Matthew Siderhurst
Mattss@lamar.colostate.edu
16 April 2001
Insect Behavior
Abstract
Chaos is often either poorly or totally misunderstood among the public at large and
unfortunately among many biologists as well. This is unfortunate because its implications
for understanding biological systems are wide-ranging. Chaotic behavior in systems
represents a shift to a new paradigm in understanding nonlinear systems that are being
increasingly recognized throughout the physical world. Much of the problem in
understanding chaos stems from the Newtonian worldview in which systems are largely
understood as linear, independent, closed and at equilibrium. Chaos in a physical sense is
not merely randomness or disorder but instead nonlinear interdependent systems that are
highly deterministic. Chaotic systems can be governed by fairly simply rules, attractors,
but still appear highly random. In biology the fields of ecology, physiology, population
dynamics, epidemiology and animal behavior have been of particularly interest. Studying
chaos in natural systems can be very difficult due to problems of noise and sample size.
These problems are now beginning to be overcome by the use of modern technology in the
form of computing power and digital visual equipment. Key to this is the development of
models that guide the collection of data and show what is adequate to describe behavior.
Chaos theory was first applied to insects in looking at the behavior of entire
populations. Populations of the flour beetle Tribolium have been studied in the laboratory
and modeled on computer simulations and represent a classic example of chaos in ecology.
Activity cycles in individual ants have now been shown to be chaotic; the first time chaos
has been observed in behavior. Experimental work has been carried out in two ways; work on
live colonies of ants by using video imaging over time with environmental modifications,
and by employing computer models such as the edge of chaos model, FNN (fluid neural
networks), LPA (nonlinear interactions among larva, pupa and adults), and CNN (cellular
neural network). Much work has been done to explain the mechanism by which ants move from
chaotic individual behavior to periodic collective behavior and the adaptive advantages of
such mechanisms. From showing models adequate to describe behavioral changes to guiding
data collection, chaos is allowing insect behaviorists to understand and describe complex
nonlinear systems, like insect behaviors, at a higher level then ever before.
Introduction:
It is unfortunate that many words in use today have different means in common usage and as
defined in scientific fields of study. A common example of this is the relationship
between the words 'velocity' and 'speed'. In common usage they have roughly the same
meaning. In terms of physics however they have precise definitions and while related have
distinct meanings. It would not be appropriate to tell a police officer that you were not
in fact 'speeding' although your 'velocity' may have been rather high. This difference in
definitions is nowhere more evident then with the term 'chaos'. In common usage the word
chaos refers to disorder, randomness or abyss. Chaos in this sense is the end product of
entropy, a state without order or control. The scientific definition of chaos stands in
stark contrast to this the common usage of this word. Chaotic systems are not random at
all, although they may appear to be at first glance, they are instead deterministic
nonlinear interdependent systems that are neither periodic nor quasiperiodic, but are
confined in "phase space" by an "attractor" (Logan and Allen, 1992).
The first evidence for chaos in natural systems came from Edward Lorenz in 1963. His
accidental discovery of chaotic behavior in a simple atmospheric weather model opened a
field which has now touched, in some way, almost every field of the natural and social
sciences (Logan & Allen, 1992) (Robertson & Combs, 1995) (Degn et al., 1986).
Chaos has been implicated in phenomena ranging from quantum mechanics (Fox, 1991) to
planetary orbits (Kerr, 1992), from nonlinear dynamics in physiology (Glass, 1999) to
psychological theories on depression (Hector et al., 1995). The development of more
sophisticated techniques for analysis and great computing power has allowed science to
find patterns and understand systems that were hither-to thought of as simply random or to
complicated to analyse.
To understand chaos and its implications for insect behavior and biology as a whole we
must first look at the broader field of nonlinear dynamics. Many scientists trained in the
reductionist thinking of modern science seem to regard the world from a largely Newtonian
stand point (Frank and McCoy, 1991). This worldview sees the natural universe as being
made up of independent, closed, equilibrium systems that exhibit linear dynamics (Goerner,
1995). Linear dynamics refers to the situation where either n > n + 1 > n + 2... or
n < n + 1 < n + 2..., where the increase in x, is reflected by a proportional
increase or decrease in y. Because of the complexities of many systems, biological systems
being a prime example, it has been easier for scientists to break down systems to their
smallest components and regard them in terms of the Newtonian, linear worldview mentioned
above. This approach can simplify systems making them more readily understandable and this
has yielded much of the scientific progress achieved over the last several hundred years
(Goerner, 1995). However, it is becoming increasingly accepted in recent years that
natural systems are as a whole, highly interdependent (meaning they interact), open, often
far from equilibrium and, as a consequence, exhibit a high degree of nonlinearity
(Liebovitch, 1998) (Goerner, 1995). With a nonlinear system the input is not proportional
to the output, this is, an increase in x, does not mean a proportional increase or
decrease in y. Nonlinear systems comprise basically all relationships that are not
straight lines. Chaotic systems are a subset of nonlinear systems and arise from some of
the more complicated solutions of these systems.
So what do chaotic systems look like and how do they differ from simply random systems? If
one considers two data sets, one generated by a random mechanism and the other by the
equation x ( n + 1 ) = 3.95 x ( n ) [ 1 - x ( n ) ]
(Liebovitch, 1998). Just by looking at the data sets it is hard to see any differences.
The differences between these two are only revealed when the data are transformed into a
"phase space" diagram. Phase space is a system-variable space where one or more
system variables are plotted against each other as they move through time, i.e. there is
no time axis (Liebovitch, 1998). The equations used to transform data to phase space are
called Lorenz equations. Typically phase space diagrams are , or 3-dimentional. The
transformation of chaotic data into phase space shows that n + 1 is dependent on n and is
deterministic. The randomly generated data set is not determinist but is rather
"space filling" in the phase space, n and n + 1 have no relationship.
Another thing to notice about the chaotic data in phase space is that it is constrained to
a area or path by what is referred to as an attractor. An attractor is the behavior toward
which a system tends as time increases, a time-asymptotic behavior of a system
(Liebovitch, 1998). Comsider a chaotic data set in 3-dimentional phase space
constrained by an attractor. If a system is started at a fixed point off the attractor it
quickly approach the attractor is what is term "transient behavior" For a system
to be chaotic the fractal dimension (a measure of self sameness) of the attractor must be
small, it cannot be 0 or infinity. Most workers consider 6 to be the upper limit of small.
In describing chaos it might appear at this point that if given initial conditions and the
equations governing a system, one could predict the entire behavior of the system. This
however is not the case because of chaotic system’s sensitivity to initial
conditions. While chaotic systems are constrained by attractors to a finite area they can
take on a wide range of values in phase space. Small differences in initial conditions are
quickly magnified as time increases. This sensitivity puts limits on the predictions that
can be made about the future behavior of chaotic systems and also requires highly accurate
data sets when studying chaotic behavior.
Systems may not show chaotic behavior throughout all phase space. The system may be
constrained to one path for given values of t, but as time increases there may appear
bifurcation forks, which allow two paths. This bifurcation (splitting) can lead to areas
of chaotic behavior interspersed with more constrained areas.
Discussion:
The classic example of chaotic behavior in an insect population: The flour beetle,
Tribolium castaneum.
There are many problems that arise when trying to study chaos in naturally occurring
systems. Ecological data sets are often very short in terms of analysing for chaotic
behavior (Vikas & Schaffer, 2001). They may also be corrupted by observational error
and process noise that doesn't allow accurate characterization of the underlying dynamics,
chaotic or otherwise (Vikas & Schaffer, 2001). A very important tool is then the
ability to extract deterministic signals from noisy data sets.
A system that has shown itself to be very helpful in developing method for data extraction
has been the population dynamics of the flour beetle, Tribolium castaneum. The flour
beetle system was first investigated by Jillsonin (1980). While chaos has been observed in
the population dynamics of many insects (Perry et al., 1997) (Wilder et al., 1995) (Harnos
et al., 2000), the flour beetle has proved an outstanding system to work on (Cushing et
al., 1998). The beetles can be raised in the laboratory in containers of flour and their
population dynamics investigated. The large number of beetles and the ease of observation
has lead to the generation of data sets that are large and robust. This is a rather simple
system ecologically speaking but the population fluctuations are highly interesting.
Developing models that accurately describe the flour beetle system and demonstrate chaos
has largely been the work of J.M. Cushing and a team of other scientists. Costantino et
al. published a paper in Science in 1997 in which they used a nonlinear demographic model
to predict the population dynamics of the flour beetle in a laboratory population. By
setting the adult mortality in their model to a high level, and experimentally changing
the adult-stage recruitment, the dynamics of beetle abundance changed from equilibrium to
quasiperiodic to chaotic. These transitions correlated well with their mathematical models
and phase space graphs developed provided considerable evidence for the existence of chaos
in the system (Costantino et al., 1997) (Henson et al., 1999). Computer modeling has
proved an important tool, used in several ways in studying chaos. As a first step they are
helpful in showing the type of models that are adequate to describe behavior changes
(Costantino et al., 1997) (Cole, 1991) and also to guide the collection of data suitable
for the models (Henson et al., 1999) (Cushing, 2001) (Cole, 1991). Both of these cases
came into play with the documentation of chaos in a living laboratory population
correlated with the "LPA model" (Cushing et al., 2001). The LPA model is based
on the nonlinear interactions among the life cycle stages of the beetle (larva, pupa and
adult). In controlled laboratory experiments, cultures of the flour beetles underwent
bifurcations in their dynamics as demographic parameters are manipulated. The
well-validated deterministic LPA model accurately predicted these bifurcations, including
a specific route to chaos. A stochastic version of the model accounts for the deviations
of the data from the deterministic model and provided the means for rigorous statistical
validation. The attractor of the deterministic LPA model and the stationary distribution
of the stochastic LPA model describe the experimental data in phase space with great
accuracy.
Chaotic systems and ant foraging behavior:
As noted before, determining chaos in biological systems faces multiple obstacles. Add to
this the fact that behavior can be highly complex with simply linear models and the
magnitude of the task at hand becomes evident. Insect behavior has long been a model for
animal behavior in general because it is complex enough that mechanisms can be scaled to
other organisms and yet it is also simple enough that mechanisms can be deduced. In fact
chaos in behavior has only been documented in one other organism, the goldfish, Carassius
auratus (Nepomnyashchikh, 1998 and 2000), outside of insects.
The first work showing chaos in insect behavior came from experiments done on activity
cycles in ant species, Leptothax allardycei (Cole, 1991). "Ants, refuting popular
fables, do not work untiringly all day long," noted Delgado and Sole (2000). They may
actually spend much time inactive. According to a study by Herbers (1983) on Leptothorax
longispinosus and L. ambiguus, "ants spent two-thirds of their time apparently doing
nothing at all". Cole used an automatic digitizing camera to take pictures of an ant
colony as a whole and individual isolated ants. Then comparisons were made between
successive images so that the activity of individual ants could be determined.. Cole came
up with two very interesting behavioral observations:
1) The activity of the colony as a whole was periodic with an activity cycle that ranged
from 15-37 minutes.
2) The activity cycles for individual, isolated ants however is not periodic. Individual
ants have activity patterns characterized by what appear to be spontaneous bursts of
activity followed by comparatively long periods of inactivity. When analysised the
experimental data showed evidence of low-dimensional deterministic chaos. The attractor
was reconstructed from the time series and was shown to have a mean dimension of 2.43,
which lends itself to construction of 3-dimentional phase space diagrams.
Cole speculates that, "The existence of chaos in animal behavior can have several
important implication. Variation in the temporal component of individual behavior may not
be due simply to chance variations in the stochastic world, but to deterministic processes
that depend on initial conditions." If shown in other animal’s behavior we may
be on the verge of opening a new door into the mechanisms that drive behavior.
We have seen that models are very important to understanding the underlying dynamics at
play in biological systems. It is not surprising then that multiple models have been
proposed to explain the ant behavior observed by Cole. Sole et al. (1993) proposed a
"neural network model" of interactions between ants to explain how individuals
could have chaotic behavior and the colony periodic behavior. This model arises from both
theoretical studies of neural organization and experimental observations. Using this
simple model, emergent properties of the dynamics of some ant colonies could be
reproduced. In particular the global oscillations and synchronization (entire colony
activity cycles) observed by Cole (1991) were able to be reproduced. The colony wide
periodic activity cycle emerges from the local interactions of two or more non-periodic
(chaotic) elements in this model. Such oscillations were shown to be present in several
species of ants and are suspected in other related situations.
Another model that was proposed in light of Cole’s discovery to describe activity
synchronization in ants was the CNN model (Nemes & Roska, 1995). The cellular neural
network model is a spatial lattice on which ‘ants’ interact, become active or
inactive, and move about according to the constraints of the system. This model was also
able to reproduce the synchronization effects seen between individual ants. The parameters
that were produced were slightly different from Cole’s observations but the authors
explained this as, "differences between selection and displacement techniques",
and "different initial conditions". Once again the source of the emergent
synchronization appeared to come from, "interactions within a local
neighborhood".
The question of how individual chaotic behavior might be a selective advantage for ants is
still somewhat unknown. There are however several possibilities (Bonabeau, 1997) (Delgado
& Sole, 2000) (Sole & Miramontes, 1995). One idea is that ant colonies exhibit a
characteristic known as "edge of chaos" (Bonabeau, 1997) (Sole & Miramontes,
1995). Edge of chaos implies that the ant colony gains an adaptive advantage by existing
at a point of instability. "An ant colony exhibiting an appropriate combination of
group and mass recruitment can adaptively switch to a newly introduced food source if it
is richer (by being having a certain amount of instability): this is precisely the case of
some species, such as Tetramorium caespitum, whose behavioral parameters are argued to be
those characterizing the edge of chaos (Bonabeau, 1997)."
Sole has used the edge of chaos idea in conjunction with the fluid neural network model to
explain the adaptive advantages of the observed self-synchronized activity in ant
colonies. In particular he suggests that ants may benefit from an ordered temporal pattern
of behavior arising from the chaotic behavior of individuals. "(The) …model of
self-synchronized activity (the fluid neural network) suggests that with self-synchronized
patterns of activity a task may be fulfilled more effectively than with non-synchronized
activity, at the same average level of activity per individual." A key to this idea
is that synchronized behavior of this type must start from a basis set of individuals with
chaotic behavior.
Conclusion:
Studying chaos in natural systems can be very difficult due to problems of noise and
sample size. These problems are now beginning to be overcome by the use of modern
technology in the form of computing power and digital visual equipment. Key to this is the
development of models that guide the collection of data and show what is adequate to
describe behavior. Population dynamics have lead the way in understanding chaos in
biological systems but there is now greater interest in insect behavior as well. With the
example of chaotic behavior in individual ants we may be poised to enter into a new
understand of the underlying forces in behavior.
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