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Biological Conservation78 (1996) 143-148
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M O D E L S OF SPATIAL SPREAD: A S Y N T H E S I S
Alan Hastings
Division of Environmental Studies, Institute of Theoretical Dynamics
and
Center for Population Biology, University of California, Davis, CA 95616, USA
Abstract
other species in the community which has been
invaded, either through competition, predation, parasitism, or more complex indirect dynamical effects; (3)
the spatial spread of the invading species from a single
location.
All three phases are important, and have been
the subject of theoretical investigations. The theory
of biological control (e.g. Mackauer et al., 1990) in
effect looks at the possibility of establishment of
species and their effect on communities, for example.
However, as the difficulties with the theory of biological control illustrate, making theoretical predictions
of the success of invaders and of the effects of the
invading species on the community invaded can be
very difficult.
In this paper I focus on the final phase of a biological
invasion: the spatial spread of the invading species.
Models of spatial spread, an area where we can hope
for robust predictions, which may have extreme practical
implications. For example, what is the rate of spread of
rabies across Europe (Murray et al., 1986), or of
Africanized bees (e.g. Matis et al., 1994)? Knowing the
rate of spread may allow preparation for the consequences. Knowing how the rate of spread depends on
the parameters of a simple, robust, model may help in
the design of control measures which would slow the
rate of spread.
Examples of spatial spread with long-term, relatively
reliable estimates of range spread have been reviewed
by van den Bosch et al. (1990, 1992), Andow et al.
(1990) and Okubo (1988). In particular, the spread of the
muskrat in central Europe provides a means of judging
the adequacy of the theories that have been developed.
As the first step towards understanding the dynamics
of invasions, I discuss the different models which can
be used to describe the dynamics of spatial spread in
invading species, focusing primarily on the approach
embodied in the Fisher (1937) equation and related
approaches. I outline the models and approaches which
can be used to predict rates of spatial spread of invading
species. Mollisson (1991) has provided an outstanding,
more mathematically oriented, review of this topic,
without however covering all the complications induced
by density dependence, or including models with more
than one species.
I review models describing the dynamics of range expansion (spatial spread) of invading organisms, emphasizing
two apparently robust results. First, there appears to be
a linear rate of spread with time, and second, this rate is
proportional to the per capita growth rate of the population when the invading species is rare. Both results hold
for a variety of single-species and two-species models. I
then present two models, one stochastic, and one with an
Allee effect, that demonstrate that the constant linear
rate of spread may only hold after an initial period of
slower spread. Finally, I emphasize that the dependence
of the rate of spread on the per capita growth rate need
only hold when this is maximized when the species is
rare. This last observation may have important implications for understanding the rate of spread for precisely
those species which are likely to cause the most disruption to the communities they invade. Copyright © 1996
Elsevier Science Limited
Keywords: spread, invasion, theory, diffusion, competition,
predation.
INTRODUCTION
Theory in ecology and population biology has many
potential applications, ranging from explaining natural
phenomena to suggesting experiments to making predictions. In looking at questions of applied interest in
ecology, a reasonable hope is that we will be able to
turn increasingly to theory to help make decisions
about future behavior of populations and ecosystems.
One class of problems where theoretical predictions are
potentially very useful are questions related to the
dynamics of invading species (Hengeveld, 1989; Andow
et al., 1990; Mollison, 1991).
To delineate the areas where I believe theory can
play a role, I divide the dynamics of invasions into
three phases: (1) the establishment of the invading
species at a single spatial location; (2) the consequence
of the growth in numbers of the invading species for
Correspondence to: A. Hastings. Tel.: 916-752-8116; Fax:
916-752-3350; e-mail: [email protected] or amhastings
@ucdavis.bitnet
143
A. Hastings
144
Prediction of spatial spread
To make predictions, rather than simply understanding
the dynamics of invasions from a basic theoretical
point of view, the parameters in models need to be
determined. I do not emphasize the statistical problems
associated with this question, but instead present two
conceptually different ways of approaching the problem
of using invasion models to predict the spread of an
invader.
The first way to predict rates of spread is to use previous observations to estimate a future rate. These
could come from the earlier stages of the same invasion, or potentially from a different invasion of the
same species.
A second way to predict rates of spread would be to
estimate the parameters in a particular model from
observations of the species in question. For the simplest models, these parameters are fecundities (per
capita growth rates), and rates of movement of individuals, which in the simplest case would be mean squared
distance per generation (Kareiva, 1983; Andow et al.,
1990). Solutions of the model with these parameters
then give estimates of rates of spread.
where N(x, t) is the (density function for) the population size at x at time t, D(x) is the diffusion coefficient,
and the function f(N, x) is the per capita rate of
increase of the population at position x.
For two spatial dimensions an analogous model is
used, with x now a vector, and the partial derivatives
replaced by gradients. In most applications, radially
symmetric solutions are used to describe spread. Thus
the problem once again reduces to one where there is a
single spatial variable, r, the distance from the point of
introduction. If the diffusivity is a constant, D, the
equation in this case is
~N D ~ (r~N]
~ - - r Dr ~ ~-r ] + Uf(U, r).
MODELS OF SPATIAL SPREAD
I begin by describing approaches which can be used to
model the spatial spread of individual species. It is
important to underline some of the assumptions that
are made in these models so that the robustness of the
biological conclusions can be considered.
Fisher equation
Among the earliest models of spatial spread are ones
describing the dynamics of a single species in a continuous habitat. In words the model takes the form
rate of change of local population size
= change due to random movement
+ change due to local dynamics.
The verbal model can be translated into equations as
follows. First, assume that the random movement is
described by a large number of small steps leading to a
description of movement by diffusion (Okubo, 1980).
For one spatial dimension, such as movement along a
river or coastline, the equation becomes
The Fisher equation and its analogues make specific
assumptions about the form of movement. For instance,
cases where long-range dispersal may play a role are
difficult to reconcile with the diffusive description of
movement (Okubo, 1980) embodied in the Fisher equation. One way to loosen these assumptions is to look at
a discrete space or discrete space-discrete time model.
A discrete space-continuous time analogue takes the form
dN~ _ ~ Dij(N~- N~) + N~f(N~),
dt
(1)
Some care is needed in developing the mathematical
formulation of the term describing change due to
random movement, as described in Okubo (1980).
However, the difficulties only arise in the cases where
the habitat is not uniform and the rate of movement is
not constant and depends on habitat quality. Although
habitat heterogeneity is likely to be very important in
practice, it is not usually incorporated in the model
because it would both make analytical solutions very
difficult if not impossible and would greatly complicate
any parameter estimation approach. Thus, I present
just one description of random movement.
Also, this model implicitly assumes that movement
and reproduction occur randomly during the lifetime of
the individual and makes analogous assumptions about
the effects of density dependence. In the next section, I
will consider how deviations from some of the assumptions in this model can be included.
(3)
(4)
j=,.,,
where Ni(t) is the population size at position i at time t,
D~ = Dji is the rate of exchange between patches i and j,
and the function f ( N ) is the per capita rate of
increase of the population at position i.
Solutions to these models will be discussed below.
However, in the spirit of considering applications of
these models, it is important to emphasize that the form
of the density dependence, i.e. the function f, may play
a critical role in determining the rate of spatial spread.
More complex single-species Fisher-type equations
As emphasized in van den Bosch et al. (1990, 1992), the
diffusion model (2) is based on the assumption of random movement throughout the life of an individual,
which is certainly false for many species. Rather than
introducing complications step by step, I now describe
the formulation developed by van den Bosch et al.
(1990, 1992), which allows arbitrary timing of reproduction and movement. This formulation includes the
possibility of long-range movement as well. The cost of
145
M o d e l s o f spatial spread
this generality is that the resulting model cannot be
solved in general. However, simplifying assumptions
can then be exactly specified which will lead to models
of intermediate complexity, which can in fact be solved
analytically.
Building upon earlier work by Weinberger (1982),
van den Bosch et al. (1990, 1992) begin with a more
general model that describes the probability of a birth
at location x as the sum over all locations z and ages u
of the probability that an individual born at location z,
and now u years old, produces a newborn at x. Thus,
for muskrats, one looks at both the movement of
young from the territory of their parents and the agespecific fecundity. Note that even this general model
still makes the simplifying assumption of no underlying
spatial variability.
The translation of this general statement of movement
and reproduction into mathematical terms is relatively
straightforward (van den Bosch et al., 1990, 1992). The
resulting model takes the form
b(t, x)
=gofo'fg_b(t - a, z)fl(a, x - z) dz da,
(5)
where b(t, x) is the number of births per unit of area and
per unit of time at position x at time t, and Rofl(a, x - z )
is the density of newborns produced per unit time at
position x by an individual of age a born at z. The two
integrals represent sums over ages and sums over space,
respectively. By writing the model in this form, not
including dependence on population size in the functions, we are ignoring density effects on both births and
movements, although they could, in principle, be included.
The Fisher model can be shown to be a special case of
this model, under the assumptions that neither movement nor fecundity depend on age (van den Bosch et
al., 1990, 1992).
A major difficulty lies in solving the resulting model.
In its general form the model is so unwieldy that it is
almost useless. Thus, in applying the general model
to any particular invasion, a knowledge of the biology
of the species is necessary not only for determining
the parameters in the model, but also for simplifying
the model to reduce it to a tractable form.
Two-species approaches
The theory just presented has the implicit assumption
that the dynamics of the invading species can be viewed
as a one-species problem. However, interactions between
indigenous and invading species may play a crucial role
in determining the rate of spatial spread. To understand these interactions, models which include more
than one species must be used. Additionally, if biological
control measures are to be used to slow the rate of
spread of a species, interactions between species must
be studied. However, aside from models of the spread
of diseases (e.g. Murray et al., 1986), invasion models
with more than one species have received limited attention (e.g. Okubo et al., 1989; Gardner, 1982, 1984;
Dunbar, 1983, 1984).
In studying the spatial spread of the introduced grey
squirrel Sciarus carolinensis in Britain, Okubo et al.
(1989) felt compelled to use a two-species model to
account for the role played by the replacement of the
native red squirrel S. vulgar& by the grey squirrel.
Thus, in one spatial dimension, the model they used was
OSi/Ot = D i0x-~2Sl
5- + alSl(1 - blSl - ctS2)
(6)
OS2/Ot = D ~2S2
2~0 x + a2S2(1 - b2S2 - c2St),
(7)
where a; are intrinsic rates of increase, b; represent
intraspecific competition, ci represent interspecific competition, Di are diffusion coefficients, and S; are the
population densities of each species.
Analagous models for predator-prey dynamics have
been studied as well (Dunbar, 1983, 1984; Gardner,
1984). Here, there are two different cases of biological
interest. One could look at the rate of spread of a
predator in an environment where its prey is already at
carrying capacity, as has been done by Dunbar (1983,
1984) using the Lotka-Volterra model to describe the
interactions between species,
OSl/Ot = D
02SI +
n~Ox
aiSl(1 -
blS1 ClS2)
02S2
OS2/at = D2-~x 2 - a2S 2 + c2StS2,
-
(8)
(9)
where all the parameters are positive, St is the density
of prey and $2 is the density of the predator. Alternatively, one could examine the simultaneous rate of
spread of a predator and its prey, where both are
invading a habitat they were previously absent (Gardner,
1984).
Stochastic and simulation approaches
In all the approaches we have discussed so far,
the approach has been deterministic. Even though the
process of spread assumed in these models is random
at the level of the individual, the resulting models at
the population level, and the descriptions of population growth, are deterministic. Yet, intuitively, one
would expect that stochasticity plays an important role
in the dynamics of invasions. At the boundaries of the
current range of an organism, the density of organisms
must be small so that stochastic effects might play a
role.
The role of stochasticity in determining the dynamics
of invading organisms can be explored via birth-death
models (e.g. Matis et al., 1994), simulations (e.g. Hastings,
1996) or analyses of diffusion approximations (e.g.
Mollison, 1977). The point emphasized by Mollison
(1977, 1991) is that stochastic models with density
dependence are the ones that give results different from
deterministic models. I discuss this in more detail
below.
A. Hastings
146
MODEL ANALYSIS AND RESULTS
I begin the description of the dynamics of invasion
models with the simplest models. The goal is to demonstrate what the common outcomes are, and where the
outcome depends on the specific model chosen from
those described above.
Basic result for the simple model
The analysis of the simplest model (2) breaks down
into two different approaches, depending on whether
density dependence is included in the description of the
population dynamics. I describe both the approach and
the outcome for these different cases. The first case presented is density independence, where the rate of
spread is given by the change in the area of habitat in
which the population is above some detection level.
The next case is the dynamics of spatial spread with
density dependence, where the rate of spread can be
described by a terms of a travelling wave. Here, it is
important to distinguish between the logistic description
of density dependence and more complex descriptions.
Before examining non-logistic descriptions, I describe
results for the two-species and age-structured models
described above.
In the analysis of a model without density dependence,
~N - ~-x~ (D(X)~x ) +
(10)
the rate of spread is defined as the change in the area
bounded by the line at which the population is above
some small, specified density (Skellam, 1951). For a
large enough time, the rate of spread of this detection
level is essentially linear in time for one spatial dimension.
For the case where the diffusivity is a constant, this
rate of spread is 2(rD)~ in one spatial dimension. For
two spatial dimensions, it is the square root of the area
occupied by the species that increases linearly with time.
How does the rate of spread with density dependence
compare with the rate of spread with exponential growth?
The analysis of the model (2), with F(N, x) = rN(1 - N),
is now a classic result (Fisher, 1937). The idea is based
on looking for travelling wave solutions, i.e. solutions
where the density N does not depend on t and x separately, but only in the combination x - ct. Thus the
density at x at time t will be exactly the same as the
density at x + c at time t + 1. Hence, the speed of the wave
is given by c.
Using fairly straightforward arguments, one can
show that there is a unique minimum speed, Cmi,, for
which a travelling wave can exist (Fife, 1979). It is
much more complex to show that in fact this minimum
wave speed is stable, and virtually all initial conditions
approach this minimum value (Fife, 1979). What is perhaps more surprising is that an analysis of this model
with density dependence yields exactly the same estimate for the asymptotic rate of spread, 2(rD)~, as the
corresponding model without density dependence. The
intuitive reason for this correspondence may be that
the rate of spread is determined by dynamics 'near the
front', and density dependence is unimportant there
because population levels are low. However, as we shall
see below, the exact correspondence in this case is
somewhat deceptive.
These results on the dynamics of spatial spread
would not be of much interest if they only held for the
specific eqns (3) and (10). Thus, I consider to what
extent these results carry over to more complex models,
some of which have already been introduced above.
Two-species models
The analysis of the first two-species model presented,
the model with competition (eqns 6, 7) leads to the
intuitively expected result that a competitor slows
down the rate of spatial spread (Okubo et al., 1989). In
the case where one species replaces the other, with
b2 > cl
(11)
c2 > bl,
(12)
in one spatial dimension, one can find, in a special
case, a rate of spread (wave speed) that is
2( 1 -
cl/b2)~(Dla
I )!.
(13)
Note that this speed is slower than the rate in the
absence of species 2 by the factor (1 - cl/b2)~. One
might have guessed this by noting that this is exactly
the reduction in the per capita rate of increase of the
invading species when it is rare. As noted above, it is
the per capita increase rate when rare that determines
the rate of spread.
For predator-prey models, the rate of spread has
been computed by Dunbar (1983, 1984) for the case
where a predator is invading a habitat already occupied
by its prey, when the dispersal (diffusion) rate of the
prey is not greater than that of the predator. In this
case, the rate of spread is once again determined by the
initial per capita rate of increase of the predator, in
exact analogy to the one-species logistic model. For the
case where both predator and prey are invading, Gardner
(1984) has demonstrated that there is a linear rate of
spread, but has not computed it.
Age-structured models
The case of age-structured models has been treated by
van den Bosch et al. (1990, 1992). Their approach
emphasizes determining approximations to the general
model (5) for which the linear rate of spread can be
easily calculated. They emphasize that there are two
ways this could be done. One could estimate fecundities
and compute an intrinsic rate of increase, and estimate
overall movement. These parameter estimates could
then be plugged into the simple models of the form (2).
The alternative approach would take into account the
age specificity of the movement or fecundity. They
Models of spatial spread
demonstrate that the answers obtained by the two
approaches are roughly equal as long as the population
is not growing too rapidly. For rapidly growing populations, however, including age-specific parameters can
cause large changes in predicted rates of movement. All
of this theory is based on density-independent growth.
Role of stochasticity
A point emphasized by Mollison (1977, 1991) is that
density-independent stochastic models yield the same
rates of spread as those predicted by the deterministic
models, for which density dependence of the logistic
form does not alter the results. However, Mollison does
emphasize that density-dependent stochastic models
may yield different results. Here, analytic results will be
hard to come by and simulations will be necessary. As
a preliminary example of this approach, I (Hastings,
1996) have used a stochastic simulation to show that it
is likely that there may be a significant time of slower
spread before the ultimate linear rate of spread is
reached. Much further work is needed in this area.
Role of density dependence of different forms
Finally, I emphasize how the particular choice of the
function describing density dependence may play a
role. These results have recently been discussed in the
context of a specific model by Lewis and Kareiva
(1993), based on earlier work by Fife and McLeod
(1975), Fife (1979), Hadeler and Rothe (1975) and
Rothe (1981), which is able to encompass general
descriptions of density dependence. For a model with
F(N) = rN(l - N ) ( N - a)+ a population clearly will
spread if a < 0. The population can still spread, even if
a > 0 but 1/2 > a, provided that the initial number of
organisms is large enough. The rate of spread is given by
2(-arD)~
(14)
for a < -1/2. The rate of spread is given by
(1/2 - a) (2arD)~
(15)
for -1/2 < a < 1. Lewis and Kareiva (1993) emphasize
that the presence of an Allee effect embodied in this
description of density dependence slows down the rate of
spread. They also emphasize that this model can
exhibit slower initial rates of spread.
Here, I emphasize a different viewpoint, that in this
case the rate of spread is much faster than would be
predicted by examining the rate of increase of the population when rare. Obviously, if there is an Allee effect,
a > 0, the rate of increase when rare will be negative,
but the population can spread. Now look at the case
when -1/2 < a < 0. Even if the population can increase
when rare but the per capita growth rate of the population increases with density, the rate of spread can be
larger (by an arbitrarily large factor if a is close to 0)
than the rate of spread that would be implied by the
initial per capita rate of growth. Thus, what appeared
to be a general rule, that the per capita rate of increase
147
when rare determined the rate of spread, is not true. As
demonstrated in the papers cited above, the actual rule
depends both on the maximum per capita rate of
growth of the population and the per capita rate
of growth of the population when rare - - it is true that
the rate of spread is determined by the growth rate of
the population when rare if the growth rate when rare
is the maximum growth rate. | discuss the implications
of this result for multiple-species models below.
CONCLUSIONS
The theory of spatial spread has yielded some robust
predictions, as well as others which appear to be
strongly model-dependent. The robust predictions
appear to be that the asymptotic rate of spread is linear
with time. Predicting the actual, numerical, rate of
spread appears to be more difficult, as I discuss below.
Given that the prediction of an ultimate linear rate
of spread is robust - - all the models presented here
yield this result - - how well has this result held up in
practice? An important issue is that the linear rate of
spread is only asymptotic and is reached after some
initial period of transient behavior. The question then
is: what is the time scale of this transient behavior?
Several recent reviews (Okubo, 1989; Andow et al.,
1990; van den Bosch et al., 1992) emphasize that this
linearity prediction holds for long times. However, as
emphasized by Okubo (1989), transient, slower, rates of
spread may be common. Two of the models discussed
above provide possible explanations for this behavior
- - it may be due to an Allee effect or to stochastic
effects. In either case, however, this observation provides a strong cautionary note when using initial rates
of spread to predict future rates of spread. Both in the
cases reviewed by Okubo 0989) and in the models
reviewed here incorporating either stochasticity or an
Alice affect, the initial rates of spread are always
slower. Thus initial estimates of rates of spread may
underestimate final rates.
Predicting the numerical rate of spread may be complicated by two factors discussed above. In the simplest
case without age-specific fecundity and movement, it
appears that the rule of determining the intrinsic rate
of increase and movement may allow prediction of the
rate of spread. In fact, the two-species results imply
that this result holds in the more complex situation,
as long as the rate of increase is determined in the
community being invaded. This may provide an explanation as to why a species may spread rapidly in one
habitat, and only slowly in another. The first complication is that the rate of spread may be strongly affected
by the age specificity of the fecundities and movement
rates.
There is a second, potentially much larger, numerical
problem with using per capita growth rates in small
populations to predict rates of spread. If the maximum
per capita growth rate of the population is much
148
A. Hastings
greater than the per capita growth rate at low population levels, the rate of spread can be significantly
greater (even orders of magnitude) than would be
predicted from the per capita growth rate at low population levels. Although this has been demonstrated
rigorously only in the single-species models, it is
reasonable to assume that a similar result must hold in
a multiple-species context. This observation may be
very troubling from the standpoint of conservation
because the invading species which are most troublesome are those which cause large changes in the
communities they invade, and those are precisely the
species most likely not to have their largest population
growth rates when rare.
What does the theory have to say about potential
control measures? Reduce both the movement rate of
individuals from their point of birth and the maximum
per capita growth rate of the population as much as
possible to slow the rate of spread. Thus, controlling
the rate of spread of a non-indigenous species yields a
different biological problem than controlling the impact
of a non-indigenous species in a single location. Controlling the impact of a non-indigenous species usually
means reducing its equilibrium population level. However, the rate of spread does not depend directly on the
equilibrium level of the invader, but directly only on
the maximum per capita growth rate.
Although a substantial theory of the rate of spread
o f single species has been developed, much more work
will be needed to understand the spread of species in
the context o f larger communities. Solutions to this
difficult problem will yield important new results for
understanding the impacts of invading species on
natural communities throughout the world. Solutions
to these questions will also provide further insight into
designing measures, both biological and non-biological,
for controlling the spread of non-indigenous species.
ACKNOWLEDGEMENTS
I thank Sergey Gavrilets and Akira Okubo for helpful
conversations. Comments on an earlier draft by Ted
Grosholz and Dan Simberloff were particularly helpful.
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