Ecological Complexity 10 (2012) 52–68
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Ecological Complexity
journal homepage: www.elsevier.com/locate/ecocom
Original research article
Scale transition theory: Its aims, motivations and predictions
Peter Chesson
Department of Ecology and Evolutionary Biology, The University of Arizona, Tucson, AZ 85721, USA
A R T I C L E I N F O
A B S T R A C T
Article history:
Received 16 May 2011
Received in revised form 24 October 2011
Accepted 11 November 2011
Available online 23 December 2011
Scale transition theory is an approach to understanding population and community dynamics in the
presence of spatial or temporal variation in environmental factors or population densities. It focuses on
changes in the equations for population dynamics as the scale enlarges. These changes are explained in
terms of interactions between nonlinearities and variation on lower scales, and they predict the
emergence of new properties on larger scales that are not predicted by lower scale dynamics in the
absence of variation on those lower scales. These phenomena can be understood in terms of statistical
inequalities arising from the process of nonlinear averaging, which translates the rules for dynamics from
lower to higher scales. Nonlinearities in population dynamics are expressions of the fundamental biology
of the interactions between individual organisms. Variation that interacts with these nonlinearities also
involves biology fundamentally in several different ways. First, there are the aspects of biology that are
sensitive to variation in space or time. These determine which aspects of a nonlinear dynamical equation
are affected by variation, and whether different individuals or different species are sensitive to different
extents or to different aspects of variation. Second is the nature of the variation, for example, whether it is
variation in the physical environment or variation in population densities. From the interplay between
variation and nonlinearities in population dynamics, scale transition theory builds a theory of changes in
dynamics with changes in scale. In this article, the focus is on spatial variation, and the theory is
illustrated with examples relevant to the dynamics of insect communities. In these communities, one
commonly occurring nonlinear relationship is a negative exponential relationship between survival of an
organism and the densities of natural enemies or competitors. This negative exponential has a biological
origin in terms of independent actions of many individuals. The subsequent effects of spatial variation
can be represented naturally in terms of Laplace transforms and related statistical transforms to obtain
both analytical solutions and an extra level of understanding. This process allows us to analyze the
meaning and effects of aggregation of insects in space. Scale transition theory more generally, however,
does not aim to have fully analytical solutions but partial analytical solutions applicable for
circumstances too complex for full analytical solution. These partial solutions are intended to provide
a framework for understanding of numerical solutions, simulations and field studies where key
quantities can be estimated from empirical data.
ß 2011 Elsevier B.V. All rights reserved.
Keywords:
Nonlinearity
Negative binomial
Fitness-density covariance
Aggregation
Competition
Host–parasitoid dynamics
Laplace transform
1. Introduction
It is widely recognized that the highly variable environment
experienced by organisms in nature not only affects their evolution
but also profoundly influences population and community
dynamics. Since the environment directly affects the survival
and reproduction of individuals, of necessity population sizes vary
in time and space in response to this variation. Our concern,
however, is not with such obvious and immediate effects
environmental variation, but the hypothesis that such variation
also has a profound effect on larger spatial and temporal scales
than the scale on which the variation is generated and is most
evident. For example, variation in the physical environment might
E-mail address: [email protected].
1476-945X/$ – see front matter ß 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecocom.2011.11.002
cause species densities to vary in space, but on the larger spatial
scale, where this variation is not so evident, several species coexist
with one another as a consequence of the smaller scale variation
(Holt, 1984; Chesson, 1985, 2000; Amarasekare and Nisbet, 2001;
Muko and Iwasa, 2003; Snyder and Chesson, 2004).
Populations are connected both in space and time. Connection
over time comes from the first rule of biology that new organisms
are derived by birth or survival from organisms existing at earlier
times. However, most populations are spread over highly
heterogeneous areas, and movements between areas on many
time scales are the rule also. Movement of animals comes variously
from the routine activities of life within the home range, from
dispersal between habitat locations, and from long distance
migrations. Plants move on small scales by clonal growth, on
larger scales by dispersal of seeds or plant fragments, and
genetically on many scales through dispersal of pollen.
P. Chesson / Ecological Complexity 10 (2012) 52–68
Evolutionary theory considers the consequences for the
individual of movement in space, and for life in temporally
fluctuating environments (Real and Ellner, 1992; Schreiber and
Saltzman, 2009). Scale transition theory studies the consequences
for the dynamics of populations and communities of heterogeneous landscapes, and heterogeneous time. At present, there are
two parallel developments of scale transition theory, one for space
(Chesson et al., 2005) and one for time (Chesson, 2009). Though
often developed separately, they share many features, concepts
and predictions. The discussion here focuses primarily on space,
but temporal scale transition theory has been developed further
and has had much greater attention (e.g. Chesson, 1994, 2003;
Chesson and Kuang, 2010). Moreover, results from temporal scale
transition theory can often be modified to apply to the spatial case
(Chesson, 2000).
Theory of population and community dynamics mostly
proceeds in a reductionist mode. Populations and communities
consist of individual organisms, and the properties of the
individuals form the bases of the models from with population
and community dynamics are to be explained. Traditional
ecological theory considered the interactions between individuals,
both within and between species, as key explanatory features, but
ignored both spatial and temporal and heterogeneity. Variation in
time was often regarded as a disruptive factor (May, 1974).
Although variation in space was more often acknowledged as
important (Levin, 1974), it was too often assumed that a
population or community in some given small locality could be
explained by the properties of that locality alone. In scale transition
theory, the attitude is that small localities harbor subpopulations
and subcommunities of the larger populations and communities
distributed over a heterogeneous landscape (Andrewartha and
Birch, 1954). Local populations and communities are necessarily
affected by the surrounding populations through inputs from them
and losses to them. Less easy to understand are the predictions of
models that the population and community dynamics at the
landscape scale are profoundly affected by the heterogeneity
within the landscape. How this occurs is the key concern of scale
transition theory in its spatial form. In its temporal development,
scale transition theory seeks to explain long-term aspects of
population and community dynamics on the basis of shorter-term
temporal heterogeneity in dynamics (Chesson, 2009).
Whether our concern is a single species or a collection of several
interacting species (referred to as a community) key to our
discussion is the dynamics of populations. Community dynamics
are the joint population dynamics of the constituent species. A
population can be considered at multiple spatial scales. The
standard way of measuring a population is as density (numbers per
unit area), although biomass (mass per unit area) is also common.
Scale transition theory can be developed with either sort of
population measurement without change in the fundamental
concepts. Thus, it is a matter of which units are more appropriate
for the circumstances at hand. For definiteness, our development
here is in terms density. Density and biomass both have the key
property that they average as the spatial scale is changed. The total
population over a large area is the sum of the local populations
contained within it. The density, or number per unit area, on the
large area, is just the average of the densities of the local
populations. This is an average weighted by area if local
populations are defined on areas of different sizes. Regardless,
the process of getting from local densities to the landscape (largerarea) density is a simple linear process.
As a consequence of the averaging property of density, in a
simple sense, population dynamics at larger scales are averages of
dynamics at local scales (Chesson, 1998a). This fact, although
critically important, does not lead straightforwardly to predictions
at the landscape scale from understanding at the local scale. For
53
example, it is not possible to model each local population as
isolated, but depending on different parameters, or converging on
different equilibria, or fluctuating asynchronously with other local
populations, and obtain a correct prediction at the landscape scale
when the local populations are actually connected and exchanging
individuals (Chesson et al., 2005). Similarly, it is not possible to
substitute average values for the local-scale environment and
local-scale population density and accurately predict population
dynamics at higher scales (Chesson, 2009). The fact that
predictions based on the dynamics with localities on any given
scale must change as the scale is enlarged, is called the scale
transition, and scale transition theory is the attempt to understand
it.
Within population and community ecology, the concerns taken
up by scale transition theory are understanding population
persistence, population stability, joint stability of interacting
species and species coexistence. Metapopulation theory early
tackled the issue that local populations might go extinct, yet the
population persists at the landscape scale (Levins, 1969). More
subtly, the outcomes of interactions between species can be
changed by spatial heterogeneity: unstable interactions between
predators and prey, and between competitors, can become stable
interactions and species coexistence (Hassell, 2000; Bolker et al.,
2003; Chesson et al., 2005). Scale transition theory focuses on the
fact that a good deal of this explanation can be traced in a relatively
simple way to an interaction between nonlinearity in population
dynamics and heterogeneity in space. This heterogeneity can be
either in environmental factors critical to population dynamics or
in population densities. In nature, it is necessarily in both, although
models may emphasize just one of these.
When population dynamics are linear (an unlikely occurrence),
the averaging process that translates local densities into landscape-scale densities commutes with the equations for population
dynamics, and thereby renders landscape-scale level predictions
from the local-scale equations, substituting landscape-scale
variables for local-scale variables (Chesson, 1981, 1998a). More
commonly, however, dynamics are nonlinear and the averaging
process to obtain landscape-scale dynamics must take place over
nonlinear functions. It is well understood that the average of a
nonlinear function, over varying values of the argument, is
different from the nonlinear function of the average of the
argument. This fact is rendered most simply in Jensen’s inequality
where the nonlinear function in question is convex or concave
(Needham, 1993). In these cases, a precise prediction of the sign of
the inequality is possible, but inequality between averages of
functions and functions of averages applies with few exceptions
whenever the functions are nonlinear. Hence, commutativity
between averaging and dynamics does not apply in the usual case
of nonlinear dynamics.
Scale transition theory builds on these ideas showing how
changes in population and community dynamics with scale can be
understood in terms of the nature of the nonlinearities in
population dynamics and the nature of the variation that is being
averaged (Chesson et al., 2005). The task is to make the
understanding of the nonlinearities – what they are like, and
why – into biological understanding, rather than leaving them as
uninterpreted mathematics (Chesson, 1998a). I show here that this
is entirely possible. The nonlinearities in population dynamics are
expressions of the fundamental biology built into them. Scale
transition theory then creates a biological theory of scale from how
these biologically determined nonlinearities interact with spatial
heterogeneity to modify dynamics on larger scales.
Spatial heterogeneity, not just nonlinearity, is itself a biological
story if only to the extent that different spatially varying quantities
differ in their significance to populations. But populations also
affect the heterogeneity that they respond to. Spatial heterogeneity
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P. Chesson / Ecological Complexity 10 (2012) 52–68
or spatial variation has several forms and several causes. First,
there are spatially varying physical environmental factors that are
relatively independent of the populations themselves, but
nevertheless profoundly affect those populations. Examples are
atmospheric temperature, solar radiation, rainfall, and soil
properties. In reality these are not completely independent of
biology, but change in ways and on scales that are not of concern
here. Second is the nature of local population dynamics. Where
they are unstable, they generate variation either independently or
by interacting with environmental variation. Unstable local
population dynamics have the potential to amplify the effects of
environmental variation and lead to greater overall spatial
variance (Chesson, 1998a). Nonlinearities in population dynamics
can be the cause of instabilities, and in this way, they have two
separate and dependent roles in the scale transition: they interact
with variation through the nonlinear averaging process and shape
the variation that they interact with as well. However, as we shall
see, the role of nonlinearities in generating spatial pattern and in
interacting with pattern via nonlinear averaging, can, in various
useful circumstances, be partially decoupled.
It is most important to recognize that scale transition theory is
a biological theory. It is not intended to be a way of solving
models, but to explain them. Sometimes it does lead to solutions
of models, and can provide relatively simple approximate and
exact techniques, as illustrated in this article. However, its chief
value and main intention is theory that allows understanding of
population and community dynamics at landscape scales in
terms of spatially varying biological and physical properties.
Scale transition theory applies mathematics to build ecological
concepts that are useful for understanding how spatial heterogeneity affects landscape-level dynamics. Critical to these
developments is that the concepts have a high degree of
generality, and are quantitative, measuring and explaining
how dynamics change with scale. To create understanding, key
quantities derived from the theory should be presented in
functional form i.e. the formula for the quantity should explain
the concept that it quantifies. Finally, the concepts, quantities
and their formulae should be as applicable to field and
experimental data as they are to models, and so provide a route
to testing theoretical ideas (Chesson, 2008).
How scale transition has met these goals is the subject of this
article. This is done primarily by illustration using one particular
class of ecological models. A by product of the goals of scale
transition theory is a hybrid approach to ecological models that
breaks the dichotomy between solving a model analytically, versus
simulation or numerical solution. The scale transition approach
often leads to a mixture of analytical mathematics and numerics.
The study of species coexistence is where this approach is best
developed. It provides formulae that solve the model, but these
same formulae also define key concepts, including the mechanisms
of species coexistence. The formulae are in functional form, and so
quantitatively exhibit the processes behind the mechanisms. The
formulae are determined analytically and can be partly evaluated
analytically. Analytical approximation to the full formula is often
possible, but greater accuracy is obtained from numerical methods,
with minimal loss of understanding, in this hybrid approach
(Snyder and Chesson, 2004). Components requiring numerical
evaluation for sufficient accuracy are spatial variances and
covariances, but the specific roles of these variances and
covariances are determined by gross features such as life-history
traits. This hybrid approach leads to far greater understanding than
is available by numerics alone, and demonstrates the power of
scale transition theory. Moreover, the same features that allow
extra understanding from simulation facilitate translation of the
results into field measurable quantities (Sears and Chesson, 2007;
Chesson, 2008).
Scale transition theory intersects with moment closure
approaches to solving spatial models (Bolker and Pacala, 1999),
but the focus is different. In scale transition theory, the focus is
understanding, not solution of models. Scale transition theory is
not about better ways of performing calculations, but about how to
develop ecological theory where quantitative concepts become
ecological concepts. In working with ecological models, there is
often an emphasis on directly solving the model, or determining its
behavior. Scale transition theory is about quantitative concepts
that explain the behavior of ecological systems. Many other
sciences have such concepts. Physics naturally provides numerous
examples in the areas of statistical mechanics, but of perhaps more
interest are the concepts in the sister fields of population genetics
and quantitative genetics, with numerous quantitative concepts
forming the basis of the subject and providing tests of ideas in
nature (Hamilton, 2009). Although ecology is rich in quantitative
concepts (Legendre and Legendre, 1998), they are not as integrated
with the dynamical models and theory of the subject as in these
other fields. I hope the developments discussed here demonstrate
that there is potential for rich integration of quantitative concepts,
dynamical models and theory in ecology too.
2. Illustrated outline of the theory
Scale transition theory can be developed in discrete or
continuous time and discrete or continuous space (Chesson
et al., 2005). The developments and concepts are similar and the
choices between them relate to the question of which approach is
more suitable to answer the question at hand. The simplest
approach for illustration of the fundamentals is discrete time and
discrete space, and we will follow that here, basing this outline on
Chesson et al. (2005). See Chesson et al. (2005) and Melbourne and
Chesson (2006) for continuous time developments and Snyder and
Chesson (2004) for continuous space.
The discrete-time development begins with the fitness, lx;t , of
an individual organism as a function of its spatial location, x, and
the time t. The quantity lx;t is defined as the expected
contribution of an individual of a given species at location x at
time t to the population on the landscape at time t + 1. The easiest
way to think about this is to imagine that an individual at x and
time t experiences the environment at x (including both physical
conditions and biological conditions) applying during the time
interval t to t + 1, and may as a consequence reproduce or die,
according to a probability distribution determined by x and t.
During that same interval, the individual in question, or its
offspring, may disperse to other locations. These dispersing
individuals are counted in lx;t , but they experience the effects
of other locations during dispersal. In subscripting lx;t merely by x
and t, the assumption is made that the probably distribution
defining where dispersing individuals may go, and what they
may experience on the way, is fixed by x and t. This assumption
means that lx;t is potentially a function of the physical and
biological environments of all locations on the landscape.
However, in the illustrations presented here, the environment
on x at t is all that must be known for lx;t to have a well-defined
value.
The definition of lx;t means that if N x;t is the population density
at location x and time t, the expected output from locality x one
time unit later is
N 0x;tþ1 ¼ lx;t N x;t :
(1)
It is important to note that this expected output, N 0x;tþ1 , is not in fact
the local population size at time t + 1 for two reasons. First, and
most important, individuals disperse from and to the local site x,
and so the population at x reflects immigration to x and emigration
P. Chesson / Ecological Complexity 10 (2012) 52–68
from x, which N0x;tþ1 does not. In general, lx;t (and hence N0x;tþ1 )
must account for mortality that takes place during dispersal: it
includes surviving dispersers from x that are at other locations at
time t, but does not include dispersers from x that die during
dispersal. Second, the actual output from site x will deviate from
the expected output by chance due to demographic stochasticity
and other randomly varying factors that are not accounted for in
the formula for lx;t , and so the expected output is not the same as
the actual output. In general, lx;t depends on species, and is a
function of environmental characteristics of the local site, as is
illustrated below.
The fitness, lx;t , defines the local scale dynamics through Eq. (1).
Dynamics at the larger or landscape scale are given by the equation
N̄tþ1 ¼ l̃t N̄t ;
(2)
where N̄t is the landscape-level population density: the total
population on the landscape divided by the area. Here, the
assumption is made that dynamics at the landscape scale are
closed, and output at the landscape scale at time t + 1 is also the
next input. The landscape level fitness, l̃t , is implicitly defined by
the equation
N̄
l̃t ¼ tþ1 :
N̄t
(3)
However, it is more naturally defined as
P
P 0
Nx;tþ1
x N x;tþ1
¼ P
N
x;t
x
x N x;t
l̃t ¼ Px
(4)
or the ratio of the total output at time t + 1 to the total input on the
landscape at time t. The sums are over all spatial locations, and are
naturally replaced by integrals in formulations where Nx is a
density function in continuous space rather than an actual local
density (Snyder and Chesson, 2004). It is also worth noting that Nx
could be a stochastic point process (Isham, 1981). To accommodate
all of these situations, the sums would simply be interpreted as the
appropriate Lebesgue–Stieltjes integral with respect to a random
measure, Nx.
In equating the two ratios in (4) we are explicitly recognizing
the fact that the outputs at time t + 1 are merely redistributed on
the landscape, and so the total output is equal to the total of the
inputs at time t + 1. Note that also, by equating the two ratios,
demographic stochasticity is assumed to have averaged out over
space: N0 is the conditional expected output, not the actual output.
Averaging over enough space means that expected and actual are
the same. To be correct in the model, this outcome requires an
infinite landscape, and the ratios in (4) are then defined as a limit as
the total area involved in the sum goes to infinity. The same
process is necessary to define the landscape-level densities in
Eq. (3). Such technical issues need not concern us here. We shall
simply assume that the landscape is large enough that all three
ratios in (3) and (4) can be equated.
Following from these preliminaries, we can now use formula (1)
for the output from the patch to see that the landscape-level
fitness, which defines landscape-level population dynamics by
Eq. (2), can be written as a weighted average over the landscape of
the local-scale fitness, lx;t . To do this, we define the local relative
density,
vx;t ¼
N x;t
;
N̄t
(5)
of the population at location x. In terms of local relative density, the
landscape-level fitness becomes
l̃t ¼ lt vt ;
(6)
55
where the bar on the right means the spatial average over the
implicit spatial coordinates x of the product lx;t vx;t . This result is
simply derived by noting from Eq. (1) that
lx;t vx;t ¼
lx;t N x;t
N̄t
¼
N 0x;tþ1
N̄t
:
As N0x;tþ1 averages to N tþ1 in space, the spatial average of the RHS
above is l̃t by Eq. (3), while the LHS is lt vt , giving (6). Note that vx;t
averages to 1 over space, which means that the average (6) can be
regarded as a weighted average with weights being the relative
densities, vx;t .
It is at this point where scale transition theory starts to draw
some conclusions. A product is a nonlinear function of the
arguments of the product. It is in fact a quadratic nonlinearity.
The average of a product of two variables is simply the product of
the averages plus the covariance between these two variables (see
box 12.2 of Chesson et al., 2005), and since nx,t averages to 1 over
space, we see that Eq. (6) becomes
l̃t ¼ l̄t þ covðlt ; vt Þ:
(7)
Thus, the landscape-level fitness, l̃, differs from the spatial-average
fitness, l̄, by an amount equal to the spatial covariance between
local fitness and local relative density, i.e. fitness-density covariance
(Chesson et al., 2005). Simply put, landscape-level fitness would be
increased if a population tended to be concentrated in areas of higher
fitness, and decreased if it were concentrated in areas of lower
fitness. This equation has some similarities with the Price Equation
of quantitative genetics (Frank, 1995), but the application here is
quite different, despite superficial similarities.
2.1. The role of fitness-density covariance
It is now very simple to see how biological factors come into
play affecting distributions on the landscape relevant to fitness.
The theoretical idea of an ideal-free distribution says that species
should distribute themselves on landscapes in such a way that
fitness is equal everywhere (Cressman and Krivan, 2006).
Individuals seek places with higher fitness, but as density builds
up in favorable places, fitness is reduced by intraspecific
competition, and so tends to be evened out everywhere. As a
consequence, fitness-density covariance would be zero under an
ideal-free distribution. Then spatial-average fitness and landscapelevel fitness would be the same. However, individuals rarely have
such perfect freedom of movement in relation to fitness, and
fitness can be constrained by many factors preventing changes in
local density from evening out its value.
Two cases are of particular importance in understanding
fitness-density covariance. First in the extreme case where the
physical environment is homogeneous everywhere, and the
species’ fitness is merely a function of its own local density,
chance variation in local density leads to nonzero fitness-density
covariance. If species experience net intraspecific competition,
then fitness-density covariance is negative. Thus, fitness-density
covariance slows population growth and leads to a lower carrying
capacity on a landscape, as first recognized many years ago by
Lloyd (1967). On the other hand, social factors might lead to an
increasing relationship between local density and fitness (Allee
effects, Amarasekare, 1998a,b), at least for some ranges of
densities, and so cause positive fitness-density covariance,
elevating landscape-level fitness (Chesson, 1998b).
The second simple case to consider is where the physical
environment or other species vary in space, but the fitness of the
species in question is independent of its own density. The case
where a species is independent of its own density is critical in
analyses of community dynamics as the limit of the case where the
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P. Chesson / Ecological Complexity 10 (2012) 52–68
species has low abundance everywhere on the landscape (Chesson,
2000). The question is whether the species has a tendency to
recover from such low density situations and thus persist in the
system. In this case, it is a reasonable expectation that fitnessdensity covariance will be positive. First, if a species can actively
select habitats, then it is likely to become concentrated in locations
where its fitness is higher, while its absolute density, as opposed to
its relative density, remains low everywhere. Second, many
species, especially plants, have leptokurtic dispersal kernels,
meaning that the kernel peaks at no dispersal (Clark et al.,
2001; Chesson and Lee, 2005). Thus, in many cases, significant local
retention of individuals is to be expected. Over time, such retention
means that a population builds up in relative terms in locations
where fitness is highest (Chesson, 2000; Snyder and Chesson,
2003).
These considerations suggest that species that are at uniformly
low absolute densities on a landscape will often benefit from
positive fitness-density covariance. However, as a species becomes
more abundant and experiences intraspecific competition, its
ability to benefit from positive fitness-density covariance will
likely become reduced because then local fitness is reduced by
local-population buildup, weakening fitness-density covariance
and perhaps even driving it to negative values (Chesson, 2000;
Snyder and Chesson, 2003). As noted above, an ideal free
distribution potentially applying in these circumstances would
give zero fitness-density covariance. We shall see later that such
changes in fitness-density covariance figure importantly in species
coexistence mechanisms.
2.2. Nonlinearities in the fitness function
So far we have considered simply a natural and unavoidable
bilinear nonlinearity: the product of fitness and relative density,
which appears in Eq. (6) for the landscape-level fitness, and leads to
the fitness-density covariance term in Eq. (7). However, the first term
in Eq. (7), viz the spatial average fitness, l̄t , has a role in the scale
transition because local fitness is commonly a nonlinear function of
various spatially varying quantities, including density. In the
notation of Chesson et al. (2005) fitness can be expressed generically
as a function of a vector of spatially varying fitness factors:
lx;t ¼ f ðWx;t Þ;
(8)
where Wx;t is a multidimensional fitness factor defining quantitatively everything of importance at location x and time t to the
fitness of individuals of the species in question. In the simplest
cases in models, Wx;t consists of population densities, both of the
given species and of other species. Realistically, it should also
contain physical environmental factors, for example, as developed
in Snyder and Chesson (2004) and Chesson et al. (2005). Because of
the nonlinearity of f, the spatial average of Wx;t will in general fail
to predict, l̄t i.e.
l̄t ¼ f ðWt Þ 6¼ f ðWt Þ;
(9)
causing deviations from the prediction of the lower scale model (8)
when extrapolated to the landscape scale.
It is important to emphasize that the inequality (9) depends
critically on the nonlinearity of the function f. To understand why,
it is worthwhile recalling that a linear function has the
fundamental property that
p f ðw1 Þ þ q f ðw2 Þ ¼ f ð pw1 þ qw2 Þ;
(10)
for any scalars p and q and any two values w1 and w2 of the
argument of f. If we assume here that p and q are nonnegative
numbers summing to 1, they define a probability distribution on
the two points w1 and w2. In this present context, this means that
over the landscape, Wx;t takes on just these two values, and with
relative frequencies p and q. The property (10) for linear functions
means that for distributions on two points,
f ðWt Þ ¼ f ðWt Þ:
(11)
Generalizing this result to arbitrary probability distributions is a
straightforward matter of an iteration to demonstrate it for all
finite distributions, and a limiting process to generalize it to all
probability distributions (Billingsley, 1995). Most important, this
same process proves strict inequality in (9) for two broad cases of
nonlinearity, namely strictly convex and concave functions. Such
functions are defined by the properties
p f ðw1 Þ þ q f ðw2 Þ > f ð pw1 þ qw2 Þ ðconvexÞ and
(12a)
p f ðw1 Þ þ q f ðw2 Þ < f ð pw1 þ qw2 Þ ðconcaveÞ:
(12b)
For differentiable functions of a single variable, convex functions
are simply those functions, such as f(w) = w2, with a positive
second derivative, and concave functions are those with a negative
second derivative (Bertsekas et al., 2002). For continuous functions
f, these inequalities also lead to the statements
f ðWt Þ > f ðWt Þ ðconvexÞ and
(13a)
f ðWt Þ < f ðWt Þ ðconcaveÞ;
(13b)
as can be seen by following the same procedure as described for the
proof of (11) in the linear case. The inequalities (13) are known as
Jensen’s inequality (Needham, 1993). In general, a nonlinear
function may be convex over part of its domain and concave over
other parts. This means that the inequality (9) changes from
greater than to less than as the distribution of Wx;t shifts from
convex to concave parts of the domain of f, with the potential for
complex effects of spatial variation on the dynamics of a spatial
model.
Concrete examples of the dynamical implications of the
inequality (9) are conveniently illustrated by several ecological
models in which f is a negative exponential function of a local
population density (potentially a different species) or a linear
function of the local densities of several species (Table 1). These
densities or linear functions of densities serve as W x;t . For example,
in the single-species Ricker model, W x;t is just the species’ own
local density (De Jong, 1979; Ives and May, 1985); in the
Nicholson–Bailey host–parasitoid model, W x;t for the host is the
parasitoid density, P x;t (Hassell, 2000); and for the multispecies
Ricker model, W x;t is a linear function of the densities of several
species in the same guild, including the given species (Ives and
May, 1985; Ives, 1988). These examples are all covered by the
generic local fitness function
lx;t ¼ l0 eaW x;t :
(14)
For this fitness function, l̄t is always greater than f ðW t Þ. A negative
exponential leads to severe declines in fitness as a function of the
Table 1
Various models with a negative exponential fitness function.
Model
Wx,t: fitness factor
Nicholson–Bailey
Ricker
Multispecies Ricker
Px,t: parasitoid density
Nx,t: intraspecific density
b1N1,x,t + b2N2,x,t: a linear combination
of both species’ densities
For the Nicholson–Bailey model, local the parasitoid output is the local number of
hosts killed: P 0x;tþ1 ¼ N x;t ð1 eaPx;t Þ; thus, study of host fitness is sufficient to
understand the scale transition for host–parasitoid system as a whole. For the
a
multispecies Ricker, the species have unique a’s, and l0’s. Thus, li,x,t = li,0 e iWx,t.
P. Chesson / Ecological Complexity 10 (2012) 52–68
57
Table 2
Probability distributions for fitness factors, W.
Probability distribution
Probability density function, fW(w)
Gamma with constant k
ðk=WÞ
G ðkÞ
Gamma with k ¼ uW
uuW uW1 ewu ; w > 0
G ðkÞ w
Poisson
eW ðWÞ
w!
Negative binomial with constant k
G ðkþwÞ
ðW=kÞ
G ðkÞw! ð1þW=kÞwþk ; w ¼ 0; 1; 2; . . .
Negative binomial with k ¼ uW
G ðuWþwÞ
uw
; w ¼ 0; 1; 2; . . .
G ðuWÞw! ð1þu1 Þwþk
k
w
k1
wk=W
e
; w>0
w
Laplace transform of W, wW(u)
k
1 þ Wu
k
Laplace transform of vðWÞ; ’vðWÞ ðuÞ
k
1 þ uk
1 þ uu
uW
u Þ
w
u Þ
1 þ Wð1e
k
1 þ ð1eu
uW
u
uW
u=W
eWð1e
; w ¼ 0; 1; 2; . . .
1þ
k
uW
u Þ
Þ
eWð1e
k
u=W
Þ
1 þ Wð1ek
uW
u=W
1 þ ð1e u Þ
For continuous distributions, the integral of the probability density over a set of possible values gives the probability of that set of values. For discrete distributions, the
R1
integral is replaced by a sum, and the probability density has the elementary interpretation PðW ¼ wÞ ¼ f W ðwÞ. The Laplace transform is ’W ðuÞ ¼ 0 euw f W ðwÞdw; with the
integral replaced by a sum in the discrete case.
fitness factor, often creating instabilities and chaos in population
dynamics (May and Oster, 1976; Hassell, 2000). Moderation of this
exponential decline at the landscape scale through the scale
transition can have enormous effects on dynamical stability and
the outcomes of species interactions, as we shall see below.
Negative exponentials arise naturally as biological models for
the survival of an organism in the face of numerous small
independent opportunities for death. The Nicholson–Bailey host
parasitoid model provides an example (Nicholson and Bailey,
1935). If we think of Px;t as the actual number of parasitoids present
locally, rather than their local density, and exp(a) as the
probability of evading parasitism by any individual parasitoid,
then exp ðaPx;t Þ is simply the probability of surviving parasitism:
it is the product of the independent probabilities of evading each
individual parasitoid. Competition, both intraspecific and interspecific, in some circumstances may work similarly (Appendix A).
Negative exponentials lend themselves to at least partial
analytical solution of the scale transition because averaging a
negative exponential yields the Laplace transform of the argument,
which is known for many standard probability distributions (Table
2). Detailed discussion of Laplace transforms and their applications
to scale transition theory is given in Section 3, below. Here, we give
the results of applying Laplace transforms to illustrate scale
transition theory for the models in Table 1. Fundamentally, the
Laplace transform of a spatially varying quantity, Wx, is the spatial
average
’W x ðuÞ ¼ euW x ;
(15)
where the bar over the entire expression on the right means the
average of the exponential on the right over all locations, x. The
quantity u is just any positive number, and the average on the right
is just a function of u. Hence the notation ’W x ðuÞ. In terms of the
fitness function (14), we see that
l̄t ¼ l0 ’W x;t ðaÞ:
(16)
So with a negative exponential fitness function, the Laplace
transform describes the effect of the fitness at the landscape scale
as function of the coefficient a defining its per unit effect. The
Laplace transform thus allows us to calculate this average fitness if
the probability distribution defining the spatial variation has a
known form. Using the Laplace transforms in Table 2, Table 3 gives
the value of l̄t for the specific case of the Nicholson–Bailey host–
parasitoid model. Of most note is that in some cases, the negative
exponential is replaced by a negative power of a linear function,
while in other cases, a negative exponential is retained, but with a
lower coefficient for the per unit effect of parasitoids. The
corresponding formulae for l̄t for the Ricker model are obtained
by simply substituting N̄t for P̄t in Table 3. However, for the
Nicholson–Bailey model, it is possible for l̄t to equal l̃t and
therefore define the full landscape-scale dynamics of the system.
For this to occur, fitness-density covariance must be zero, which
means here that the degree of parasitism found at a locality must
not be correlated with the host density there. In nature, this
situation is common, although both negative and positive
correlations do occur too (Cronin and Strong, 1990).
The scale transition, being an interaction between nonlinearities
and spatial variation, naturally depends not just on the nonlinear
fitness function, here exemplified by the negative exponential, but
also on the nature of the variation. For this purpose, the relevant
aspects of variation are summarized by the probability distribution
defining the frequencies with which various values of the fitness
factor, W x;t , are encountered as the spatial landscape is explored. For
W x;t ð¼ P x;t Þ in the Nicholson–Bailey model, May (1978) proposed a
gamma distribution, with a fixed value for the shape parameter, k,
but a mean equal to P̄t . With this assumption, spatial variation has a
very strong effect on the dynamics of the host–parasitoid interaction. As shown in Table 3, the negative exponential, applying at the
local scale, is replaced by a negative power of a linear function at the
landscape scale. Highly patchy parasitoid distributions, as are often
seen in nature, mean that k is small (Pacala and Hassell, 1991). As a
consequence, this negative power implies that the landscape-level
fitness function, l̄t , is much milder than the local scale fitness
function. In particular, when k < 1, the highly unstable dynamics at
the local scale give way to a stable equilibrium at the landscape scale.
However, such stability does not arise with all distributions of Px,t
(Table 3). Indeed, although it is always true that the landscape-level
fitness function is milder than the local-scale fitness function, a
negative exponential relationship between l̄t and P̄t is retained at
the landscape scale in some cases, and as a consequence the
instability of the Nicholson–Bailey model under spatially homogeneous conditions is retained with spatial heterogeneity too. These
two different outcomes result from different assumptions about the
probability distribution for the parasitoid’s dispersion in space. Why
are these outcomes so different?
Table 3
The Nicholson–Bailey host–parasitoid model with various probability distributions
for Px,t.
Distribution of Px,t
l̄t
Gamma with constant k
l0 1 þ akP̄t
Gamma with k ¼ uP̄t
l0 eau P̄t
Poisson
l0 ea P̄t
Negative binomial with constant k
l0 1 þ akP̄t
Negative binomial with k ¼ uP̄t
l0 eau P̄t
au ¼ u lnð1 þ a=uÞ; a0 ¼ 1 ea ; a0u ¼ u lnð1 þ a0 =uÞ.
k
0
0
0
k
58
P. Chesson / Ecological Complexity 10 (2012) 52–68
When P x;t is assumed to have a gamma distribution with
constant shape parameter, k, the relative density of the parasitoids,
vðPx;t Þ ¼ P x;t =P̄t , is invariant over time, and other distributions that
have this same property give essentially the same results (Chesson
and Murdoch, 1986). This invariance is a particular biological model
for how parasitoids disperse (see Section 3, below). It is most
consistent with parasitoids cuing into, or being constrained by
features of the environment, unaffected by their own density. They
must have high mobility on the spatially varying landscape so that
dispersal from their natal location does not affect their ability to cue
into or be constrained by environmental features. This means that
the scale of environmental variation needs to be comparable to, or
shorter than, the dispersal distance. This assumption is not likely to
remain true over a very large area in nature, and so is applicable only
to relatively small landscapes (Hassell, 2000). The gamma distribution is a continuous distribution, which means strictly speaking that
it cannot model actual numbers of parasitoids at a location. Instead,
it is often interpreted as the amount of parasitoid-time there.
Parasitoids are assumed to visit several to many locations in their
lives, and are counted fractionally at each location visited. Most
important, however, is that parasitoids do not choose patches
independently of one another because all parasitoids are responding
to the same environmental factors. This lack of independence is the
reason behind the deviation of the landscale-level fitness function
from exponential (Appendix A).
Almost the same landscale-level fitness function is possible
with a discrete distribution of parasitoids too, viz, the negative
binomial distribution. With a constant k, in this case the
‘‘clumping parameter’’, a parasitoid would visit only one spatial
location in its life, but the process of finding it would be identical
to that described above for the gamma with a constant k (Section
3, below). Again, the dependence between individual parasitoids,
induced by their common responses to spatially varying
environmental conditions, causes the landscape-level fitness
function to depart from the exponential form, with dramatic
effects on population dynamics. On the other hand, independence
between parasitoids is retained for the Poisson distribution and
also when k is proportional to P̄t ðk ¼ uP̄t Þ in the gamma and
negative binomial distributions (Section 3, below). As a consequence, an exponential fitness function is retained at the
landscape scale (Appendix A).
While parasitism is assumed to be uncorrelated with host
density, the actual distribution of hosts in space, and indeed the
nature of their dispersal from one patch to another, are all irrelevant
to the outcome. The reason is very simply the nature of the fitness
function for the Nicholson–Bailey model, which expresses no effect
of the host’s own density on individual fitness. Any departure from
this assumption or the assumption of a lack of correlation between
parasitism and local host density, would mean that host distribution
in space would become an issue. In particular, the dynamics of the
system would not be determined by l̄t because l̄t would not be
equal to l̃t : fitness-density covariance would be nonzero. For the
Nicholson–Bailey model, these issues have been discussed at length
elsewhere (Chesson and Murdoch, 1986; Hassell et al., 1991;
Chesson et al., 2005). However, we illustrate the effects of fitnessdensity covariance with the Ricker and multispecies Ricker models
in the next subsection.
2.3. Combining nonlinear fitness and fitness-density covariance
For the case of the Ricker model and the multispecies Ricker
model, the fitness function depends on intraspecific density. Thus,
fitness-density covariance is commonly nonzero and so l̃t does not
equal l̄t . For exponential fitness functions, l̃t is available from the
derivative of the Laplace transform of the fitness factor, as
explained in Section 3, below. In particular, when W x;t ¼ Nx;t , as in
the Ricker model,
l̃t ¼
l0 ’0Nx;t ðaÞ
N̄t
(17)
:
Solution of the multispecies Ricker model is a straightforward
extension of this idea (Section 3), but for the case where the
different competing species are distributed independently in
space, the landscape-level fitness for any species i can be found
simply as the product of the Ricker solution and the Laplace
transforms for other species, to give
l̃it ¼
li0 ’0Ni;x;t ðaii Þ Y
N̄i;t
’N j;x;t ðai j Þ;
(18)
j 6¼ i
where ai j ¼ ai b j .
Formulae (17) and (18) are evaluated for the negative binomial
distribution for two contrasting situations, viz for case for which
the parameters k remain constant, and the case for which they are
proportional to the mean abundance of the species on the
landscape (Table 4). As explained for the case of dispersing
parasitoids, constant k means that all individuals of species
disperse in relation to environment features, which causes
correlations between dispersing individuals.
When k is proportional to the mean ðk ¼ uW t ; ki ¼ u i N̄i;t Þ;
dispersing organisms do so independently. Both the Ricker and
multispecies Ricker are applicable to particular sorts of insects,
especially certain groups that lay their eggs in patches of organic
matter, such as leaves, fruit, patches of dung, mushrooms, or small
dead animals (De Jong, 1979; Atkinson and Shorrocks, 1981; Ives,
1991). In this particular situation, we can think of the different
spatial locations as being as food patches. As these disappear and
are renewed, their properties, which define the environment of a
locality, vary simultaneously in time and space. This being the case,
the environment encountered by a dispersing individual need not
be related to the environment of its natal patch, even if dispersal
distance is short. Thus, in this case, ignoring distance effects in
dispersal is reasonably justified.
The case where k is proportional to the density of the organisms
has the special interpretation that each dispersing organism lays
eggs in batches in the food patches that it visits (Section 3, below).
In general, these batches vary in size, but the same kind of results
(although not a negative binomial distribution) occur if they do
not. The input, N x;t , at food patch x, is defined as the number of eggs
of the species there, and consists of the sum of the batches of eggs
produced by females visiting that patch. Outputs, N0x;tþ1 , are
numbers of eggs produced by females born in that patch. The
females disperse randomly, and so female visits to a patch have a
Poisson distribution. Arrival at a patch is independent between
females, but individual eggs in the same batch are not independent, as they are correlated by being delivered to the same location.
The independence that is present at the level of the dispersing
females imposes structure on the probability distribution of N x;t
that leads to the retention of exponential fitness functions in that
case (Table 4).
2.3.1. The Ricker model
The Ricker model (Table 1) is a discrete-time analogue of
logistic population growth (Appendix A) and thus represents
density-dependent dynamics of a single species (May and Oster,
1976). Due to the fact that the fitness factor is the local density of
the organism itself, fitness-density covariance necessarily arises if
there is any spatial variation in this factor. The simplest way to
understand the role of fitness-density covariance is not through
the absolute value of that quantity (the difference l̃t l̄t ), but
P. Chesson / Ecological Complexity 10 (2012) 52–68
59
Table 4
Ricker and multispecies Ricker models.
l̃t
Distribution
Model
Ricker
Multispecies Ricker
l̃t =l̄t
Negative binomial, constant k
k1
0
l 1 þ a kN̄t
1
0
ea 1 þ a kN̄t
Negative binomial, k ¼ uN̄
l00u eau N̄t
Negative binomial, constant kl
ki 1 a0 N̄ j;t k j
a0 N̄
l0i0 1 þ iik i;t
1 þ i jk
0
0
0
i
0
i0u
Negative binomial, kl ¼ ul N̄l
l
j
a0 N̄ a0 N̄
e iiu i;t i ju j;t
ea
ð1þa0 =uÞ
1
a0 N̄
eaii 1 þ iik it
i
eaii
ð1þa0 =ui Þ
ii
Parameters as in Table 3 with additional parameters a0il ¼ 1 eail ; a0ilu ¼ ul lnð1 þ a0il =ul Þ; l ¼ i; j; l0i0 ¼ li0 eaii ; l0i0u ¼ li0 eaii =ð1 þ a0ii =ui Þ for the multispecies Ricker,
with subscripts, i, l merely absent for the Ricker model.
through the ratio, l̃t =l̄t , given in Table 4. Note that
l̃t
1
lx;t Nx;t
1 ¼ covðlx;t ; vx;t Þ ¼ cov
;
:
l̄t
l̄t
l̄t N̄t
(19)
Thus, the ratio l̃t =l̄t simply gives the magnitude of fitness-density
covariance relative to spatial average fitness. It gives the relative
effect of fitness-density covariance on population growth, with
values equal to 1 meaning no fitness-density covariance. This
relative form indicates proportionately how much the phenomenon of fitness-density covariance affects the landscape-level
fitness.
For the case of a negative binomial with constant k, there is a
strong effect of fitness-density covariance: l̃t =l̄t declines strongly
with N̄t (Table 4). The net result is that landscape density at time
t + 1 ðN̄tþ1 ¼ N̄t l̃t Þ, is always a hump-shaped function of N̄t . In
contrast N̄t l̄t is a monotonic function of N̄t whenever k 1.
Fundamentally, variation in density in space means that individuals experience a broad range of conditions, moderating the
behavior of l̄t , but this outcome for l̄t is partly counteracted in
l̃t by fitness-density covariance. Fitness-density covariance can be
seen as taking account of the fact that more individuals are in highdensity locations than low-density locations in comparison to the
frequency of high- and low-density locations on the landscape
(Lloyd, 1967).
When k is proportional N̄t , not only is still an exponential
function of N̄t , but fitness-density covariance is no longer a
function of N̄t . This fact is quite easy to understand in terms of the
origin of the negative binomial with k proportional to N̄t in terms of
batches of individuals. Since ovipositing females disperse independently of one another and at random, the location of an
individual egg is only correlated with other eggs in the same batch
from which it is derived. An increase in density increases the
expected number of batches of individuals arriving at any location,
but since batch size does not change with density, a given egg is not
correlated in location with a greater number of other individuals.
Relative fitness-density covariance is then independent of density,
N̄t , on the landscape. In contrast, when k is constant, all individuals
are correlated in where they are likely to be found. Fitness-density
covariance therefore has stronger relative effects at higher density.
May and Oster (1976), noted that Ricker model is prone to
unstable dynamics, and even chaos, whenever the maximum value
of l, viz l0, is large. In contrast, sufficient patchiness in the
distribution of the organisms in space can dramatically change this
outcome (De Jong, 1979; Ives and May, 1985; Chesson, 1998a). This
effect is very strong in the case of a negative binomial with a
constant k because the exponential fitness function is replaced by a
negative power of a linear function at the landscape scale.
However, in the case of a negative binomial with k proportional
to the mean, some stabilization of the dynamics still occurs due to
the reduction in value of l0 from the local to the landscape scale
(Chesson, 1998a), an effect that is entirely due to fitness-density
covariance from the laying of eggs in batches (Table 4). This model
has been studied also by Hastings and Higgins (1994) showing how
limited dispersal (an issue not addressed here) interacts with
unstable population dynamics, creating a patchy distribution in
space. The outcome is changes in landscape dynamics through the
scale transition. Although the authors make the point that the
patchy distribution of organisms in space may take a long time to
stabilize, landscape-level dynamics are still much stabler than
predicted by the local dynamical equation in the absence of spatial
variation.
2.3.2. Multispecies Ricker
The multispecies Ricker is a model of competitive interactions
between species, and indeed is a natural discrete-time analogue of
Lotka–Volterra competition. In Table 1, the multispecies Ricker in a
variable environment is represented as having a single fitness
factor for each species, which is a linear function of the densities of
each species. Alternatively, the multispecies Ricker might be
represented as has having a vector-valued fitness factor
Wx ¼ ðN1;x;t ; N2;x;t Þ. This latter situation is more convenient here
because statistical independence of the components of this vector
is an important case to consider. The fitness function can be
written as
li;x;t ¼ li;0 eaii Ni;x ai j N j;x :
(20)
The formula for W x;t in Table 1 corresponds to ai j ¼ ai b j . This case
is important because it defines a situation where the species are
limited by a common competitive factor (i.e., W x;t in Table 1 does
not have a species subscript), and in this case, coexistence is
impossible without spatial variation. This fact is easily verified by
the relationship
ln l1;t
a1
ln l2;t
a2
¼
ln l1;0
a1
ln l2;0
a2
;
(21)
applying in this case. This fixed linear relationship between the
ln l’s means that whichever species has the larger value of ln l=a
will exclude the other (Levin, 1970; Chesson and Huntly, 1997) in
the absence of spatial variation.
As in the Ricker model, a negative binomial has commonly been
used to specify spatial variation, and most commonly, N1;x;t and
N2;x;t are assumed to be distributed independently in space. This is
of course a strong assumption, and Ives (1988) has studied the case
where the distributions are correlated. Fundamentally, however, it
is the degree to which species are different in their spatial
distributions that leads to coexistence, and for the purposes of
illustration here, the independent case is sufficient. The results for
l̃t for independent negative binomials are given in Table 4.
Ives and May (1985) and Ives (1988) sought to understand the
hypothesis of Atkinson and Shorrocks (1981) that independent
aggregation of organisms in space promotes their coexistence. This
idea has been controversial in terms of the kind of aggregation that
is sufficient for coexistence (Hartley and Shorrocks, 2002).
Independent negative binomial variation with a constant k has
60
P. Chesson / Ecological Complexity 10 (2012) 52–68
long been known to have strong effects on species coexistence.
Then, l̃t splits into a product of negative powers (Table 4),
reflecting this independent aggregation of the two species to
environmental conditions. That coexistence is promoted is well
known and not controversial (Ives and May, 1985; Green, 1986).
For simplicity, consider just the case where a single species would
come to a stable equilibrium, which is guaranteed when the l0’s
are not too large (May and Oster, 1976). Then, invasion analysis
(Kang and Chesson, 2010) can be used to demonstrate stable
coexistence. Using the formula in Table 4, it can be seen that when
species j is at its single-species equilibrium and species i is at zero
density, species i can increase (i.e. l̃i;t > 1) when
l0i0 > 1 a0i j a0i j 0 1=ðk j þ1Þ
þ
l
a0j j a0j j j0
!k j
:
(22)
The simplest way to see how coexistence is promoted is to
consider the case where all the a’s are the same. In this case, an
individual of any species added to a locality causes exactly the
same proportionate reduction in local fitness, lx;t , at that locality,
regardless of species. In a constant environment, the species with
the larger value of l0 then excludes the other, but in the presence
of spatial variation, condition (22) for invasion implies two-species
stable coexistence whenever
1þ
1
k2
1
<
r 010
1
< 1þ
;
0
k1
r 20
(23)
0
where r 010 ¼ ln l10 . Small values of the k’s naturally mean that the
coexistence region is broad in terms of the species differences in r 00
values compatible with coexistence. This outcome should not be
surprising because this case means that the species are highly
intraspecifically aggregated due to spatially varying environmental factors that the two species respond to independently,
enhancing competition within species relative to competition
between species when the whole landscape is considered. The full
expression (22) shows that coexistence is promoted quite
generally in this way, but it becomes very simple to understand
when the r 00 values are small, because then (22) reduces to the
following approximate condition
r 0i0 > r 0j0
0
k j ai j
;
0
kj þ 1 ajj
(24)
which rearranges to
a0i j =r 0i0
1
<1 þ :
kj
a0j j =r 0j0
(25)
When this is true for both (i,j) = (1,2) and (2,1), the two species
coexist stably. Note that with k j ¼ 1; there is no spatial
environmental variation and the species are distributed in space
according to independent Poisson distributions. The case of a
completely even distribution in space – or just a single homogenous patch – simply removes the primes from expression (25) with
k j ¼ 1. Fundamentally, the a=r ratios in (25) measure interspecific
and intraspecific competition relativized to the maximum per
capita growth rate r, comparing interspecific competition, j on i,
with intraspecific competition, j on j. When this is less than 1,
stable coexistence occurs in the absence of partitioning of spatial
environmental variation. However, spatial partitioning ðk j < 1Þ,
allows coexistence to occur with interspecific competition
exceeding intraspecific competition. There is no fundamental
change in the concept that stable competitive coexistence requires
intraspecific competition to exceed interspecific competition, but
on a larger scale: it does not occur within individual patches, but on
the landscape as a whole due to the fact that the two species tend
to be concentrated in different localities.
Note that Eq. (25) produces the essentially the same result as a
continuous-time, discrete-space development of the effects of
spatial aggregation presented in Chesson et al. (2005) as Eq.
(12.20), although Chesson et al. (2005) is more general in
accounting for correlations between species in their dispersal.
The promotion of coexistence in Chesson et al. (2005), and in the
discrete-time model considered here with k independent of
density, can be attributed to fitness-density covariance. Here, if
the conditions for invasion are calculated using l̄t instead of l̃t , the
1/k terms in both (23) and (25) disappear and the primes on the r’s
vanish as well. The result is that conditions (23) do not allow
coexistence in the absence of fitness-density covariance, and
conditions (25) are the same as those applying in the absence of
spatial variation.
The case of constant k considered above has the natural
interpretation that the organisms partition the environmental
spatially. When k is proportional to density on the landscape, the
outcome is very different. Most striking is that spatial variation has
no effect on the form of the equations at the landscape scale—only
the parameters of these equations are affected by spatial variation.
The coexistence conditions for this case are variations on
conditions (25), but do not require the assumption that the r 00
are small. The exact result is simply
a0i ju =r 0i0u
< 1;
a0j ju =r 0j0u
ði; jÞ ¼ ð1; 2Þ and ð2; 1Þ;
(26)
where the r’s are the natural logs of the corresponding l’s. Without
the primes and u subscript, this is just the condition for coexistence
in the absence of spatial variation. How are the quantities on the
LHS of (26) affected by spatial variation? The first thing to note is
that for small values of the a’s, this condition just reverts to the
case with no spatial variation – all the primes and the subscript u
disappear from (26). Only with very strong competition, such that
the individuals present in a single batch of eggs substantially
depress the success of other individuals, is there any effect of
spatial variation. This outcome is in strong contrast to (23) and
(25), for the case of constant k, where the key effect of spatial
variation, as measured by k, is independent of the local population
size. Second, in the case where the u’s are the same and the a’s are
all the same, the LHS of (26) is satisfied if and only if r j =r i < 1, which
is of course impossible for both (i,j) = (1,2) and (2,1), and
coexistence is impossible, again in strict contrast to the case of
constant k where the effects of spatial variation remain strong
regardless of such conditions.
Although unequal u’s or a’s in the condition (26) do affect
whether or not it will be satisfied, the primed and u subscripted
parameters in (26) are monotonic transformations of the
parameters from the nonspatial situation. Although in some
instances the transformed parameters might enter the coexistence
region, to each of these instances are others where the transformed
parameters leave the coexistence region. In the absence of
tradeoffs that mean entering the coexistence region is more likely
than leaving it, such spatial variation can hardly be claimed to
promote coexistence. Fundamentally, when followed for a full
generation, this mechanism has no general tendency to concentrate intraspecific competition relative to interspecific competition
(Ives, 1991).
A particular focus in the literature has been the case where a
tradeoff leads to a smaller u value (larger clutch size) for a stronger
competitor (Heard and Remer, 1997). When individuals are more
tolerant of competition, a plausible scenario is that they tend to be
dispersed in larger batches. Although not exactly the formulation
of Heard and Remer (1997), when ai j ¼ ai b j , coexistence is not
possible under spatially homogeneous conditions, but two species
coexistence is possible for some ranges of r0 values when eggs are
P. Chesson / Ecological Complexity 10 (2012) 52–68
dispersed in batches and a smaller u i corresponds to a smaller ai
(larger clutch size means lower sensitivity to competition).
However, it is not clear how important this scenario might be in
nature, and it seems quite unlikely that it could lead to coexistence
of more than two species. Moreover, for this mechanism to be of
any importance at all, there must be enough individuals in a typical
batch of eggs for strong competition to result.
61
3.2. The cumulant generating function and closeness of l̄t to
exponential
To study ln l̄t we take the natural log of the Laplace transform.
That natural log is the cumulant generating function, with a
negative sign substituted in the argument (Johnson et al., 1992).
Specifically, for any random variable X, the cumulant generating
function is
3. Insights to the scale transition from Laplace transforms
cX ðuÞ ¼ ln ’X ðuÞ:
Laplace transforms are an important tool in understanding the
scale transition. Although their clearest applications is to models
with exponential fitness functions, they can also be helpful in other
cases too (Ives and May, 1985). The sections above give results
from applying Laplace transforms. In this section, these results are
justified, and further insights are derived. For these purposes, it is
useful to introduce some formal probability-theoretic notation.
We treat W x;t as a random variable, which is easily done by simply
thinking of x as randomly chosen from all spatial locations. This
process makes the location, x, a random variable, and all locations
then become equivalent to each other in a probabilistic sense,
simplifying the derivations. To begin with, we can equate spatial
average values with expected values, for example, N̄t ¼ E½N x;t , and
of course W t ¼ E½W x;t .
3.1. Basic use of the Laplace transform
The Laplace transform of W x;t is by definition the function ’W x;t
given by the formula
’W x;t ðuÞ ¼ E½euW x;t ;
(27)
and the relevant probability distribution for evaluating this
expectation is the distribution giving the relative frequency of
the different values of W x;t over all locations on the landscape.
Direct application of the Laplace transform allows l̄t to be
calculated in the case of exponential fitness functions (Table 2).
However, more insight is available using the Laplace transform of
the relativized variable vðW x;t Þ ¼ W x;t =W t , which is a generalization of relative density, vx;t ¼ N x;t =N̄t . It thus expresses the
magnitude of the fitness factor, W x;t , relative to its average value
on the landscape, providing the information remaining on W x;t
after its spatial average has been specified. With this definition,
landscape-level and local-level information are given separately in
W t and vðW x;t Þ respectively.
In terms of the Laplace transform of vðW x;t Þ, the exponential
fitness function (14) gives the spatial average fitness,
l̄t ¼ l0 E½eaW t vðW x;t Þ ¼ l0 ’vðW x;t Þ ðaW t Þ:
(28)
Thus, the argument of the Laplace transform depends on W t
(landscape-level information), while the Laplace transform, i.e. the
function ’vðW x;t Þ , depends on the local-level information. Although,
a clean separation of two levels of information appears to have
been achieved, often ’vðW x;t Þ depends on W t through its parameters, not just its argument, as we shall see below. Nevertheless, we
can now ask the critical question of how different (28) is from a
negative exponential in W t . In other words, to what extent does the
scale transition modify the functional form of the relationship
between spatial average fitness and the fitness factor at the
landscape scale? This question can be rephrased in terms of how
different from linear is the relationship between ln l̄t and W t .
Ultimately, of course, we want to understand the relationship
between ln l̃t and W t , but we can approach this question first
through study of ln l̄t .
(29)
Eq. (28) can now be written,
l̄t ¼ l0 ecvðW x;t Þ ðaW t Þ :
(30)
In the absence of spatial variation, cvðW x;t Þ reduces to the identity
function, reproducing the local scale relationship, and therefore to
define the scale transition here, we ask how different cvðW x;t Þ is
from the identity function. Going further, we ask how different
cvðW x;t Þ ðaW t Þ is from a linear function, because this difference
defines important qualitative aspects of the scale transition,
fundamentally affecting system behavior in the kinds of models
that we have studied.
The Taylor expansion of the cumulant generating function
specifies the cumulants of the distribution (Johnson et al., 1992),
which are characteristics like moments, but of more value for
understanding the properties of the distribution. The first two
cumulants are equal to the mean and variance of the distribution,
and every cumulant is a polynomial function of the moments. The
cumulants beyond the first two are zero for the case of the normal
(Gaussian) distribution. Thus, these cumulants measure various
kinds of deviation from normality. The expansion to the level of the
first two cumulants, is instructive, however:
1
2
cX ðuÞ ¼ uE½X þ varðXÞu2 þ oðu2 Þ;
(31)
for any random variable X with a finite variance (Johnson et al
1992). The combination of the first two terms on the RHS is the
exact cumulant generating function for the normal distribution.
Using this expansion, we can immediately gain a reasonable idea of
how l̄t deviates from l0 expðaW t Þ:
n
o
l̄t ¼ l0 exp cvðW x;t Þ ðaW t Þ
1
2
2
¼ l0 exp aW t þ x2t ðaW t Þ þ o½ðaW t Þ ;
2
(32)
where x2t ¼ varðvðW x;t ÞÞ or equivalently, the squared coefficient of
variation of W x;t (CV2 in the host–parasitoid literature, Hassell,
2000). Thus, the variance in space adds a positive squared term in
aW t , counteracting the simple exponential decline for the
situation with no spatial variation.
If the fitness factor were indeed normal with a constant value of
x2, then expression (32) would be exact without the approximation term, demonstrating a dramatic effect of spatial variation
because then (32) would be nonmonotonic in W t , declining in W t
at small values of this variable, but ultimately very strongly
increasing in W t . The reason is that at large values of W t both
negative and positive values of W x;t would be larger in magnitude
due to the assumption of a constant value of x2, which would mean
that the spread of values of W x;t would be fixed in relation to the
mean. As W t increased in magnitude, the increase in the negative
values of W x;t would begin to dominate the behavior of l̄t , with the
positive values converging on zero contributions to l̄t . Thus, l̄t
would ultimately increase in W t .
62
P. Chesson / Ecological Complexity 10 (2012) 52–68
3.3. Evaluating l̃t with transforms
3.4. Spatial variation, Laplace transforms, and the scale transition
Spatial average fitness, l̄t , is generally not the same as the
fitness at the landscape scale, l̃t , the difference being fitnessdensity covariance. In the Ricker model, there is no doubt that l̄t
and l̃t are different. In that model, landscape-level fitness is
The normal distribution was used above to illustrate how l̄t
might be affected by spatial variation. Although it is not a
distribution of spatial variation commonly arising in applications,
there is no reason why it might not be a model for the effects of a
physical environmental factor in some circumstances. More
commonly in models, W x;t represents a population density or
linear function of population densities (Table 1). In that case, the
assumption that the distribution of vðW x;t Þ does not vary with time
is a particular model of how organisms are distributed in space
(Chesson, 1998a; Chesson et al., 2005). For instance, in the host–
parasitoid model with a mobile parasitoid, and W x;t ¼ P x;t
(parasitoid density), a fixed probability distribution for distribution of vðW x;t Þ corresponds to host-density-independent heterogeneity of parasitism (Hassell, 2000), whose effects have been
studied by various authors (Bailey et al., 1962; May, 1978; Chesson
and Murdoch, 1986; Reeve et al., 1989; Hassell et al., 1991).
In the host–parasitoid model, a fixed probability distribution for
vðP x;t Þ is plausible if the relative amount of time parasitoids spend
at a particular spatial location, x, depends on properties of the
physical environment that do not change with population
densities. Fundamentally, if individual parasitoids do not interact
with each other in the process of searching for hosts, and are
sufficiently mobile that they can access local sites with a broad
range of physical environmental conditions, then the relative
density of parasitoid visits to a site, vðP x;t Þ ¼ P x;t =P̄x;t , should not
depend on the absolute density of parasitoids. Because it is the
physical environmental conditions rather than host density that
determines the time parasitoids spend at a location, constancy of
the probability distribution of vðPx;t Þ is the natural outcome.
Although parasitism rates that depend on local host densities fit
intuition, in fact in many cases in nature, parasitism rates are
independent of host-density (Cronin and Strong, 1990), justifying
focus on this case. Moreover, this case seems to have the most
important effects on the outcome of the host–parasitoid interaction at the landscape scale (Hassell, 2000).
In this case, the approximation (32) applies with a constant x2
(variance of relative parasitoid density vðP x;t Þ). Thus, for small aP̄t
there is a quadratic correction to the linear term in the exponent
for the landscape-level fitness. However, for larger aP̄t , higher
order terms prevent nonmonotonicity of the landscape-level
fitness in aP̄t . It is easy to see this directly from Eq. (28) where,
under the assumptions here, the Laplace transform ’vðPx;t Þ is a fixed
function (does not change with time or P̄t ) and is necessarily
monotonic in its argument when vðP x;t Þ is nonnegative. The exact
result is given for the gamma distribution with shape parameter k
in Table 3. It serves usefully for illustration here. In the gamma
distribution, x2 = 1/k, but in the exact formula the negative
exponential is replaced by a negative power of a linear function
parasitoid density. In terms of the cumulant generating function,
this case gives
l̃t ¼ l0 E½vx;t eavx;t N̄ ;
(33)
which is the average in space of the fitness function (14) weighted
by the relative density, vx;t ¼ Nx;t =N̄t . As it is always possible to
differentiate a Laplace transform, and, moreover, exchange the
order of differentiation and expectation (Feller, 1971), differentiating the middle expression in (28) with respect to aW t shows that
expression (33) is equal to
l̃t ¼ l0 ’0vx;t ðaN̄t Þ;
(34)
where the prime means derivative. (This equation was written
above in terms of ’Nx;t as Eq. (17), which is derived similarly.)
Although easy to calculate, expression (34) does not have strong
intuitive content. However, the first derivative of the cumulant
generating function can be used because of the relationship
c0vx;t ðuÞ ¼ ’0vx;t ðuÞ=’vx;t ðuÞ, which means that (34) can be
rewritten as
l̃t ¼ l̄t c0vx;t ðaN̄Þ;
(35)
and thus, we see that the ratio l̃t =l̄t , which specifies relative
fitness-density covariance by Eq. (19), can be written as the
derivative of the cumulant generating function:
l̃t
0
¼ cvx;t ðaN̄t Þ:
l̄t
(36)
Critical to whether l̃t remains exponential in N̄t is whether
c0vx;t ðaN̄t Þ is constant in N̄t . Expression (32) above shows that this
is impossible if the distribution of vx;t does not change over time,
0
for then the variance of vx;t will be constant, and cvx;t ðaN̄t Þ must
be approximately quadratic in N̄t for small N̄t . However, for the
Poisson and the negative binomial with k proportion to the mean,
the distribution of vx;t changes over time in a manner that makes
expression (36) a constant: the parameters of the distribution
depend on N̄t . Expression (36) is not equal to 1 in these cases
because fitness-density covariance is present, but relative fitness
density-covariance is independent of population density (Table 4).
We shall see how such situations arise in the next subsection.
Expression (36) generalizes beyond the Ricker equation by
using the appropriate multivariate cumulant generating function
and differentiating it with respect to the argument involving the
relative density of the species whose l̃t is being evaluated. For
example, with the two-species Ricker model, we use the joint
cumulant generating function of vi;x;t and v j;x;t to give
l̃i;t ¼ l̄i;t cð1;0Þ
vi;x;t ;v j;x;t ðaii N̄i;t ; ai j N̄ j;t Þ;
(37)
where the superscript (1,0) means the partial derivative with
respect to the first argument. The relationship (36) for relative
fitness-density covariance clearly generalizes to this case. Importantly, species j will only contribute to relative fitness-density
covariance to the extent that species i and species j are distributed
dependently in space. When they are independent, the joint
cumulant generating function is simply the sum of the separate
generating functions, and the partial derivative in (37) is just the
derivative of the cumulant generating function of vi;x;t alone. This is
evident in Table 4 where relative fitness-density covariance for
species i in the two-species Ricker is independent of species j and is
in fact identical to the value in the single-species Ricker model.
cvðPx;t Þ ðaP̄t Þ ¼ k ln 1 þ
aP̄t
k
;
(38)
and expanding ln in terms of aP̄t yields the exponent in Eq. (32) as
it must. For small k, i.e. large x2, the resulting l̄t is a much milder
function of P̄t than a negative exponential, greatly reducing the
magnitude of population fluctuations over those of the Nicholson–
Bailey model. Indeed, for x2 > 1 they convert divergent oscillations
to a stable equilibrium point (Bailey et al., 1962; May, 1978).
Critical to the outcome here is the constancy of the variance x2
of relative parasitoid density as the landscape-scale parasitoid
density changes. Alternative assumptions can give strikingly
different results. Rather than assume that all parasitoids respond
P. Chesson / Ecological Complexity 10 (2012) 52–68
identically to environmental conditions, we can make the opposite
assumption that the amount of time that an individual parasitoid
spends in a patch is independent between parasitoid individuals.
When this is the case, var(Px,t) is proportional to P̄t , and
furthermore, x2 is inversely proportional to, P̄t , x2t ¼ 1=uP̄t , say,
and then the first two terms in the exponent of l̄t in Eq. (32) are
aP̄t ð1 a=2uÞ. Thus, a negative exponential is retained at the
landscape scale, but at a reduced rate. However, we can be more
precise about this, because it is not just variances that are additive
over sums of independent variables, but cumulant generating
functions too. This additive property means that
cPx;t ðuÞ ¼ c1 ðuÞP̄t ;
(39)
where c1 is the cumulant generating function for the case P̄t ¼ 1.
Now, cvðPx;t Þ ðuÞ ¼ cPx;t ðu=P̄t Þ, which means that
cvðPx;t Þ ðaP̄t Þ ¼ c1 ðaÞP̄t :
(40)
Hence, this additivity of cumulant generating functions over
independent variables means that l̄t indeed remains a negative
exponential in P̄t , but with a replaced by c1 ðaÞ, a reduction in
magnitude, but a negative exponential nevertheless. In particular,
at the landscape scale, the model remains the Nicholson–Bailey
model. The unstable parasitoid and host equilibria are increased by
this change as they are both inversely proportional to a, but the
stability properties of the system are completely unaffected, as the
dynamics of host and parasitoid densities relative to their
equilibrium densities are unaffected by a, and hence unaffected
by the change from a to c1 ðaÞ from the local to landscape scale.
For Eq. (39) to make sense regardless of the landscape density of
parasitoids, P̄t , the probability distribution of P x;t has to be
infinitely divisible (Billingsley, 1995), in other words, no matter
how small P̄t is made on the RHS of (39), the equation still defines
the cumulant generating function of a probability distribution. Not
all distributions have this property, but, as we shall see below,
there are natural biological conditions that lead to it. The gamma
distribution is infinitely divisible (Johnson et al., 1994), and so it
can be used in this case as well as in the situation discussed above
where the distribution of parasitoid relative density, vðPx;t Þ, is fixed
as P̄t changes. The parameter k, which is inverse to the variance,
must then be proportional to P̄t . In Table 2, the gamma distribution
is reparameterized for this situation by the substitution k ¼ uW
(k ¼ uP̄t in the present example). As in the general case defined by
Eq. (39), the patchiness of the parasitoid in space, as measured by
x2 , is inversely proportional its density, P̄t , on the landscape, and
patchiness increases as abundance decreases.
Greater patchiness at lower abundance makes sense under the
biological scenario assumed here because the patchiness arises
from independent individual behavior, which is then averaged
over increasing numbers of individuals as their density increases.
Thus, P x;t is relatively less variable, as measured by x2 , the more
parasitoids there are contributing to it. The reduced relative
variation in x2 means also that the nonlinear averaging of the
exponential fitness function (14) becomes relatively less severe as
the density of parasitoids increases, naturally explaining why the
scale transition is less dramatic in this case. However, there is also a
more fundamental explanation. As discussed above, the exponential fitness function arises from the net outcome of the independent activities of individual players, for example, parasitoids.
Patchiness can still arise because individual parasitoids behave
nonrandomly in space, but when these nonrandom patterns are
independent between individuals, the exponential fitness function
for the hosts is retained at the landscape scale because even at that
scale it represents the outcome of independent behavior affecting
host survival multiplicatively (Appendix A).
63
The examples discussed so far treat the fitness factor, W x;t , as a
continuous variable, which is quite reasonable in the case of the
host–parasitoid model when it reflects the amount of time
parasitoids spend at a spatial location. However, often this fitness
factor will be discrete because it is the actual number of organisms
present at that locality. In the case of the host–parasitoid model,
this could be because the parasitoids are mobile for a short
dispersal period, but then are confined to one locality. If individual
parasitoids select locations independently and at random, then this
discrete variable is the Poisson distribution (Table 2). The Poisson
distribution is infinitely divisible, which can be understood from
its common derivation as the collective outcome of the independent actions of an essentially infinite number of players, each with
a small probability of being present at any one location (Johnson
et al., 1992). As expected from the development above, l̄t does
indeed remain exponential in this case, in other words the
Nicholson–Bailey model and its predictions of an unstable
interaction remain true in this case (Table 3).
From this Poisson beginning, we can add the postulate that
locations vary in their attractiveness to organisms, given some
local environmental factor, Ux, say. For parasitoids, this means that
Px;t would be conditionally Poisson, given Ux, which for the sake of
argument we can assume has mean 1. The conditional Laplace
transform is the Poisson Laplace transform
u Þ
eU x P̄t ð1e
:
(41)
The expected value of this conditional Laplace transform is the
Laplace transform of the actual (unconditional) distribution of P x;t ,
which is a Poisson mixture distribution in general (Johnson et al.,
1992), but a negative binomial distribution in the particular case
where Ux has a gamma distribution (Johnson et al., 1992). The
parameter k reflects the variation in the physical environment,
which we might assume to be time invariant, and the negative
binomial inherits this parameter as the so-called clumping
parameter (Table 2). The principal effect of this negative binomial
distribution on l̄t is through the gamma component (Table 3)
converting the negative exponential to a negative power of a linear
function. However, the value of a is changed by the Poisson
component from a to the value a0 ¼ 1 ea . Note, however, it is
the value of k that determines stability, as it does with the gamma
distribution, but in this case, x2 ¼ 1=k þ 1=P̄t , and so x2 is not
completely independent of time. Stability is determined therefore
not by x2 this case, but by the component, 1/k, of x2 that reflects
the common response of the parasitoids to their environment, with
the Poisson component having no effect on whether the system
will be stable (Hassell et al., 1991). The Poisson component, by
changing a to a0 does reduce the rate of parasitism at the landscape
scale, but this effect is compensated for by higher parasitoid
abundance at equilibrium.
In the models in Section 2, above, we considered also the
negative binomial with k ¼ uW t . This is possible because the
negative binomial, like the Poisson, is an infinitely divisible
distribution. This situation is most reasonable for insects laying
eggs in ephemeral food patches, as described for the Ricker and
multispecies Ricker models in Section 2, above. In this case, visits of
egg-laying females, of a given species, to a patch, Lx;t have a Poisson
distribution. If the sizes of these are B1;x;t ; B2;x;t ; . . . ; for batches,
1,2,. . ., then the number of eggs laid by that given species is
Nx;t ¼
Lx;t
X
Bl;x;t :
(42)
l¼1
Assuming that the Bl;x;t have Laplace transform ’B and mean mB ,
the Laplace transform of N x;t given Lx;t is
½’B ðuÞLx;t ;
(43)
64
P. Chesson / Ecological Complexity 10 (2012) 52–68
because the Laplace transform of a sum of independent random
variables is simply the product of their Laplace transforms. The
unconditional Laplace transform is the expected value of (43),
which is obtained by using ln ’B ðuÞ as the argument of the
Poisson Laplace transform (Table 2), to give
eN̄t ð1’B ðuÞÞ=mB :
(44)
Note from (42) that the expected value of the Poisson random
variable Lx;t has to be N̄t =mB . The probability distribution of N x;t is
clearly infinitely divisible, a property that it inherits from the
Poisson variable Lx;t . In the case where the B’s have a log series
distribution (Johnson et al., 1992), with Laplace transform
’B ðuÞ ¼
lnððð1 eu Þ=ð1 þ u ÞÞ þ ðu=ð1 þ uÞÞÞ
;
lnðu=ð1 þ u ÞÞ
(45)
the negative binomial with k ¼ uN̄t results. This distribution gives
a mean clutch size mB according to the formula
1
mB ¼ E½Bl;x;t ¼ ½u lnð1 þ u1 Þ ;
(46)
which is a decreasing function of u. As it must, this infinitely
divisible case simply gives back a negative exponential fitness
function at the landscape scale once more.
4. Discussion
Scale transition theory seeks to understand how the equations
for population dynamics change with the spatial or temporal scale.
It seeks to understand how the phenomena predicted by equations
applying on a small scale are changed as the scale changes. The
only reason that such changes occur is because population
densities or environmental variables vary from one unit of space
or time to another on any given scale. However, such variation in
these fundamental variables is not enough for changes in dynamics
and predictions beyond those available directly by substituting
average values of local scale variables into the equations for local
scale dynamics. For substantive changes, the dynamical equations
must be nonlinear. Nonlinearity is common, and so substantive
changes are expected to be common. It is simply a tautology that
these changes stem from an interaction between nonlinear
dynamics and variation. However, like many scientific tautologies,
there is much be learned by examining the details.
Here we have focused on the spatial form of the scale transition
and have used nonlinearities defined by exponential fitness
functions as our chief example. These are not arbitrary functions
but have a biological origin, as we have seen here as a model of
independent risks. Their exponential form means that they lend
themselves to analysis by means of Laplace transforms, allowing a
detailed understanding of the interaction of this form of
nonlinearity and spatial variation. Despite their relative simplicity,
exponential fitness functions have a rich interaction with spatial
variation. For example, we have seen how some forms of spatial
variation interact with the exponential fitness function in a way
that retains the independent risk property at the landscape scale.
These forms arise from independent variation at the level of the
individual organism. As a consequence, an exponential fitness
function emerges at the landscape scale too. The exponents of the
fitness functions on the landscape and local scales are different, but
many qualitative predictions, such stability of dynamics, or species
coexistence, are unaffected or little affected. On the other hand,
when a varying physical environment has common effects on the
different individuals of a species, the landscape-level fitness
function is profoundly different from the local-scale fitness
function. An intermediate situation occurs when the organisms
disperse in batches, and individuals within the one batch go to the
same location. This leads to limited correlations between
individuals. The fitness function at the landscape scale retains
its exponential form due to an assumption of independent
dispersal of batches of organisms, but the correlation between
individuals within a batch leads to stronger quantitative effects on
this exponential fitness function.
Our analysis of when an exponential fitness function is retained
at the landscape scale illustrates an essential feature of the scale
transition. It is an interaction between the nonlinearities and the
variation. This means that the nature of the variation is just as
important to the scale transition as is the nature of the
nonlinearities (Chesson, 1998a,b). Laplace transforms, and their
relatives, cumulant generating functions, provide efficient ways of
characterizing the properties of spatial variation of importance to
the scale transition, especially when combined with exponential
fitness functions. Here they have also characterized critical
properties of fitness-density covariance defining when coexistence
of competitors will be promoted by spatial variation. Using the
joint properties of the exponential fitness function and the
properties of spatial variation, and applying Laplace transforms
as a key tool, we have explored several standard models to sharpen
insights on their behavior. In particular, we have illustrated when
the host–parasitoid interaction should be stabilized by spatial
variation in searching behavior of parasitoids, and when instead its
exponential fitness function retains the exponential property at
the landscape scale, and remains unstable despite spatial variation.
These insights are then extended to the Ricker model, where
fitness-density covariance has important effects that are not
always present in the host–parasitoid model.
In addition, we have reexamined the controversy about when
aggregation of competing insects in space will allow them to
coexist. Two different types of negative binomial distribution arise,
with quite different effects on species coexistence. The first of
these (constant k) implies that species aggregate independently in
relation to features of the environment, and in the second (k
proportional to the mean), species aggregate intraspecifically, but
not interspecifically because individual ovipositing females lay
their eggs in batches. Our analysis shows quite clearly that the first
form of aggregation powerfully promotes species coexistence,
while the second can only do so when local competition is strong
and when species with larger clutch sizes have lower sensitivity to
competition. Its general application remains in doubt, as discussed
in more detail below.
An important concept in scale transition theory that features
strongly in these developments is fitness-density covariance
where organisms are nonrandomly distributed relative to variation
in fitness in space. Although this phenomenon is to be expected
commonly, in some important host–parasitoid models it is absent
due the dependence of host fitness on parasitoid density alone and
the lack of a relationship between parasitoid density and host
density (host-density independent parasitism). In that case, the
landscape-level fitness, l̃t , reduces to the spatial average fitness,
l̄t . The study of the effects of spatial variation on l̄t has led to the
concept of aggregation of risk (Chesson and Murdoch, 1986;
Hassell, 2000), where variation from host to host in the risk of
parasitism is critical to stability of the host–parasitoid interaction
on the landscape scale. Risk to an individual is measured as the
value of aP x;t applicable to it compared with the average of this
value across all spatial locations. In the developments here, we
have seen how correlations between individual parasitoids in their
dispersal is critical to transforming the exponential fitness
function of the hosts into a much stabler negative power of a
linear function.
The host–parasitoid models considered here have no immediate density feedback to the host organisms. Instead, density
feedback occurs through the parasitoid on a multi-generation
timescale. Direct aggregation of parasitoids to locations of high
P. Chesson / Ecological Complexity 10 (2012) 52–68
host density (Hassell et al., 1991) would change that, as would
correlations between host and parasitoid dispersion in space due
to aggregation to common environmental factors (Reeve et al.,
1989). In such instances fitness-density covariance does emerge at
the landscape scale in host–parasitoid models (Chesson et al.,
2005). However, with the Ricker model of density-dependent
dynamics, variation in the species own density in space of
necessity leads to fitness-density covariance. It is a critical
component of the landscape-level fitness, and has the effect in
the Ricker model of reducing the magnitude of the variation that
individual organisms experience relative to the variation present
on the landscape as a whole. This occurs simply because dense
localities have more individuals to experience the conditions in
those places. The variation in density measured by choosing
locations at random is very different from the variation that is
measured by choosing individual organisms at random because
individual organisms of necessity are not found at random relative
to their own density (Lloyd, 1967).
The effect of fitness-density covariance in the single-species
Ricker model, however, is nowhere near as striking as it is in the
multispecies Ricker model. Then the density feed back associated
with fitness-density covariance provides a very strong distinction
between resident and invader states of a population, and thus the
ability of a species to recover from low density. This case also
highlights a critical distinction between different sorts of spatial
variation: does it result from the independent action of ovipositing
females, which cause dependence in their offspring by laying their
eggs in batches, or does it result from females seeking or being
entrained by properties of the environment that are spatially
variable? In the both cases, the different species can be distributed
in space independently as negative binomials, but only in the case
where the environment is the cause of their spatial patchiness is
coexistence generally promoted. There are limited cases where
coexistence might result from batch laying due to density
dependence that comes from a single batch of eggs, and the
limited fitness-density covariance that it produces, but the
emphasis on this case appears misplaced, as it appears only able
to allow coexistence of two species, not multiple species. Although
we have studied competition between species using the exponential fitness function, a variety of different models and approaches
have produced similar results, likely due to the fact that most of the
effects of aggregation on coexistence hinge strongly on the
properties of fitness-density covariance, not on an exponential
fitness function (Ives and May, 1985; Kretzschmar and Adler, 1993;
Heard and Remer, 1997; Hartley and Shorrocks, 2002).
The analysis here highlights some issues with empirical tests of
the sufficiency of observed spatial patterns of species for
coexistence. Ives (1991) and Sevenster (1996) have both developed statistical methods, which are applied to distributions of
species in the field to see if they are more strongly aggregated
intraspecifically than interspecifically. These methods are capable
of distinguishing independent negative binomials from completely dependent negative binomials. However, they do not generally
examine the change in the distribution with changing regional
densities of the species, and so cannot distinguish between
negative binomials with constant k values and those with k
proportional to the mean. Thus, they cannot in fact test whether
there is sufficiently spatial segregation of the species for
coexistence. Ives (1991) makes the point that these measures
should be applied not to eggs or larvae but to aggregating females.
This procedure certainly greatly lessens the problems that might
arise with incorrectly counting aggregation from egg-laying in
batches, but nevertheless it is not robust in the face of
idiosyncratic of behavior of individual females (the ‘‘personalities’’ that individual organisms, even insects, are increasingly
being shown to have, Cote et al., 2010), perhaps making multiple
65
visits to a patch and mimicking true species-level aggregation.
Part of the scale transition theory program is robust tests of
coexistence mechanisms, allowing discrimination between variation that promotes coexistence, and that which does not, as
discussed below.
The specific examples of the scale transition program discussed
here have been restricted to relatively simple movement processes
where most organisms disperse during every time period, and the
properties of the locations to which they disperse are independent
of the properties of the locations from which they dispersed. This
outcome does not require that all of the landscape be equally
accessible to all individuals within one unit of time, but just that an
unbiased sample of the full range of variation present on a
landscape be accessible to a dispersing individual (Comins and
Noble, 1985). This property is a restriction on the scale of variation,
not the spatial extent of the system in question, although as
landscape size increases, the scale of variation is likely to increase
too (Lavorel et al., 1993). Different scales of variation allow more
complex development of spatial pattern (Levin, 1992; Hassell,
2000), but the fundamental themes of interactions between spatial
variation and nonlinearity are retained (Reeve, 1988; Durrett and
Levin, 1994; Bolker and Pacala, 1999; Hassell, 2000; Bolker et al.,
2003; Snyder and Chesson, 2004).
In these spatially more elaborate models, full analytical solution
in general is impossible. However, scale transition theory retains
an important role. This is true also in developments that have not
explicitly used the scale transition framework, but have implicitly
done so (Bolker and Pacala, 1999). Following a scale-transition
approach, measures of nonlinearity are defined analytically, and
are combined with measures of spatial variation to define the
change in dynamics with the change in scale, in other words the
scale transition. The best example in the present article is simply
Eq. (7) which shows how spatial variation causes l̃t to differ from
l̄t . This equation applies generally, regardless of the relative scales
of variation and dispersal. The measure of variation applying in this
case is simply fitness-density covariance. What is not so evident is
the measure of nonlinearity that multiples fitness-density covariance in expression (7) because it is equal to 1. However, it is has the
2
elaborate formula @ lx;t vx;t =@lx;t @vx;t ; which comes from treating
the quantity lt;x vx;t (averaged in space to obtain l̃t ) as a function of
the two spatially varying quantities lx;t and vx;t . In this case, the
nonlinearity measure is trivial, but it is there nonetheless. As we
have seen, fitness-density covariance can have a critical role in
species coexistence, which is quantified by a measure denoted Dk
and compares cov(l, v) between resident and invader states
(Chesson, 2000; Snyder and Chesson, 2004; Chesson et al., 2005).
This quantity Dk then defines how much the differences between
resident and invader fitness-density covariance contribute to the
value of l̃t for a species in the invader state.
In the study of species coexistence, fitness-density covariance
captures one aspect of spatial partitioning. Another aspect is
called the spatial storage effect, and results from the phenomenon that a spatial location where individuals perform well will
also experience higher demand for resources (Chesson, 2000;
Snyder and Chesson, 2004; Chesson et al., 2005). This
phenomenon is measured as a covariance across spatial
locations between the direct response of a species to the
physical environment, and the competition that it experiences
there. Its importance to coexistence is then measured by a
quantity DI that compares the change in environment-competition covariance between invader and residents states, multiplied
by a nonlinearity measure defining the importance of the
covariance in a given setting. A full expression for the invader l̃t
is made up of measures of various mechanisms affecting species
coexistence that arise from scale transition theory (Chesson,
2000; Snyder and Chesson, 2004; Miller and Chesson, 2009).
66
P. Chesson / Ecological Complexity 10 (2012) 52–68
Often the covariances are not available analytically, or by
satisfactory analytical approximations, but they can be evaluated numerically or by simulation (Snyder and Chesson, 2004),
which means that understanding from the overall framework of
scale transition theory is available even though specific
quantities cannot be determined by analytical means.
An important feature of this approach to understanding
mechanisms is that the measures of mechanism magnitude are
field measureable, which leads to strong approaches to testing the
mechanisms (Sears and Chesson, 2007; Chesson, 2008; Chesson
et al., 2011). The idea is to show that a quantity critical to the
functioning of the mechanism, such as fitness-density covariance,
or environment-competition covariance, changes between invader
and resident states. Other approaches use a model to infer what the
change between resident and invader states should be, quantifying
the mechanism from field data without directly studying resident
and invader states (Angert et al., 2009; Chesson et al., 2011). These
other approaches are only as good as the assumptions of the model
and risk some of the same kinds of issues discussed above on the
comparison of aggregation between versus within species
(Chesson, 2008; Siepielski and McPeek, 2010).
Scale transition theory has naturally led to field quantification in
other areas also such as density-dependent population dynamics of
individual species in a patchy environment (Melbourne and
Chesson, 2006), and also measurement of the critical squared
coefficient of variation (x2 ) for assessing its effects on the stability of
host–parasitoid interactions (Pacala and Hassell, 1991). In general,
direct density manipulations have not been involved, and so these
measurements stand only as indicating the potential for spatial
variation to have the purported effects on landscape scale dynamics.
Practical methods including density manipulations are needed in
these areas for strong tests of scale transition theory.
Although this article has been illustrated with models that are
solved analytically, as discussed in this last section, it goes well
beyond analytical approaches alone, and the intention is not the
solution of models as such. With modern computing power,
solution of models is not a limiting step in ecology, but
understanding is and always will be. The agenda of the scale
transition program is to develop and test ecological theory for the
changes in dynamics and predictions that occur with a change in
scale. Naturally, carrying out this agenda does involve solution of
models, but the critical issue is how those models are solved. The
scale transition program has identified the interaction between
variation and nonlinear dynamics as key, and has set about
understanding this interaction with the aim of producing
ecological concepts and associated quantities to produce quantitative ecological theory designed to be tested rigorously in nature.
Acknowledgments
I am grateful for comments on the manuscript by Andrew
Morozov, and anonymous reviewer. This work was supported by
NSF Grant DEB-0542991.
Appendix A. The origin and conservation of exponential fitness
functions
A.1. Derivation of the Ricker model
Whenever the individuals of a given species independently
harm a given individual of either the same or a different species by
a constant multiplicative amount, a negative exponential form for
their total effect on that individual’s fitness results, as presented in
the text for the effects of parasitoids on host survival. The case for
independent harm is less easy to countenance when W x;t is
intraspecific density. For example, if the harm caused is death, then
the decline in density over time as individuals die would seem to
preclude independence. The idea that density dependence is time
delayed, as it often is, can rescue the independence argument,
however. Specifically, the Ricker model can be derived from the
following equation:
1 dN tþu
¼ rðuÞ aðuÞNt ;
N tþu du
(A1)
where t counts discrete-time, such as a generation or year, and u is
time within the year or generation. Time dependence of r and a
simply allows development, competition and reproduction to take
place at time varying rates within the generation or year.
Integrating over time leads to
!
Z 1
Z 1
N tþ1
¼ exp
rðuÞdu Nt
aðuÞdu :
(A2)
Nt
0
0
R
R1
1
Now defining l0 ¼ exp 0 rðuÞdu ; and a ¼ 0 aðuÞdu, we obtain
the Ricker fitness function
lt ¼ l0 eaNt :
(A3)
A.2. Preservation of the exponential fitness function at the landscape
scale
To understand how the exponential fitness function is preserved
when patchiness derives from the variable activities of the
individuals whose density is the fitness factor, consider the host–
parasitoid model. Assume that patchiness results because individual
parasitoids respond differently to each patch, spending very
different amounts of time in these different patches. The amounts
of time spent in the patches will be independent from one parasitoid
to another. Suppose that the number of parasitoids is p, and that
parasitoid j spends Tj units of time in patch x, and in doing so reduces
the survival probability of any individual host by the multiplier
expðaT j Þ. The parasitoids each independently have this effect, and
so the expected survival rate of hosts in patch x, given T1, T2,. . .,Tp, is
p
Y
a
eaT j ¼ e
Pp
j¼1
Tj
¼ eaPx;t ;
(A4)
j¼1
where Px,t is identified here implicitly as the amount of parasitoid
time spent in the patch. Because the Tj are independent, the
expected value of this quantity based on the product on the LHS is
2
3
p
p
Y
Y
a
T
j5 ¼
E½eaT j ¼ ecT ðaÞ p ;
(A5)
E4 e
j¼1
j¼1
where cT is the assumed common cumulant generating function of
the T j . The expected value of the product on the left splits into the
product of the expected values in the center due to independence, and
we see a negative exponential preserved in the final result on the RHS.
The uncertainty present at the level of the individual parasitoid leads
to the change in the parameter a to cT ðaÞ, at the landscape scale,
but otherwise there is no difference in this formula between variable
and constant parasitoid behavior. Indeed, the argument for the
exponential form presented above does not depend on whether
individual parasitoids are variable or constant in behavior. The issue is
whether they are independent of one another. The situation changes
dramatically when the behavior of the parasitoids is dependent, for
example, because some locations are more accessible or attractive
than others to all parasitoids. Then the expected value of the product
in (A5) does not split into the product of the expected values. The case
P
of complete dependence means that j T j ¼ T p ¼ Px;t , where T is the
P. Chesson / Ecological Complexity 10 (2012) 52–68
amount of time that each parasitoid spends in a given patch. In that
case, the right hand side of (A5) becomes
expfcT ða pÞg ¼ expfcvðTÞ ðaT̄ pÞg ¼ expfcvðPx;t Þ ðaP̄t Þg;
(A6)
where it should be noted that v(T) = v(Px,t), and T̄ p ¼ P̄t . This result
is the same as Eq. (30) in the text for aggregation to environmental
variation that all actors treat the same.
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