8)x2<x3<x4<x5<x0<x1.The corresponding transition
matrix is(3.9)M(f)=[0100000100000100000111111],which has the characteristic
polynomial(3.10)p(λ)=det(M(f)−λI)=1+λ+λ2+λ3+λ4−λ5.The growth number s(f) (the spectral radius of matrix M(f)) is 1.96595….
Therefore, the value of the topological entropy can be given by(3.11)htop(f)=logs(f)=0.675975….Several situations of the variation of the topological
entropy with each of the parameters are plotted in Figures6 and 7. We
emphasize that the logistic-like family of Poincaré return maps associated to
the Rosenzweig-MacArthur model for tritrophic chains exhibit positive
topological entropy, which is a signature of its chaotic behavior. Indeed, the
consideration of a Poincaré section of the type y=k led us to identify a large region of the
parameter space, associated to logistic-like maps, where chaos occurs.Figure 6
Variation of the topological entropy forϵ∈[0.36,1.0[, with δ2=0.662.Figure 7
Variation of the topological entropy forδ2∈[0.585,0.686], with ϵ=0.5.Figure 8
Periodic orbits(n≤5) of the turning point C in the Ω region. From right to left, the corresponding
kneading sequences are as follows: C∞, (RC)∞, (RLRC)∞, (RLRRC)∞, (RLC)∞, (RLLRC)∞, (RLLC)∞,
and (RLLLC)∞.
The parameter values in the grey region do not correspond to unimodal maps and
are beyond the scope of our study.It is interesting to notice that with the study of the
kneading sequences it is possible to represent the curves, in the regionΩ of the parameter space, corresponding to the
periodic orbits of the turning point C.
The diagram of Figure 8 shows how the periods (n≤5) are organized throughout the region Ω of the parameter space considered (whose pairs
of values (δ2,ϵ) correspond to logistic-like Poincaré return
maps). From right to left in Figure 8, the corresponding kneading orbits are as
follows: 1-period—C∞,
2-period—(RC)∞,
4-period—(RLRC)∞,
5-period—(RLRRC)∞,
3-period—(RLC)∞,
5-period—(RLLRC)∞,
4-period—(RLLC)∞,
and 5-period—(RLLLC)∞.
The parameter space ordering of the kneading sequences leads to the
identification of different levels for the topological entropy, which remains
constant over each curve. Table 1 represents some kneading
sequences and the corresponding topological entropy.This is an example of how our
understanding of the parameter space can be enhanced by the techniques of symbolic
dynamics.Table 1Kneading sequencesCharacteristic polynomialTopological entropyRC1−t0RLRC−1+t+t2−t30RLRRC−1+t−t2−t3+t40.414013…RLC−1−t+t20.481212…RLLRC1−t−t2−t3+t40.543535…RLLC1+t+t2−t30.609378…RLLLC−1−t−t2−t3+t40.656256…


## 4. Final Considerations

In this paper, we have provided new insights into the
study of a very well-known model in the field of theoretical ecology: the
Rosenzweig-MacArthur model for tritrophic food chains. In the available
literature, the detailed examination of this remarkable model involves many
different issues, both in the biological and mathematical domains, and its
scientific investigation is still an active research area. As pointed out in the
introduction, a plausible and compelling motive to study chaos in ecology is
due to the principle (which may have a direct biological relevance and
significant management implications) that the average top predator biomass, in
various tritrophic food chain models, is maximum at the onset of chaos. From
the point of view of mathematics, chaos in food chain models is not clearly
understood. Indeed, the first theorem on the existence of chaotic food chain
dynamics was recently obtained in [4]. The characterization and analysis of chaos generating
mechanisms for the Rosenzweig-MacArthur tritrophic food  chain model
are one of the recent significant achievements in
mathematics applied to ecology (see [4–6, 17]). These
studies proved the existence of different one-dimensional Poincaré return maps
generated by the model and left open a collection of questions pertaining to
chaotic attractors in terms of symbolic dynamics and various measurements of
complexity (see [5]).
In this context, the work carried out in the present article addresses a
contribution for the comprehensive mathematical study of the chaotic behavior
associated to a specific class of one-dimensional return maps (see [18], for the analysis of a
different class of maps).In the field of life sciences, where quantitatively
predictive theories are rare, the use of powerful tools for the analysis of
dynamic models, such as the symbolic dynamics theory, stands out to be
extremely effective for the computation of an important numerical invariant
related to the exponential orbit growth—the topological entropy. In fact, a
rigorous study of the iterated maps, that incorporate the salient dynamical
properties of the system, became possible analyzing the variation of this
measure of complexity with the two control parametersϵ and δ2.
Our analysis reveals that when the reproduction rate of the top predator and
the reproduction rate of the predator become closer (which means an increase in ϵ), the topological entropy decreases. In a
similar way, when the per capita natural death rate of the top predator becomes
closer to its reproduction rate (which means an increase in δ2), the topological entropy also decreases.
Therefore, high values of these control parameters tend to stabilize the food
chain. To each value of these control parameters corresponds a value of the
topological entropy which is a quantifier for the complex orbit structure and
an attribute efficiently used to identify different chaotic states.The representation of the isentropic curves
(corresponding to the periodic orbits of the turning pointc) in the region Ω allowed us to introduce the parameter space
ordering of the dynamics. In fact, this construction gives insights about the
behavior of the topological entropy in all the parameter space considered.Indeed, the family of maps associated to the model
exhibits positive topological entropy, which demonstrates its chaotic nature.
The techniques of symbolic dynamics allowed us to quantify the orbit complexity
and to distinguish different chaotic regimes (extracting order from chaos) in a
significant region of the parameter space.



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