Exploring dynamical systems and chaos using the logistic map model of population change

Prévia do material em texto

Exploring dynamical systems and chaos using the logistic map model
of population change
Jeffrey R. Groffa)
Shepherd University, Institute of Environmental and Physical Sciences, Shepherdstown, West Virginia 25443
(Received 31 August 2012; accepted 22 June 2013)
The logistic map difference equation is encountered in the theoretical ecology literature as a
mathematical model of population change for organisms with non-overlapping generations and
density-dependent dynamics influenced solely by intraspecific interactions. This article presents the
logistic map as a simple model suitable for introducing students to the properties of dynamical
systems including periodic orbits, bifurcations, and deterministic chaos. After a brief historical and
mathematical introduction to models of population change and the logistic map, the article
summarizes the logistic map activities I teach in my introductory physics laboratories for non-
physics majors. The logistic map laboratory introduces the many bioscience students in my courses
to a foundational model in population ecology that has inspired ecologists to recognize the
importance of nonlinear dynamics in real populations. Although I use this activity in courses for
non-majors, the logistic map model of population change could also be taught to physics majors to
introduce properties of dynamical systems while demonstrating an application of mathematical
modeling outside of traditional physics.VC 2013 American Association of Physics Teachers.
[http://dx.doi.org/10.1119/1.4813114]
I. INTRODUCTION
The use of differential equation-based models of dynami-
cal systems is a cornerstone of Newtonian mechanics. In in-
troductory physics, students use differential equations
including equations of motion and the wave equation. But
the complicated behaviors of dynamical systems composed
of coupled differential equations evolving with different
time constants, such as periodic orbits, bifurcations, and the
emergence of deterministic chaos, are not typically taught to
these students, who are learning Newtonian mechanics for
the first time. These advanced topics are largely beyond the
scope of the introductory course and are reserved for more
advanced courses in classical mechanics, biophysics, or
mathematical modeling, intended for physics or mathematics
majors only.
However, these interesting topics are accessible to intro-
ductory students, even in algebra-based courses, if presented
using difference equations instead of differential equations.
In addition, exposure to these topics could prove valuable to
the large number of bioscience students who, at many insti-
tutions, are a significant fraction of the students enrolled in
introductory physics. Biology students may wrongly assume
that mathematical models of dynamical systems are largely
irrelevant to their discipline; in reality such models, both dif-
ferential and difference-equation-based, are important tools
for biologists. For example, the field of neuroscience is
founded on the work of biophysicists Hodgkin and Huxley,
who formulated a differential equation-based model of
action potential generation in 1952, earning them a Nobel
Prize.1 At the same time, the field of theoretical population
ecology relies heavily on both differential and difference
equations.2–5
This article focuses on introducing physics students to dy-
namical systems and the fascinating concepts of periodic orbits,
bifurcations, and deterministic chaos through experimentation
with a difference equation known as the logistic map. The
logistic map is widely encountered in the population ecology
literature as a model of population change for species with
non-overlapping generations and density-dependent dynamics
influenced solely by intraspecific interactions.6–8 After a brief
historical and mathematical introduction to models of popula-
tion change and the logistic map, I summarize the logistic map
laboratory activities that I perform with my introductory
physics students. This summary is followed by a discussion of
real population dynamics with examples of logistic-map-like
dynamics and a discussion of the pedagogical importance of
teaching the logistic map as a foundational model in theoretical
ecology.
II. CLASSIC MODELS OF POPULATION CHANGE
AND THE LOGISTIC MAP
The father of population modeling, Thomas Malthus, pub-
lished his seminal work, “An Essay on the Principle of
Populations,” in 1798.9 In this essay, which is more a cri-
tique of the morality and sustainability of 18th century urban
society than a systematic study of population modeling,
Malthus argues that when resources are abundant, human
populations tend to grow in geometric fashion; that is, popu-
lations have a fixed doubling time. While Malthus doesn’t
discuss differential equations in his work, his statement
describes the behavior of the classic exponential growth
model that often bears his name. In differential equation
form, the Malthusian growth model is
dN
dt
¼ rN; (1)
where N is the number of individuals in the population and r
is the Malthusian growth rate. This rate should be thought of
as a combination of the birth and death rates of a population:
r¼ b – d where b is the birth rate and d is the death rate. For
r > 0 (that is, b > d), N grows exponentially without bound;
for r < 0 (that is, b < d), the population asymptotically
approaches zero representing extinction.
Malthus hoped to convince his readers that unrestrained
human population growth would put ever-increasing pres-
sure on resources and result in a vicious cycle of despair and
suffering for the poor. However, his assumption of largely
725 Am. J. Phys. 81 (10), October 2013 http://aapt.org/ajp VC 2013 American Association of Physics Teachers 725
unrestrained growth is unrealistic in a world where organisms
(humans included) compete for limited resources. In 1838,
Verhulst modified Malthus’ model to account for the reasona-
ble condition that the growth rate would slow as the popula-
tion increases and approaches the carrying capacity of the
environment.10 While Eq. (1) has just a single steady-state so-
lution, namely the trivial case where N¼ 0 and the population
is extinct, Verhulst’s formulation can be written as
dN
dt
¼ rN 1� N
K
� �
; (2)
which is perhaps one of the simplest models having two
steady-state solutions, N¼ 0 and N¼K, where K is the carry-
ing capacity of the population’s environment. Note that
K < 0 makes no ecological sense and would represent
extinction. Equation (2) has come to be referred to as the
logistic equation and has been successfully used to model
bacteria population growth in laboratory settings.11 It has led
to the practice among ecologists of referring to organisms as
r-strategists (those that focus on high growth rates) versus
K-strategists (those that focus on living close to the carrying
capacity of the environment). The quantity r(1 – N/K) of Eq.
(2) represents a growth rate that decreases linearly with N
and is zero when N¼K.
The logistic map, or logistic difference equation, was pro-
moted by the ecologist Robert May as an example of a sim-
ple ecological model with very complicated dynamics.6,12
The logistic map can be written as
Niþ1 ¼ r^Ni 1� Ni
K
� �
; (3)
where r^ is a dimensionless population growth factor and Ni
is the population of the ith generation. The parameter K is
the largest possible value the population can attain and is of-
ten called the carrying capacity like the K of the logistic
equation, even though in this case the population cannot
remain near K indefinitely. Note that if N is much smaller
than K, then 1 – N/K is approximately 1 and the population
grows in proportion to r^; but if NuK, then the right-hand
side of Eq. (3) is close to zero and the population will dra-
matically decline. While the logistic map [Eq. (3)] resemblesthe logistic equation [Eq. (2)] and has been called a discrete
analog to the logistic equation,4,13 the two models are mathe-
matically very different. For example, notice that r in Eq. (2)
has dimensions of 1/time while r^ in Eq. (3) is dimensionless.
Also, we require 0 � r^ � 4 in order for the logistic map to
remain bounded between 0 and K.
To highlight the differences between these two mathemat-
ical models, Fig. 1 shows representative trajectories gener-
ated with Eq. (2) [Fig. 1(a)] and with Eq. (3) [Fig. 1(b)]. In
both panels, the simulations use K¼ 1000 and an initial pop-
ulation value of Nð0Þ ¼ N0 ¼ 250. The resulting dynamics,
however, can be quite different. Figure 1(a) shows that as the
r used by the logistic equation increases from 0.75 (dashed-
dotted) to 1.75 (dashed) to 2.75 (solid line), the population
grows more rapidly initially, but regardless of the value of r
used the population always reaches a fixed-point steady state
equal to the carrying capacity K. On the other hand, Fig. 1(b)
shows that as the r^ used by the logistic map increases in the
same fashion, the population reaches different fixed-point
steady-state values. In fact, when r^ ¼ 0:75 (dashed-dotted
line), the population goes extinct after about 10 generations.
Extinction can also be observed using the logistic equation if
r is negative. The step-like shape of the trajectories in Fig.
1(b) compared to the smooth trajectories in Fig. 1(a) reflects
the discrete nature of the logistic map.
This discreteness can yield complicated dynamics. For
example, notice the transient damped oscillations before the
steady state is achieved in Fig. 1(b) when r^ ¼ 2:75 (solid
line). In fact, if r^ is sufficiently large, steady-state periodic
orbits of period 2 or greater are observed, as seen in Fig.
2(a). For example, the dashed line with r^ ¼ 3:1 shows a
period-2 solution, while the solid line with r^ ¼ 3:5 shows a
period-4 solution. Further increases in r^ lead to ever more
complicated solutions. Figure 2(b) demonstrates that r^
¼ 3:75 results in deterministic chaos, where even a 0.01%
change in the initial conditions used (dashed line, N0 ¼ 400;
solid line, N0 ¼ 400:04) results in drastically diverging tra-
jectories after about 15 generations.
To highlight the rich dynamics exhibited by the logistic
map, Fig. 3(a) shows the well-known bifurcation plot for Eq.
(3), summarizing the steady-state solutions—the attractors—
of the model as a function of r^ . Figure 3(b) shows detail of
the parameter region exhibiting periodic orbits and chaotic
dynamics. In the periodic regime, the attractor is a set of
two or more values between which the population oscillates.
In the chaotic regime, the system does not asymptotically
Fig. 1. (a) Representative trajectories of the logistic equation (2) showing
the population N as a function of time t with N(0)¼ 250, K¼ 1000, and
r¼ 0.75 (dashed-dotted), 1.75 (dashed), or 2.75 (solid). (b) Representative
trajectories of the logistic map (3) showing the population Ni versus genera-
tion number i with N0 ¼ 250, K¼ 1000, and r^ ¼ 0:75 (dashed-dotted), 1.75
(dashed), or 2.75 (solid).
726 Am. J. Phys., Vol. 81, No. 10, October 2013 Jeffrey R. Groff 726
approach any finite set of values. The black dots show a sub-
set of the seemingly random values visited by the model af-
ter many generations.
“Cobweb” plots like those shown in Fig. 4 are another
way to visualize the dynamics of the logistic map. Unlike the
trajectories of Figs. 1(b) and 2, the generation number is not
plotted. Instead, the plot connects the current value of the
logistic map (Ni, horizontal axis) to the subsequent value
(Niþ1, vertical axis) using a line segment. Note that for each
Ni, the Niþ1 value of the logistic map falls on a parabola
defined by Eq. (3). A second line segment is drawn from the
parabola to the line Niþ1 ¼ Ni to show that the Niþ1 value is
used as Ni to carry out the next iteration of the map. On cob-
web plots, fixed-point solutions are sinks that are sometimes
approached by spirals as shown in Fig. 4(a), which uses the
same parameters as the solid trajectory in Fig. 1(b). Periodic
solutions appear as closed-loop cycles; for example, Fig.
4(b) shows a period-4 oscillation using the parameters of the
solid trajectory in Fig. 2(a). Chaotic solutions appear as
seemingly random cycles that never repeat, visiting a wide
range of values; Fig. 4(c) shows the chaotic solution of the
solid trajectory in Fig. 2(b).
While the properties of Eqs. (2) and (3) are drastically dif-
ferent, a mathematical connection between the two models
can be shown in a variety of ways involving transformation
of variables or redefinition of parameters. Here, I take the
approach of showing that the logistic map is a special case of
a more general recursion relationship similar to the formula
obtained by discretizing the logistic equation. Defining this
new recursion model provides insight into the interpretation
of the parameters of the logistic map and logistic equation
and the ecological situations where each is applicable.
By the definition of the derivative, the logistic equation
can be approximated as
rNðtÞ 1� NðtÞ
K
� �
� Nðtþ DtÞ � NðtÞ
Dt
(4)
if Dt is sufficiently small. Thus
Nðtþ DtÞ � rDtNðtÞ 1� NðtÞ
K
� �
þ NðtÞ; (5)
which, using i ¼ t=Dt to represent the generation number
and switching to subscript notation, can be written as
Niþ1 � rDtNi 1� Ni
K
� �
þ Ni: (6)
Fig. 2. (a) Representative trajectories of the logistic map (3) showing the
population Ni versus generation number i with N0 ¼ 400, K¼ 1000, and r^ ¼
3:1 (dashed) or 3.5 (solid). (b) K¼ 1000, r^ ¼ 3:75, and N0 ¼ 400 (dashed)
or N0 ¼ 400:04 (solid).
Fig. 3. (a) Bifurcation diagram for the logistic map (3) with K¼ 1000. The
parameter r^ is the dimensionless population growth factor. (b) Detail for
3:4 � r^ � 4 showing the region of parameter space where the logistic map
exhibits periodic solutions and deterministic chaos.
727 Am. J. Phys., Vol. 81, No. 10, October 2013 Jeffrey R. Groff 727
Now consider the more general recursion relationship
Niþ1 ¼ rDtNi 1� Ni
K
� �
þ zNi: (7)
Note that in the limit where Dt is small and z¼ 1, this recur-
sion relation is an implementation of Euler’s forward differ-
ence formula applied to Eq. (2) and can be used to
numerically integrate the logistic equation. In the limit where
z¼ 0, this relation is the logistic map with r^ ¼ rDt. A ver-
sion of the logistic map similar to Eq. (7), but with the nota-
ble absence of the parameter z, has been presented by several
authors4,14 but to my knowledge, the inclusion of z has not
been published elsewhere.
Figure 5 summarizes the steady-state solutions of Eq. (7)
as a function of r^ and z. Importantly, Fig. 5 demonstrates
that Eq. (7) can exhibit all of the complicated dynamics of
the logistic map: the white region denotes parameters result-
ing in extinction; dots indicate parameters resulting in stable
fixed-point solutions; shades of gray denote regions with pa-
rameters yielding periodic orbits of period 2 (light gray) or
higher (increasingly dark shades of gray); the region with
horizontal lines indicates parameters resulting in chaos; and
cross-hatching indicates parameters yielding numerical insta-
bility where solutions diverge to positive or negative infinity.
Note that while Eq. (7) models the logistic equation if z¼ 1
and r^ is sufficiently small (Dt is small, r is small, or both Dt
and r are small), this model is capable of generating all of
the complex dynamics of the logistic map when r^ is suffi-
ciently large even if z¼ 1.
Equation (7) gives us insight into the meaning of the pa-
rameters in the logistic equation and the logistic map.
Assume that Dt in Eq. (7) represents the amount of time dur-
ing each generation that new individuals are being added to
the population. Althoughthis “birthing time” may represent
only a fraction of the breeding season, I will refer to this pe-
riod of time as the breeding season for convenience. Then
rð1� Ni=KÞ represents the net new additions (births minus
deaths) to the population per unit of time as a fraction of the
population at the beginning of the breeding season. Thus,
rDtNið1� Ni=KÞ is the net new additions during the breed-
ing season. The parameter z represents the fraction of the
population at the beginning of the breeding season (does not
include net new additions) that survives to the subsequent
breeding season. Thus, zNi is the total number of individuals
that overlap from one generation to the next. Therefore, with
z¼ 1 and Dt small, the logistic equation is suitable for popu-
lations that reproduce nearly continuously with completely
overlapping generations, that is, populations that have no
gap between periods of breeding. Since humans lack well-
defined breeding seasons, humanity’s breeding habits align
with this description. On the other hand, the logistic map
with z¼ 0 is suitable for populations that reproduce season-
ally with no individuals at the beginning of the breeding pe-
riod surviving to the beginning of the subsequent breeding
Fig. 4. “Cobweb” plots showing (a) a steady-state fixed-point solution, (b) a
steady-state period-4 oscillation, and (c) deterministic chaos. Parameters are
the same as the solid trajectories in Fig. 1(b) (fixed-point solution), Fig. 2(a)
(period-4 oscillations), and Fig. 2(b) (chaos).
Fig. 5. The steady-state dynamics of Eq. (7) are summarized as a function of
r^ and z. Dots denote stable fixed-point solutions, increasingly dark shades of
gray denote periodic solutions of increasing period, white denotes extinc-
tion, horizontal lines denotes chaos, and cross-hatching denotes numerical
instability yielding solutions that diverge toward infinity or negative infinity.
728 Am. J. Phys., Vol. 81, No. 10, October 2013 Jeffrey R. Groff 728
period. That is, generations are non-overlapping.14 For this
reason, recursion relations like the logistic map have been
traditionally used to model populations of insects with com-
pletely non-overlapping generations.14,15 Interestingly, the
analysis presented here demonstrates that perhaps Eq. (7) is
a better choice than Eq. (2) or Eq. (3) for modeling most
populations, since most organisms fall between the two
extremes of completely overlapping generations and com-
pletely non-overlapping generations. At the same time, this
analysis demonstrates that the dynamical complexity dis-
played by the discrete logistic map or by the logistic equa-
tion, if integrated using a method such as Euler’s forward
difference method with a sufficiently large time step, is not
simply a numerical artifact. This complexity reflects gaps
between periods of reproduction and generations that do not
completely overlap, which are properties of many real popu-
lations. As John Maynard Smith points out, delayed feedback
provided by discrete breeding periods is a hallmark of sys-
tems that yield complicated dynamics.4
III. EXPLORING THE DYNAMICS OF THE
LOGISTIC MAP
The complicated dynamics exhibited by the logistic map as
r^ is varied, combined with the mathematical simplicity of Eq.
(3), make this mathematical model of population ideally
suited for studying many interesting properties of dynamical
systems in introductory physics. Below I summarize the logis-
tic map laboratory activities that I carry out with both my
algebra- and calculus-based introductory physics classes. To
complete these activities, students use a computer spreadsheet
that I have programmed with formulas and graphs to model
the logistic map. The spreadsheet calculates a trajectory like
those shown in Fig. 2 when students supply a growth rate fac-
tor (r^), a maximum population value (K), and an initial popu-
lation value (N0). In preparation for the activities, I define the
following important concepts from dynamical systems as they
apply to the logistic map model of population change.
1. Initial condition: The initial population value used by the
model, N0.
2. Deterministic: A model in which any future value can be
determined from knowledge of the model equations, pa-
rameters, and initial conditions.
3. Trajectory: A graphical record of how the population
changes from generation to generation, that is, a plot of Ni
as a function of i.
4. Transient dynamics: The short-term behavior of the popu-
lation model at the beginning of a trajectory.
5. Steady state: A stable population state, either fixed or
oscillating, that does not change over time and is reached
after many generations.
6. Fixed point: A steady state that is a fixed, constant popu-
lation value.
7. Periodic orbit: A steady state characterized by a stable os-
cillation in the population, so that the population “visits”
a finite sequence of values repeatedly.
8. Chaotic dynamics: Seemingly random behavior of themodel,
with population values limited within a certain range but not
tending toward any fixed point or periodic orbit, and with
long-term behavior that is sensitive to the initial condition.
9. Bifurcation: An abrupt change in the behavior of the pop-
ulation model resulting from a small change in the value
of the parameter r^ .
In order to gain a deeper understanding of these concepts,
students complete the following activities.
The first activity uses an r^ value less than 1, indicating
more individuals dying each generation than being born.
Students set r^ to 0.75 and, using trajectories to support their
conclusions, describe what happens to the population after
many generations and whether the outcome depends on the
initial conditions. They find that the population goes extinct
and exhibits a fixed-point steady state of zero, and this out-
come does not depend on the initial population value [see
dash-dotted trajectory of Fig. 1(b)].
The second activity asks students to probe the dynamics
when r^ is greater than 1, which, when N is small compared
to K, indicates more individuals being born each generation
than dying. Students set r^ to 1.75 and 2.75 and again
describe how the population changes after many generations.
They find that the population reaches a nonzero fixed-point
steady-state solution indicative of a stable population [see
dashed and solid trajectories of Fig. 1(b)]. The value of this
fixed-point solution increases as r^ increases. Again, the out-
come does not depend on the initial conditions used, but if r^
is sufficiently large then transient damped oscillations are
observed. Mathematically, a steady-state fixed-point out-
come is one in which N1þ1 ¼ N1, where “1” denotes that
many generations have passed, making N1 the steady-state
fixed-point value. I ask students to substitute N1 for N1þ1 in
the logistic map recursion relation, solve for N1, then use
the resulting formula,
N1 ¼ K 1� 1
r^
� �
; (8)
to calculate N1 for r^ ¼ 1:75 and 2.75 and compare these
values to what the spreadsheet yields.
The third activity asks students to study the outcomes of
the model when the growth rate factor is increased further.
Students set r^ to 3.1 and 3.5 and describe the dynamics of
the model after many generations. They find that now the
population exhibits a steady-state periodic orbit of period 2
(for r^ ¼ 3:1) or period 4 (for r^ ¼ 3:5) [see dashed and solid
trajectories of Fig. 2(a)]. Still, the steady-state outcome of
the model is not sensitive to the initial conditions used. I also
challenge the students to find a value for r^ that results in a
period-8 periodic orbit.
The fourth activity involves exploring the consequences
of increasing r^ still further. Students set r^ to 3.75 and find
that now the model exhibits chaotic dynamics [see dashed
and solid trajectories of Fig. 2(b)]. They find that even small
changes in the initialconditions used result in trajectories
that are drastically different after several generations.
Nevertheless, students also find that this chaotic behavior is
completely deterministic. To confirm this, they can run the
model several times using the same parameters and observe
the same trajectory.
Students may also be shown that even in the periodic and
chaotic regimes, the logistic map has fixed-point steady-state
solutions defined by Eq. (8). However, these solutions are
unstable, causing trajectories to diverge from these solutions in
response to small perturbations. This diverging behavior can
be qualitatively described to students in terms of the tendency
of successive iterations of the difference equation to overshoot
the steady-state solution or for small perturbations from the
steady state to grow instead of shrink. This can be contrasted
with the behavior of the differential-equation-based logistic
729 Am. J. Phys., Vol. 81, No. 10, October 2013 Jeffrey R. Groff 729
equation, where solutions always asymptotically approach a
fixed-point steady-state value, or the logistic map near a stable
fixed-point steady-state solution, where successive iterations
also asymptotically approach the steady state.
Quantitatively, a steady-state fixed-point solution to the
logistic map (N1) is stable if an Ni having a small displace-
ment from the steady-state value (@Ni ¼ Ni � N1) is selected
and the displacement gets smaller with successive iterations.
In other words, the N1 is stable if j@Niþ1j < j@Nij. To dis-
cover the situations in which this occurs, let f ðNiÞ ¼ r^Ni
ð1� Ni=KÞ and consider the Taylor series expansion of f ðNiÞ
near N1,
Niþ1 ¼ f ðNiÞ ¼ f ð@Ni þ N1Þ
¼ f ðN1Þ þ f 0ðN1Þð@NiÞ þ � � � ; (9)
where the derivative is with respect to Ni. Assuming @Ni is
sufficiently small to ignore higher-order terms and noting that
f ðN1Þ ¼ N1 and Niþ1 ¼ @Niþ1 þ N1, Eq. (9) reduces to
@Niþ1 ¼ f 0ðN1Þ@Ni: (10)
Therefore, a steady-state fixed-point solution is stable if the
quantity
jf 0ðNiÞj ¼
���� dNiþ1dNi
���� ¼
����r^ 1� 2NiK
� �����; (11)
evaluated at N1, is less than 1. In other words, a steady-state
fixed-point is stable if the slopes of the parabolas plotted
in Figs. 4(a)–4(c) have magnitudes less than 1 at the point
where each intersects the line Niþ1 ¼ Ni. Periodic solutions
or chaotic solutions result when this condition is not met.
On the other hand, for the logistic equation, a fixed-point
steady-state solution (N1) is stable if dN=dt < 0 for small
positive displacements from N1 and dN=dt > 0 for small
negative displacements from N1. If these conditions are sat-
isfied, any small displacement from N1 will return to N1.
Therefore, a steady-state fixed point solution to the logistic
equation is stable if the derivative of the equation’s right-
hand side with respect to N, evaluated at N1, is negative. In
other words, a fixed-point solution is stable if the quantity
f 0ðNÞ ¼ df ðNÞ
dN
; (12)
evaluated at N1, is negative, where f(N)¼ rN(1 – N/K).
Since f 0ðN1Þ ¼ �r, steady-state solutions to the logistic
equation are always stable fixed-point solutions and more
complicated dynamics are never observed.
It is an extremely interesting fact that while the
difference-equation given by Eq. (3) is capable of exhibiting
periodic and chaotic dynamics, the very similar looking dif-
ferential equation given by Eq. (2) never exhibits such dy-
namics. In fact, a dynamical system defined by autonomous
first-order differential equations requires more complexity to
exhibit complicated dynamics. It follows from the Poincar�e-
Bendixson theorem that at least two state variables are
required to exhibit periodic orbits (e.g., the Lotka-Volterra
predator-prey model). Systems exhibiting chaos require at
least three state variables (e.g., the Lorenz System).16
The final activity I ask students to complete involves
the construction of a bifurcation plot for the logistic map
model of population change by systematically changing r^ ,
recording the steady-state value (or values) for each r^ , and
generating a scatter plot of this data. Following the lab, I
show my students more detailed bifurcation plots for the
logistic map similar to those shown in Fig. 3, which I gener-
ate using MATLAB. To my delight, some students genuinely
find the fine structure of the bifurcation plots not only mathe-
matically impressive but also aesthetically pleasing after
completing the laboratory.
IV. COMPLICATED DYNAMICS IN REAL
POPULATIONS
The logistic map model of population change predicts that
species having non-overlapping generations and lacking ex-
trinsic influences—species with density-dependent population
dynamics influenced solely by intraspecific interactions—
may have populations that oscillate or exhibit deterministic
chaos. In fact, all species are subject to extrinsic influences
and interspecific interactions. Nevertheless, some species do
exhibit logistic-map-like dynamics.
For example, an early survey of insect population data
concluded that of twenty-four insect species, one, the
Colorado potato beetle, exhibited a stable periodic popula-
tion. This conclusion was arrived at by fitting a regression
model given by
Niþ1 ¼ r^Nið1þ aNiÞ�b (13)
to a field population data set.15 This model, while not identi-
cal to the logistic map of Eq. (3) due to the additional parame-
ters a and b, includes the two main features of the logistic
map, namely non-overlapping generations and density-
dependent feedback. The same study demonstrated that labo-
ratory populations of Australian sheep blowflies exhibited
chaotic dynamics. It is likely that the isolation from extrinsic
confounding factors afforded by studying the blowflies in a
laboratory setting contributed to the chaotic dynamics
observed. Table I gives the r^ and b parameters used to fit the
potato beetle and blowfly population data to Eq. (13). The
dynamical regime exhibited by each fit is also shown. The a
values are not shown because they do not influence the dy-
namical regime of the fits. The data for the cabbage root fly
are also given as an example of a population exhibiting stable
fixed-point population dynamics.15 A subsequent study that
considered vertebrate populations as well as insects con-
cluded that while one species of insects exhibited chaos,
no vertebrates included in the study did. On the other hand,
there were examples of both insects and vertebrates that exhib-
ited oscillations or damped oscillations.17 While this study
also used a logistic-map-like model in that it included only
Table I. Logistic-map-like dynamics of select insect populations with non-
overlapping generations determined by Hassell, Lawton, and May by fitting
Eq. (13) to field population data sets (Colorado potato beetle and cabbage
root fly) or laboratory experiments (Australian sheep blowfly) (Ref. 15).
Species r^ b Dynamics
Colorado potato beetle
(Leptinotarsa decemlineata)
75 3.4 Stable periodic orbit
Cabbage root fly (Erioischia brassicae) 3.3 1 Stable fixed point
Australian sheep blowfly
(Lucilia cuprina)
55 >100 Chaos
730 Am. J. Phys., Vol. 81, No. 10, October 2013 Jeffrey R. Groff 730
intrinsic density-dependent feedback and non-overlapping
generations, the model included delayed density dependence,
meaning the population at generation iþ 1 depended on both
Ni and Ni�1. Such time delays are used to attempt to capture
the effects of interspecific interactions in a single species
model without explicitly modeling other species.
Because real populations experience interspecific interac-
tions and often have overlapping generations, the logistic
map and logistic-map-like models are likely inadequate for
most species. Nevertheless, the logistic map is a foundational
model in theoretical ecology worthy of learning. The compli-
cated dynamics exhibited by the logistic mapand discovered
by Robert May6 inspired population ecologists to embrace
nonlinear models.18 These nonlinear models not only capture
the dynamics of real populations, they can be predictive as
well. An especially impressive example is a model of a labo-
ratory population of flour beetles.19 The model is a recursion
relation like the logistic map. However, the model is not
logistic-map-like in that it allows for overlapping genera-
tions in a manner similar to Eq. (7) and considers three de-
mographic subgroups: (1) feeding larvae; (2) non-feeding
larvae, pupae, and immature adults; and (3) mature adults.
Importantly, the model successfully predicts the experimen-
tally observed transitions to periodic and chaotic dynamics
in response to manipulations of the adult death rate.
In addition to being a foundational model in theoretical
ecology that inspired ecologists to embrace nonlinear mod-
els, the logistic map model of population change inspired
ecologists to search for complicated dynamics, including
deterministic chaos, in real populations.18 While the compli-
cated dynamics that have been discovered are often not
logistic-map-like in that they are driven by predation, inter-
specific competition for resources, and other extrinsic fac-
tors, the search has revealed that population oscillations are
common.20 On the other hand, ecologists have discovered
that chaos in natural populations seems to be rare. This may
be due to the many food-web interactions that organisms
have with other species that may dampen complicated dy-
namics and favor steady-state equilibria.21 Of course, such
interactions are not included in logistic-map-like models that
predict chaos. Interestingly, it is more prevalent for species
to exist on the edge of chaos instead of in the chaotic re-
gime,22 and a study of voles demonstrated that populations
on the edge of chaos can be driven into and out of chaotic
regimes by external drives.23 The propensity to do so
depends on factors like predation and geographic latitude.
The best evidence for chaos in populations comes from labo-
ratory studies like the blowfly and flower beetle studies men-
tioned above. A particularly interesting long-term laboratory
experiment of a complex marine food web including several
species of phytoplankton, zooplankton, bacteria, and detriti-
vores demonstrated periodic and chaotic population fluctua-
tions.24 These dynamics were not due to logistic-map-like
dynamics but were instead attributed to interactions between
the individual populations in the experiment, indicating that
food-web interactions may sometimes destabilize instead of
stabilize populations.
The role of chaos and other complicated dynamics in natu-
ral populations continues to be debated, but even if logistic-
map-like complicated dynamics—dynamics due to intrinsic
density-dependent feedback and non-overlapping genera-
tions—is ultimately shown to play a minor role in natural
populations, the logistic map will be worth teaching to future
ecologists because this model, often with extensions,
continues to be actively researched yielding interesting theo-
retical insights. For example, work that adds stochastic ele-
ments to the logistic map has led to tools that may help to
discriminate between the intrinsic nonlinear dynamics of a
population and noise.25 Work with the logistic map and other
one-dimensional discrete maps has shown that periodic or
constant harvesting can prevent chaos in some situations and
even lead to the counterintuitive effect called the hydra effect
where increased harvesting actually increases population
size.26,27 Work with coupled map lattices, a modeling formal-
ism that adds a spatial aspect to population models, has dem-
onstrated that spatially modeled populations can exhibit
dynamics like those of the spatially-lacking logistic map.28,29
V. CONCLUSIONS
In my introductory physics courses, I emphasize that
physics is concerned with the discovery of mathematical
models that can both describe and make predictions about
physical phenomena. At the same time, I strive to convince
my bioscience students that because biological life exists in
the same physical world as falling rocks and colliding bil-
liard balls, physics can also be used to study biological phe-
nomena. This proposition is difficult for some students to
accept because introductory physics is taught in the context
of an idealized world of point masses or uniform rigid
objects having simple geometries moving with negligible
friction in straight lines, perfect circles, or parabolas. On the
other hand, biology involves the study of lumpy, squishy
objects of irregular shapes moving non-uniformly in crooked
paths with a significant amount of friction, surface tension,
and other “messy” forces. Nevertheless, I tell my students
that the quantitative mathematical modeling approach to sci-
ence we learn in physics is increasingly useful for studying
complex biological systems.
The logistic map activity helps me to advance this narra-
tive. For example, it shows that mathematical models can be
made to capture much more complicated dynamics than the
models typically taught to students in introductory physics
courses. The lesson to be learned is that complexity in bio-
logical systems does not alone preclude the use of mathemat-
ical models to study these systems and make testable
predictions. On the other hand, complexity can make practi-
cal predictive models for many biological phenomena elu-
sive. For example, even if field populations are rarely
chaotic, the logistic map and other population models have
taught ecologists that predicting population dynamics may
often require nonlinear modeling.18 A physicist’s approach
to complexity often involves making simplifying assump-
tions (e.g., projectiles move with negligible friction only
under the influence of gravity). When working with the
logistic map model of population change, I ask my students
to think about the simplifying assumptions on which it is
based. For example, the model does not include species
interactions and assumes a “well-mixed” spatially homoge-
nous distribution of organisms. I ask them to ponder the real-
world conditions under which such simplifications are likely
to break down. Giving students an opportunity to think about
the simplifying assumptions inherent in most mathematical
models teaches them to think more like physicists and to
appreciate the limitations and challenges of mathematically
modeling the physical and biological worlds.
While the logistic map activity allows me to emphasize to
bioscience students the relevance of mathematical models to
731 Am. J. Phys., Vol. 81, No. 10, October 2013 Jeffrey R. Groff 731
their discipline, it can also serve as a cautionary tale for
physics students who embrace predictive models in their dis-
cipline. The logistic map demonstrates the limitations of
using models to make predictions about systems that behave
chaotically, even if those systems are deterministic. For
example, Newtonian physics allows us to design rockets that
can land humans on the Moon but does not allow us to pre-
dict the chaotic evolution of many bodies interacting via
gravity. Studying the logistic map is a good introduction to
chaos and can prepare physics students to study the many-
body problem and other chaotic physical systems like the
analog circuits previously described in this journal.30,31
ACKNOWLEDGMENTS
The computational resources needed to conduct the logistic
map laboratory with my students were funded in part by
West Virginia Higher Education Policy Commission Division
of Science and Research Grant No. HEPC.dsr.10.10, an
EPSCoR innovation grant for the acquisition of biophysics
instrumentation for curricular enhancement, research, and
outreach.
a)Electronic mail: jgroff@shepherd.edu
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