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Mathematical and theoretical ecology: linking models with
ecological processes
Edward A. Codling and Alex J. Dumbrell
Interface Focus 2012 2, 144-149 first published online 8 February 2012
doi: 10.1098/rsfs.2012.0008
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Interface Focus (2012) 2, 144–149
doi:10.1098/rsfs.2012.0008
Published online 8 February 2012
INTRODUCTION
Mathematical and theoretical ecology:
linking models with ecological
processes
Edward A. Codling1,2,* and Alex J. Dumbrell2
1
Department of Mathematical Sciences, and 2School of Biological Sciences, University of
Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, UK
Keywords: animal movement; collective decision-making; epidemiology;
individual-based model; population dynamics; state-space models
1. INTRODUCTION
data and the theory associated with the ecological
question being addressed.
Ecology is intrinsically quantitative, with ecologists
spending much of the last 100 years enumerating the
patterns and processes underpinning the dynamics of,
and interactions within, natural populations and communities. Owing to its quantitative nature, ecology
has long recognized the need to work in parallel
with mathematical disciplines. This interdisciplinary
approach is elegantly highlighted in some of ecology’s
seminal and most highly cited papers by early ecology
pioneers, such as Fisher, Preston and MacArthur, who
used mathematical approaches to explain and analyse
ecological observations [1 – 3]. However, in addition to
being a tool to use for ecologists, the disciplines of mathematical and theoretical ecology also developed
independently of studies examining ecological data,
into a fascinating and insightful discipline in its own
right; emerging from the foundations laid by mathematicians, such as Volterra and Lotka, on which much of
current theoretical ecology is built (see Levin [4] for a
discussion of these early research leaders). Subsequently, this has produced a minor schism, where
researchers tend to envisage themselves as either theoretical ecologists or empirical ecologists. Currently,
much of the integration between the empirical and
theoretical disciplines of ecology is made informally;
empirical ecologists apply statistical models and
numerical approaches to the analysis of their data
often with little consideration for the underlying
theory, while theoretical ecologists often make biologically unrealistic simplifying assumptions in the pursuit
of beautiful and analytically tractable mathematical
models. Yet, arguably, it is now impossible to conduct
robust ecological research without considering both the
2. LINKING MODELS WITH ECOLOGICAL
PROCESSES
Without ecological theory, collecting data is a futile and
meaningless endeavour. Likewise, producing elegant
and beautiful mathematical models of ecological systems without validation against real data is an empty
achievement. These two aspects of ecology (empirical
and theoretical) are as intricately linked as the coupled
ODEs used to describe the interactions between the
predatory fox and its prey the rabbit. Theoretical ecology provides a potential mechanistic explanation of
observed ecological patterns. By following an iterative
process of evaluating models with data and discovering
which assumptions are plausible, which null hypotheses
can be rejected, and what additional theoretical developments are required to explain the data, researchers
can disentangle the underlying ecological mechanisms
regulating the natural world. A recent example of this
process is highlighted in the renewed interest in neutral
theories of biodiversity [5]. Although neutral theories of
biodiversity are based on the potentially unrealistic biological assumption of ecological equivalence across
species, via the iterative process of continued testing
and revaluation of the underlying theory ecologists
have provided major insights into the dynamics of natural communities, largely in an attempt to identify why
unrealistic assumptions do so well in explaining data.
However, this idealized view of how ecological research
should be conducted is logistically challenging and
increased integration, collaboration and communication
between empirical and theoretical ecologists is clearly
needed to overcome these logistical challenges.
In September 2011, the University of Essex hosted
an interdisciplinary conference, ‘Mathematical and
*Author for correspondence ([email protected]).
One contribution of 11 to a Theme Issue ‘Mathematical and
theoretical ecology’.
Received 12 January 2012
Accepted 12 January 2012
144
This journal is q 2012 The Royal Society
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Introduction. Mathematical and theoretical ecology
Theoretical Ecology 2011: linking models with ecological processes’ (MATE 2011), with the aim of
exploring these issues and creating discussion and
debate between theorists and empiricists working in
ecology. In this issue of Interface Focus, we present
the best contributed papers from the Essex meeting.
These papers cover a wide range of topics but all are
linked by the theme of how advances in theoretical
ecology can be integrated with ecological data.
3. ADVANCES IN POPULATION AND
COMMUNITY ECOLOGY THEORY
Three of the papers within this issue particularly focus
on some of the fundamental aspects of population and
community level ecology, and along with the other contributions, highlight some of the issues of integrating
theoretical ecology and mathematical models with
ecological data [6 – 8].
A central issue in ecology is that natural environments are patchy, and the resources used by
organisms within the local community are heterogeneously distributed [9]. In stark contrast, many
theoretical models in ecology adopt a mean-field
approach, averaging out spatial variability and largely
ignoring the spatial structure observed in almost all ecological data. This mean-field treatment of ecological
models is a simplifying assumption, which allows
analytical and numerical tractability and has been
used to produce elegant models against which ecological
data can be compared. However, whether or not a
mean-field approach, or one that accounts for spatial
variability, is adopted in ecological studies may largely
predetermine results [10]. In this special issue,
Grünbaum [6] addresses questions of whether or not
to follow a mean-field approach in ecological models
of interacting consumers and resources. Yet, in contrast
to the often assumed additional complexity associated
with spatially explicit modelling of patchy environments, Grünbaum [6] demonstrates a simple heuristic
approach that incorporates consumer – resource patchiness with little additional complexity. In addition,
Grünbaum [6] expands this approach, demonstrating
the applicability of three non-dimensional indices
(focused on an organism’s movement, reproduction
and resource consumption) in quantifying consumer –
resource interactions in patchy environments, but
which can also be used to demonstrate deviation from
mean-field assumptions; offering a potential correction
factor to large-scale ecological models that are heavily
reliant on the simplicity mean-field assumptions offer.
Often mean-field approaches provide relatively realistic biological assumptions over the temporal and spatial
scales relevant to the model and ecological question
being addressed. For example, the human population
within a densely populated city can be considered a
well-mixed homogeneous environment for highly
fecund and infectious disease-spreading microbes. However, even mean-field models with obvious analytical
solutions have problems when considering their fit to
ecological data and ability to accurately recreate
observed ecological patterns. In the simplest case, one
Interface Focus (2012)
E. A. Codling and A. J. Dumbrell
145
of the major obstacles to overcome is in parametrization;
deterministic population models by definition will ultimately produce results dependent on initial parameter
values, and even stochastic models will reflect to some
extent their starting conditions. Yet, often ecological
data are noisy or incomplete and parameter estimates
not obvious, leading to a number of issues in model
parametrization [11]. By focusing on population-based
epidemiology models, Stollenwerk et al. [7] address
issues of parameter estimates. Stollenwerk et al. [7] provide a comprehensive review of this subject by first
examining basic epidemiological models and then
expanding their examples to cover more realistic cases
focusing on models of dengue fever. Building on this
approach, Stollenwerk et al. [7] then produce a master
equation from which the transition probabilities can
be used to compute a likelihood function; linking
their theoretical models with observed data by providing
the probability of observing all points from empirical
time-series data describing infection dynamics, as a function of model parameters. Once a likelihood function
is computed it can be incorporated into a Bayesian
framework, and along with the maximum-likelihood
estimates, Stollenwerk et al. [7] then discusses the
application of these methods in parameter estimation.
Maximum-likelihood and Bayesian approaches are
increasingly being used in ecology to estimate parameters from data and select or compare between
theoretical models describing observed ecological
patterns. This use of likelihood-based statistical
approaches reflects the underlying need for ecologists
to accurately assess the performance of statistical
models when applied to their data; likelihood-based
approaches often provide a more parsimonious comparison of model performance (e.g. based on AIC scores)
than selecting models based on maximizing adjusted
r 2-values [12]. This may lead to likelihood-based
approaches providing quantitatively different results
from model-fitting analyses compared with other
approaches such as least squares, often inspiring
researchers to re-examine classic questions in ecology,
but with novel statistical techniques. In this issue,
Etienne et al. [8] tackle this challenge by applying
novel statistical methods to a classic question: why are
small-bodied taxa more diverse than large taxa, and
can this observation be explained by phylogenetic
clade age alone? Previous studies have suggested that
small-bodied taxa are more diverse simply because
they come from phylogenetic clades that originated
longer ago and thus have had longer to diversify.
Etienne et al. [8] revisit this finding, by developing a
robust statistical approach using a constant-rate
birth – death model with allometric scaling of speciation
and extinction rates. This model is then used in a maximumlikelihood approach that maximizes the likelihood of the
model, given data on extant diversity, clade age and
average body size from a compiled dataset covering a
range of Metazoan taxa. Etienne et al. [8] conclude
that clade age alone does not explain higher diversity
of small-bodied taxa, but that additional processes
are at work and that small-bodied taxa may simply
have faster ‘evolutionary clocks’. By taking the
approach in Etienne et al. [8], which robustly merges
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Introduction. Mathematical and theoretical ecology
empirical data and theoretical models, the authors
have illuminated interesting new perspectives on an
old ecological question.
4. ADVANCES IN BEHAVIOURAL
ECOLOGY THEORY
The use of statistical models to infer properties of behaviour from observed data in individual animals and
groups of animals is a topic explored by both
Schliehe-Diecks et al. [13] and King [14]. SchlieheDiecks et al. [13] give a detailed illustration of the
application of mixed hidden Markov models (HMMs)
in a case study that looks at mouse lemur behaviour.
HMMs are powerful and flexible statistical tools that
allow the user to infer and quantify sequences of behaviour. The basic rationale behind a HMM is that the
underlying behavioural or motivational states are
‘hidden’ and cannot be observed directly, but consequences or outputs of the hidden state can be
observed. In the case study of Schliehe-Diecks et al.
[13], the ‘hidden’ state is the motivational state relating
to hunger—animals are either hungry or satiated, but
we cannot observe which state an animal is in directly.
However, we can observe the ‘output’ state—whether
the animal is currently feeding or not. Schliehe-Diecks
et al. [13] develop a mixed HMM by including factors
that account for differences in the behaviour of individual lemurs within the observed group. These factors
could be covariates linked to behaviour, such as sex,
body mass or time of feeding, but the mixed HMM
also allows for ‘random effects’ to be included which
can account for different ‘personalities’ across the
study group. Schliehe-Diecks et al. [13] use their
mixed HMM to analyse behavioural time-series data
from 54 lemurs and demonstrate how feeding behaviour
is linked to both body mass (with larger lemurs staying
satiated longer) and sex (females having longer feeding
and non-feeding periods). They finish with a discussion
of how this type of mixed HMM model is easily generalized to other ecological contexts and could be used by
ecologists to identify how behavioural flexibility and
personality differences interact and lead to differences
in observed behavioural sequences.
The state-space approach reviewed by King [14] in
the context of capture – recapture – recovery data has a
very similar model structure to the HMM discussed
by Schliehe-Diecks et al. [13]. The main difference
between the HMM and state-space approach is that in
a state-space model the ‘states’ are not discrete (as in
the HMM) but are continuous. As with HMMs, statespace models are a flexible tool that can be used on
noisy datasets where there may be measurement error
or missing data, making them ideal to use with ecological datasets such as those generated from tracking
studies of animal movement behaviour [15]. In her
review, King [14] explains much of the theoretical background to the state-space approach, while highlighting
some of the common practical issues that may be
faced. For example, because a state-space model is a
continuous system process, it is generally much harder
to find the likelihood function of the observed data (in
Interface Focus (2012)
E. A. Codling and A. J. Dumbrell
contrast, this is relatively easy in an HMM). King [14]
goes on to demonstrate how the state-space approach
is an ideal method to analyse capture– recapture – recovery data. Capture – recapture studies typically involve
initial capture events where individual animals are
marked, released and then recaptured at some point
in the future (although individuals do not always need
to be physically captured—sightings and photo observations can be included). In addition to live captures
or sightings, it is also possible to ‘recover’ dead individuals. Once the capture – recapture – recovery data are
recorded, they can be used to estimate demographic
parameters such as survival probabilities and/or abundance. King [14] demonstrates how the state-space
approach can be used to analyse such data, including
extensions to the basic model to include multi-states or
mixed models such as random effects. King [14] finishes
with an informative illustration of how the solution of
a state-space model can be found in various ways using
Bayesian techniques. The key point highlighted is that
the performance and efficiency of different Bayesian
model-fitting techniques will be problem- and modeldependent. The state-space approach is clearly a useful
tool for ecological data analysis as it can combine data
from different source into a single integrated analysis.
The recent advances in statistical modelling
described by Schliehe-Diecks et al. [13] and King [14]
are likely to prove invaluable for many ecological
researchers trying to link theoretical models to data.
However, it is also important that theoretical modellers
continue to develop realistic and informative mechanistic models of behaviour to explore ecologically relevant
processes such as animal movement and predator – prey
interactions. This is the theme of the paper by Mckenzie
et al. [16] who analyse data from a detailed tracking
study of four wolves in the Rocky Mountains (Alberta,
Canada). One of the key characteristics of this type of
environment is the highly heterogeneous landscape
(see also Grünbaum [6]). Mckenzie et al. [16] explore
the effect of landscape features on wolf movement
behaviour and the consequent effect on prey interactions. In particular, they set-up and analyse a
mechanistic movement model where wolves react to
‘seismic lines’ (narrow linear stretches of forest, cleared
for energy exploration) in the landscape. Their movement model is based on a simple diffusive random
walk process that is extended to include increasing
levels of complexity such as anisotropic diffusion
along the seismic line or biased/directed movement
towards the seismic line (see Codling et al. [17], for a
review of these random walk and diffusion models).
After analysing their tracking data, Mckenzie et al.
[16] demonstrate that the wolves in their study do
alter their movement behaviour when on seismic lines
(moving faster and with more directional persistence
leading to anisotropic diffusion along the line), but
only one wolf moved towards the seismic line when
close to it (a directed or biased movement). They
then consider how the density of seismic lines and the
different movement models interact and result in different functional responses (the per capita kill rate as a
function of prey density). They show that the presence
of seismic lines can lead to an increase in prey
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Introduction. Mathematical and theoretical ecology
encounters and hence kill rate, and that the relative
effect of seismic lines on the functional response is greatest at the lowest densities of prey. The paper raises a
number of interesting questions from both a behavioural and theoretical viewpoint. For example, the authors
finish by discussing how their mechanistic model
could be made more realistic by including correlation
in movement directions (as seen in their wolf movement
data) using a transport equation approach [17] but this
would consequently make the mathematical analysis
more difficult.
One of the key advantages of a theoretical modelling
approach is that it is possible to explore a wide range of
scenarios that simply would not be possible experimentally or through field observations. Rands [18] presents
an individual-based simulation model of a host– brood
parasite interaction (e.g. a cuckoo trying to lay in the
nest of a reed warbler). In contrast to other papers in
this special issue, Rands [18] does not try to parameterize the model with real data, but instead explores the
full parameter space (within reasonable upper and
lower bounds) to determine general emergent properties
of the system. The rules of behaviour at the individual
level are very simple and are based on parameters
that govern the probability of taking various actions
(e.g. move away from nest or stay on nest in a given
time step). In his model, Rands [18] assumes that individual hosts have two possible strategies that are of
interest when a brood parasite is spotted (against a
null model of no response); either ‘mobbing’, where
the whole social group of hosts try to drive off the parasite (at the risk of one’s own nest being parasitized), or
‘sitting tight’, where individuals simply return to their
nest as quickly as possible when alarmed. The simulation includes a learning element to the host
behaviour—all hosts start off naive but learn to recognize the parasite through social interactions with
neighbours when mobbing (when sitting tight there is
a risk of not learning to recognize the parasite). After
running simulations across a wide set of parameters,
Rands [18] demonstrates that both mobbing and sitting tight are successful strategies to avoid parasites
(compared with the null model of no response). Interestingly, there was no significant difference between the two
strategies (in terms of parasite avoidance), except for the
case where the probability of a parasite appearing was
high, in which case mobbing become a more successful
strategy. The simulation model is simple and elegant
but nevertheless gives some clear predictions about
how social behaviour such as mobbing may have
emerged in a host – parasite system. Rands [18] finished
by highlighting the model assumptions that could be
modified in future studies to explore different scenarios
or problems.
In the context of animal social behaviour, the question of how animal groups collectively make decisions
has received increasing attention recently, with most
studies using a theoretical modelling approach to
address the problem. In a thoroughly comprehensive
overview of the topic, Conradt [19] considers how collective group decisions are made in the context of two key
scenarios: (i) where information available to individuals
in the group is uncertain and (ii) where individuals in
Interface Focus (2012)
E. A. Codling and A. J. Dumbrell
147
the group have conflicting preferences. In the case of
information uncertainty, Conradt [19] first details
quorum responses, where the decision at the individual
level is influenced by how many others in the group
have made a certain decision. She then discusses
the effective leadership model, where a small sub-set
of the group that are ‘informed’ about the correct
movement direction can lead a group of uninformed
individuals effectively. Finally, the independence –
interdependence model is described. In essence, all
these information uncertainty problems are examples
of the ‘many-wrongs’ principle where several decisionmakers are able to pool their personal information
and eliminate individual errors [20]. The case of conflicting preference between individuals in the group
decision-making problem is perhaps more complicated.
In this context, Conradt [19] first describes a ‘grouplevel’ model where behaviour can range from shared
decisions involving all group members to ‘dictatorial’
where decisions are made by one individual only. For
most biologically relevant assumptions, the best strategy in this model is then for the group to make shared
decisions rather than have a dictator. The ‘leader –
follower’ model is arguably the simplest group conflict
model and in this case, it is the individual who has
most need (e.g. for food) that makes the decision, an
example of ‘leading according to need’. A similar
result is obtained with the ‘pair-synchronization’
model, where leadership is effectively shared and
decisions made according to need. Conradt [19] discusses
synchronization models in small groups of three individuals and explains how in the context of movement
synchronization, both shared and dictatorial decisionmaking are evolutionary stable strategies. The ‘leading
according to need’ model is then described in larger
groups. Conradt [19] finishes with a description of two
clear open problems in this field of research. Firstly,
the problem of collective decision-making where both
information uncertainty and conflicting preferences are
present has not yet been properly addressed. Secondly,
a recurring issue with many of these models is the lack
of empirical data to verify the model predictions, and
this is where technological developments that allow
new methods of data collection for large groups of
animals may allow us to make progress [21].
5. FUTURE DIRECTIONS IN
THEORETICAL ECOLOGY
Despite many recent advances in theoretical ecology
and the associated integration with empirical data,
a number of fundamental challenges still remain to
be tackled, and as with most scientific disciplines
as new insights are gleamed, additional challenges
emerge. Broadly speaking, ecological models can be
split into two separate categories; simplistic mathematical models, which offer analytically tractable
solutions and the examination of the underlying
model properties, and simulation-based models
(either rule-based or numerical integrations of formal
mathematical models), which allow the examination
of more complex models that cannot be solved
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Introduction. Mathematical and theoretical ecology
analytically. Both of these modelling approaches have
advantages and disadvantages associated with them,
but a common theme is that as the models become
more complex, working with them becomes more computationally demanding. In particular, numerical
simulation of complex models is greatly limited by
available computer power and the questions theoretical ecologists are beginning to ask are challenging
the capacity of current computer hardware. This is
particularly evident when researchers begin to address
some of the big remaining questions in ecology, such
as those outlined by Levin [4]; for example, ‘how
does one scale from the microscopic to the macroscopic?’ In the final paper in this special issue,
Petrovskii & Petrovskaya [22] address the current
computational challenges including providing recommendations on how to deal with them. By
focusing on a number of timely and important ecological examples, including issues regarding estimating
crop pest abundances, incorporating space and scale
in models, and examining population dynamics of
pacific salmon, Petrovskii & Petrovskaya [22] identify
some key computational challenges and how they originated. A central issue identified by the authors is
that most simulation-based studies in ecology use
modelling ideas and approaches developed for application in other sciences and not specifically to
address ecological systems. Yet, ecological systems
are very different from physical and chemical systems,
notably containing more uncertainty and chaotic
dynamics, and are often influenced by stochastic processes, and ecological models must reflect this [22].
This problem is confounded by often poor accuracy
of ecological data, not due to poor sampling design
and data collection, but as a result of the complexity
of ecological interactions, making model validation difficult. Thus, computational methods in ecology need
to be developed alongside theory to account for
these issues and new computational tool kits need to
have as much rigorous testing as is applied to the original theory [22].
It is likely that these computational challenges in
ecology will continue to grow. In recent years, data
describing ecological systems have increased, reflecting
advances in automated data loggers, satellite-based
remote sensing technology and the current revolution
in ‘omics’ technologies, among other data recording
methods. A common theme of many of the papers in
this special issue is that ecologists now face the challenge of dealing with more data than has ever been
available historically, but despite this increase in
data quantity it still contains all the uncertainties,
noise and stochasticity found in all ecological systems.
This raises new challenges for theoretical ecologists as
well as empirical ecologists. For example, it is now
possible, using the latest generation of DNA sequencing
technologies,
to
quantify
communities
containing thousands of microbial species represented
by millions of individuals. Yet, developing a model,
and simulating a theoretical community representing
thousands of microbial species occupying a number
of trophic levels with biologically realistic dynamics
is at the very limit of computational ecology methods.
Interface Focus (2012)
E. A. Codling and A. J. Dumbrell
With increasing amounts of data available, ecologists
need to remain focused and continue to follow the
examples of the early pioneers of ecology and rise to
the challenge, producing theory able to match the
demands of the data. After all, without theory providing testable hypotheses, ecology could become nothing
more than data collection for its own sake. We finish
with a quote from perhaps the most famous inter-disciplinary scientist of them all:
He who loves practice without theory is like the sailor
who boards ship without a rudder and compass and
never knows where he may cast.
Leonardo da Vinci
We would like to thank the authors, reviewers and editors who
helped to create this special issue under a very tight deadline.
We are grateful to the MATE 2011 scientific committee and in
particular Sergei Petrovskii for his help and guidance. We
thank all the MATE 2011 participants and speakers for
contributing to the meeting and creating a friendly
atmosphere that allowed plenty of stimulating and exciting
discussions to take place. We also thank Hannah Lewis for
helpful comments on this introductory article. Funding for
the MATE 2011 conference was provided by the London
Mathematical Society, the British Ecological Society and
the Department of Mathematical Sciences at the University
of Essex.
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