Bulletin of Mathematical Biology (2008) 70: 1790–1826
DOI 10.1007/s11538-008-9325-2
O R I G I N A L A RT I C L E
Saddle-Point Approximations, Integrodifference Equations,
and Invasions
Mark Kota,∗ , Michael G. Neubertb
a
Department of Applied Mathematics, University of Washington, Box 352420, Seattle, WA
98195-2420, USA
b
Biology Department, MS 34, Woods Hole Oceanographic Institution, Woods Hole, MA
02543-1049, USA
Received: 15 September 2007 / Accepted: 3 April 2008 / Published online: 22 July 2008
© Society for Mathematical Biology 2008
Abstract Invasion, the growth in numbers and spatial spread of a population over time,
is a fundamental process in ecology. Governments and businesses expend vast sums to
prevent and control invasions of pests and pestilences and to promote invasions of endangered species and biological control agents. Many mathematical models of biological
invasions use nonlinear integrodifference equations to describe the growth and dispersal
processes and to predict the speed of invasion fronts. Linear models have received less
attention, perhaps because they are difficult to simulate for large times.
In this paper, we use the saddle-point method, alias the method of steepest descent, to
derive asymptotic approximations for the solutions of linear integrodifference equations.
We work through five examples, for Gaussian, Laplace, and uniform dispersal kernels
in one dimension and for asymmetric Gaussian and radially symmetric Laplace kernels
in two dimensions. Our approximations are extremely close to the exact solutions, even
for intermediate times. We also employ an empirical saddle-point approximation to predict densities using dispersal data. We use our approximations to examine the effects of
censored dispersal data on estimates of invasion speed and population density.
Keywords Saddle-point approximation · Method of steepest descent · Integrodifference
equations · Biological invasions · Invasion speed
1. Introduction
Ecologists have long used models such as the linear partial differential equation
∂ 2n
∂n
= rn + D 2
∂t
∂x
∗ Corresponding author.
E-mail address: [email protected] (Mark Kot).
(1)
Saddle-Point Approximations and Invasions
(Skellam, 1951) and the nonlinear Fisher (1937) equation,
n
∂n
∂ 2n
= rn 1 −
+D 2,
∂t
K
∂x
1791
(2)
to predict changes in the density, n(x, t), of an invading species as a function of space,
x, and time, t . Equations (1) and (2) are examples of reaction–diffusion equations,
continuous-time models for synchronous growth and dispersal. The literature on reaction–
diffusion models is quite extensive. See Britton (1986), Shigesada and Kawasaki (1997),
and Petrovskii and Li (2006) for useful books on these models.
For many invasive species, growth and dispersal occur in distinct and asynchronous
stages. As a result, many ecologists have also developed a strong interest in integrodifference equations (IDEs). Integrodifference equations are discrete-time, continuous-space
models for the growth and spread of populations (Kot and Schaffer, 1986; Kot et al., 1996).
These equations readily accommodate the varied dispersal mechanisms (Neubert et al.,
1995) and the long-distance dispersal events (Shigesada and Kawasaki, 2002) that create the leptokurtic distributions of displacements found in nature (Okubo, 1980; Willson,
1992). In recent years, integrodifference equations have been used to model the growth
and spread of a wide variety of organisms. See Neubert and Parker (2004), Cobbold et
al. (2005), Fagan et al. (2005), Mistro et al. (2005), Jacquemyn et al. (2005), Tufto et al.
(2005), Krkosek et al. (2007), and Skarpaas and Shea (2007) for examples.
Integrodifference equations, like reaction–diffusion equations, can be linear or nonlinear. Because of the general importance of density dependence, it is tempting to think that
nonlinearity in growth is an essential ingredient of all invasion models. Nonlinear invasion models do exhibit some phenomena, such as finite traveling-wave solutions, that are
absent from linear models. At the same time, linear, deterministic invasion models have
their own advantages (Mollison, 1991): Their assumptions are transparent. They often
exhibit the same velocity or rate of spread as nonlinear models (without Allee effects).
Finally, linear models are often easier to analyze than nonlinear models.
In this paper, we focus on a powerful asymptotic method for analyzing the linear integrodifference equations that model the growth and spread of populations released at
the origin. One can write formal solutions to these equations using integral transforms.
Only rarely, however, can one determine the rate of spread and shape directly from the
formal solution. One can, however, use the method of steepest descent, also known as the
saddle-point method, to determine asymptotic approximations to the solutions for large
times.
The method of steepest descent is commonly used in statistics (Daniels, 1954; Reid,
1988; Goutis and Casella, 1999; Butler, 2007) and in the study of random walks (Domb
and Offenbacher, 1978; Hughes, 1995). Radcliffe and Rass (1997) stressed that this
method can be used to determine invasion speeds for discrete-time spatial models. In spite
of this, the method of steepest descent is underutilized in the study of integrodifference
equations (but see Radcliffe and Rass, 1997 and Powell et al., 2005), perhaps because of
the absence of simple, worked-out examples. In this paper, we describe the method, use it
to determine the asymptotic density and speed of invading populations, and compare the
saddle-point approximation to the exact solution for three exactly solvable kernels in one
dimension and for two kernels in two dimensions.
Scientists often prefer to work with the empirical moment generating function, rather
than with estimates of the dispersal kernel, when predicting rates of spread from empirical
1792
Kot and Neubert
data (Clark et al., 2001; Lewis et al., 2006). We will show that one can also use an empirical saddle-point approximation (Feuerverger, 1989) to predict the asymptotic density
of an invading population under similar circumstances. Finally, we examine how censoring of long-distance dispersal data affects the empirical moment generating function and
estimates of the invasion speed and of the asymptotic population density when the true
dispersal kernel is the Laplace distribution.
In Section 2, we describe a linear, one-dimensional invasion model and show that an
exponential transform can be used to obtain a formal solution. In Sections 3 and 4, we describe the method of steepest descent and use a saddle-point approximation to determine
the population density for large times. We also show how standard invasion-speed formulas fall out of our approximation. In Section 5, we compare saddle-point approximations to
exact solutions for Gaussian, Laplace, and uniform dispersal kernels. In Section 6, we outline similar results for two dimensions. We describe a two-dimensional invasion model,
write out the formal solution, and use a bivariate saddle-point formula to approximate
the formal solution for large times. Section 6 includes approximations for the asymmetric Gaussian kernel and the radially symmetric Laplace kernel. In Sections 7 and 8, we
introduce the empirical saddle-point approximation and show how estimates of invasion
speed and density are affected by the censoring of dispersal data. Finally, in Section 9, we
summarize our results, draw some conclusions, and propose future work.
2. 1D model and solution
We begin by considering the linear integrodifference equation
nt+1 (x) = λ
+∞
−∞
k(x − x )nt (x ) dx (3)
for a point release,
n0 (x) = n0 δ(x),
(4)
of size n0 at the origin. This simple, single-species equation maps the density of a population in generation t , nt (x), to a new density, nt+1 (x), in two stages. During the first or
sedentary stage, individuals grow, reproduce, and die. At each point x, the local population, nt (x), produces λnt (x) propagules. The parameter λ is the geometric growth rate of
the population. We assume that λ > 1.
During the second stage, propagules disseminate. The dispersal kernel, k(x), is the
probability density function for the displacement of propagules. A convolution integral
tallies the contributions from all potential sources x for each destination x. Well-known
examples of the kernel k(x) include the Gaussian, Laplace, and uniform distributions.
In this section, we assume that dispersal occurs on an infinite, homogeneous, and onedimensional domain. The initial condition contains the propagule size, n0 , and the Dirac
delta function, δ(x).
For the purposes of this section, we will focus on dispersal kernels, k(x), that are
piecewise continuous and that have thin, i.e., exponentially bounded, tails. These kernels
lead to constant invasion rates (Weinberger, 1978, 1982; Lui, 1983; Kot, 1992; Hart and
Saddle-Point Approximations and Invasions
1793
Gardner, 1997). For more on fat-tailed kernels, such as the Cauchy distribution, which
generate accelerating solutions with asymptotically infinite speeds, please see Kot et al.
(1996), Lewis (1997), and Clark (1998).
To solve Eq. (3), we will make use of the exponential transform (Giffin, 1975),
f¯(s) =
+∞
f (x)esx dx.
(5)
−∞
For a piecewise continuous probability density function, k(x), with exponentially
bounded tails, the exponential transform converges in a vertical strip in the complex
s-plane (or, possibly in a half-plane or the whole plane) and is the familiar moment generating function
M(s) = k̄(s) =
+∞
k(x)esx dx.
(6)
−∞
The exponential transform is uncommon outside of probability theory, but if we replace
s with −s we obtain the better known two-sided or bilateral Laplace transform (Zayed,
1996). It follows that the inverse of exponential transform (5) is
f (x) =
1
2πi
a+i∞
f¯(s)e−sx ds,
(7)
a−i∞
where the constant a is chosen so that the integration is along a vertical line within the
vertical strip of convergence of the transform.
The exponential transform comes with a useful convolution theorem (Zayed, 1996).
Consider two functions, f (x) and g(x), whose exponential transforms, f¯(s) and ḡ(s),
converge absolutely for some common value of s. We define the convolution of these two
functions as
+∞
f (x )g(x − x ) dx ,
(8)
h(x) = (f ∗ g)(x) ≡
−∞
whenever this integral exists. The exponential transform of this convolution is simply the
product of the two exponential transforms f¯(s) and ḡ(s),
+∞
−∞
h(x)esx dx = f¯(s)ḡ(s).
(9)
This convolution theorem enables us to write out a formal solution for linear integrodifference equation (3).
The right-hand side of Eq. (3) contains a convolution. If we take the exponential transform of Eq. (3) and of initial condition (4), we obtain the difference equation
n̄t+1 (s) = λM(s)n̄t (s)
(10)
and the initial condition
n̄0 (s) = n0 .
(11)
1794
Kot and Neubert
M(s) is just the moment generating function of our dispersal kernel k(x). Difference
equation (10) is easy to solve. At each time step, we multiply our transformed density by
the geometric growth rate λ and by moment generating function M(s). It follows that
n̄t (s) = n0 λt M t (s).
(12)
After applying the inverse of our exponential transform, Eq. (7), we obtain the formal
solution
a+i∞
1
nt (x) = n0 λt
M t (s)e−sx ds.
(13)
2πi a−i∞
Although inverse transform (13) is the solution of our problem, the contour integral
in this solution can only be evaluated for a limited number of kernels. Even for these
kernels, the resulting solution may be rather unwieldy. Numerical integration of Eq. (3)
by direct methods is, in turn, computationally intensive. FFT algorithms (e.g., Andersen,
1991) can speed up this integration, but the resulting simulations often exhibit numerical
instabilities for large times. All of these problems point to the need for a simple asymptotic
approximation to our solution for large times. The method of steepest descent provides
such an approximation.
3. The method of steepest descent
The method of steepest descent is a general method for approximating integrals, in the
simplest case, of the form
(14)
I (t) = etf (z) dz
C
for large t , where C is a contour in the complex z-plane whose ends do not contribute
significantly to the integral, f (z) is an analytic function in a part of the complex plane
that contains C, and t is a positive real number. Please see Petrova and Solov’ev (1997)
for a history of this method and Murray (1974) for a detailed description of the method.
The basic idea behind the method is to deform our contour so that most of the contribution
to the integral comes from the vicinity of a single point.
To show how this can occur, let z = x + iy and
f (z) = u(x, y) + iv(x, y).
(15)
Our integrand then takes the form
etu(x,y) eitv(x,y) .
(16)
Since the modulus or magnitude of our integrand is determined by a scalar, t , and u(x, y),
we will focus our attention on the critical points of u(x, y), i.e., on points where the partial
derivatives ux and uy are both zero.
Saddle-Point Approximations and Invasions
1795
Fig. 1 A saddle point of the surface u = u(x, y). The method of steepest descent depends upon deforming
a contour of integration so that most of the contribution to integral (14) comes from the vicinity of the
saddle point.
The critical points of u(x, y) coincide with those of f (z). If we take the derivative of
f (z), by considering small changes in x, we obtain
f (z) = ux + ivx .
(17)
It follows that f (z) = 0 if and only if ux = vx = 0. The Cauchy–Riemann equations,
ux = vy ,
vx = −uy ,
(18)
are necessary conditions for the differentiability of complex functions. It follows that
f (z) = 0 if and only if ux = uy = 0. Thus, z = x + iy is a critical point of f (z) if and
only if the ordered pair (x, y) is a critical point of u(x, y).
We may determine the nature of a critical point by examining the Hessian,
u
uxy (19)
H (x, y) = xx
= uxx uyy − u2xy ,
uxy uyy at that point. A critical point z = x + iy is nondegenerate if and only if its Hessian is
nonzero, H (x, y) = 0. Because of the Cauchy–Riemann equations,
2
.
H (x, y) = −u2xx − vxx
(20)
f (z) = uxx + ivxx ,
(21)
Since
we may also conclude that critical point z is nondegenerate if and only if f (z) = 0. Since
our Hessian is then negative, we may also conclude that a nondegenerate critical point is
a saddle point (see Fig. 1).
We will deform our contour C, in Eq. (14), so that it passes through a nondegenerate
critical point of f (z). We wish to change our path in such a way that u(x, y), the real part
of f (z), attains its maximum on the contour C at the critical point and so that u(x, y)
falls off as rapidly as possible on either side of its maximum. In other words, we want the
1796
Kot and Neubert
deformed contour to be a path of steepest descent that concentrates the contribution of the
exponential in our integral over a small section of the contour.
If we take an arbitrary path through our critical point, or if we take a path through
some other point along the ridge of the saddle, then v(x, y), the imaginary part of f (z),
contributes an oscillatory term that may confound our goal of concentrating the contributions to our integral to the neighborhood of the critical point. The oscillations become
worse with increasing t . We, therefore, choose a path through the critical point that holds
v(x, y) constant, so that there are no oscillations near the critical point.
We know from vector analysis that the gradient of u(x, y) is perpendicular to the level
curve of u(x, y) that passes through the point (x, y). Likewise, the gradient of v(x, y)
is perpendicular to the level curve of v(x, y) at (x, y). Because of the Cauchy–Riemann
equations, ∇u · ∇v = 0. It follows that the level curves of u(x, y) and v(x, y) are perpendicular to each other, except possibly at critical points of f (z), where ∇u = 0 and
∇v = 0. If we choose a level curve of v(x, y) as our contour through our critical point
(see Fig. 2b), we will also follow a path of most rapid change in u(x, y) (see Fig. 2a).
There are two level curves of v(x, y) that pass through our critical point. We must be
careful to choose the level curve that gives us a local maximum in u(x, y).
For each of our examples, we will consider a single critical point, z = z0 , with
f (z0 ) = 0 and
(22)
f (z0 ) = f (z0 )ei2α = 0,
such that the endpoints of our contour lie in opposite valleys of the u(x, y) surface. If we
expand f (z) in a Taylor series about the critical point,
1
f (z) = f (z0 ) + f (z0 )(z − z0 )2 + · · · ,
2
(23)
and let
z − z0 = reiθ ,
(24)
it follows that
1 f (z0 )ei2α r 2 ei2θ + · · ·
2
1
= f (z0 )r 2 cos 2(α + θ ) + i sin 2(α + θ ) + · · · .
2
f (z) − f (z0 ) =
(25)
For a path of most rapid change in u(x, y), we need to choose θ so that the imaginary parts
of f (z) and f (z0 ) remain constant and cancel one another. In other words, we require
sin 2(α + θ ) = 0. For a path of steepest descents, we must also have cos 2(α + θ ) < 0.
We now deform our contour to the path of steepest descents. Near the critical point,
values of
1
f (z) − f (z0 ) ≈ f (z0 )(z − z0 )2 < 0
2
(26)
are now real. We, therefore, introduce a new real variable, ξ , such that
1
ξ 2 = − f (z) − f (z0 ) ≈ − f (z0 )(z − z0 )2 .
2
(27)
Saddle-Point Approximations and Invasions
1797
Fig. 2 Level curves of (top) u(x, y) and of (bottom) v(x, y). The method of steepest descent requires that
we choose a path (arrows) through the critical point (solid dot) that holds v(x, y) constant and that gives
us a local maximum in u(x, y). The plus and minus signs represent ridges and valleys above and below
the saddle point.
Our integral for I (t), Eq. (14), now becomes
I (t) = etf (z0 )
ξ2
−ξ1
e−tξ
2
dz
dξ,
dξ
(28)
where ξ1 > 0 and ξ2 > 0 are the end points of the original contour under the transformation
to the new variable ξ .
Definition (27) may be inverted to yield
z − z0 =
−2
ξ + ···,
f (z0 )
(29)
1798
Kot and Neubert
where we must be careful to take the branch of the square root that is consistent with the
direction of our contour in the complex z-plane (Murray, 1974). For example, if ξ = ξ2 >
0 and our endpoint lies in the
π
(30)
θ = −α
2
direction, we require that
π
−2
arg
= −α
f (z0 )
2
so that
z − z0 =
(31)
2
π
ei( 2 −α) ξ =
|f (z0 )|
2
|f (z0 )|
ie−iα ξ.
Integral (28) now takes the form
+∞
−2
2
tf (z0 )
e−tξ dξ + · · · ,
I (t) = e
f (z0 ) −∞
which has the asymptotic approximation
−2π tf (z0 )
e
.
I (t) ∼
tf (z0 )
(32)
(33)
(34)
The values of ξ at the endpoints, ξ1 and ξ2 , were safely replaced by infinity, if they did
not already extend to infinity, because points far away from the critical point contribute
little to the integral. The branch of the square root must be, as explained above, chosen
appropriately. If, for example, our contour passes from θ = −(π/2) − α over to θ =
(π/2) − α, our asymptotic approximation simplifies to
2π
ie−iα etf (z0 ) .
(35)
I (t) ∼
t|f (z0 )|
4. Applying the method of steepest descent
Let us now apply the method of steepest descent to the formal solution, Eq. (13), of our
linear integrodifference equation. We may put this solution into the standard form for the
method by introducing the semi-invariant or cumulant generating function
K(s) ≡ ln M(s).
The solution can now be written as
a+i∞
1
x
et[K(s)− t s] ds.
nt (x) = n0 λt
2πi a−i∞
Here, s plays the role of the complex variable z of the previous section.
(36)
(37)
Saddle-Point Approximations and Invasions
1799
We will deform our contour so that most of the integral, for large t , comes from the
vicinity of the critical point, s = s0 , of
f (s; t, x) ≡ K(s) −
x
s.
t
(38)
We will choose a path of integration through this critical point such that the real part of
f (s) has a maximum at s0 and the imaginary part remains constant.
We begin by solving
dK
x
df
=
− =0
ds
ds
t
(39)
for the saddle point s = s0 . At this point,
M (s0 ) x
= .
M(s0 )
t
(40)
For most problems of interest, Eq. (40) will have a unique real root in the domain of
convergence of the moment generating function. See Daniels (1954) for conditions and
proofs. The critical point s0 will be a function of the ratio of x and t .
We now expand f (s) in a Taylor series about s0 ,
1
f (s) = f (s0 ) + f (s0 )(s − s0 )2 + · · · .
2
(41)
This expansion may be written, in terms of the cumulant generating function, as
f (s) = K(s0 ) −
x
1
s0 + κ2 (s − s0 )2 + · · · ,
t
2
(42)
where
κ2 = K (s0 ).
(43)
As a result,
x
nt (x) ≈ n0 λt et[K(s0 )− t s0 ]
1
2πi
a+i∞
1
2
e 2 tκ2 (s−s0 ) ds.
(44)
a−i∞
Close to s = s0 , our deformed path must make the real part of
1
tκ2 (s − s0 )2
2
(45)
negative and the imaginary part of this expression zero. If we let
s − s0 = reiθ ,
κ2 = |κ2 |ei2α ,
(46)
these requirements imply that
sin 2(α + θ ) = 0,
cos 2(θ + α) < 0.
(47)
1800
Kot and Neubert
Thus,
π
θ +α=± .
2
(48)
Since κ2 is real, α = 0 or π and θ = +π/2 or −π/2. The new path of integration through
s0 is thus parallel to the imaginary axis.
If we let
1
ξ 2 = − κ2 (s − s0 )2 ,
2
(49)
we have the approximation
a+i∞
e
1 tκ (s−s )2
0
2 2
ds = i
a−i∞
=i
2
|κ2 |
2π
.
t|κ2 |
+∞
2
e−tξ dξ
−∞
(50)
In light of this integral, an asymptotic approximation for our solution for large time is
nt (x) ∼
n0 λt M t (s0 )e−s0 x
,
√
2πt|κ2 |
(51)
where s0 is determined by Eq. (40) and is a function of the ratio of x and t .
Asymptotic approximation (51) can be used to determine the invasion speed for a
population whose average spread rate converges to a positive constant. Let us imagine
that a population is first detected when it reaches the threshold density nc . We imagine
that this occurs at some x = x(t, nc ) > 0, which we, for convenience, call x. If we set
nt (x) to nc and solve Eq. (51) for the ratio of x and t , we find that
1
1
nc x
=
2πt|κ2 | .
(52)
ln λM(s0 ) − ln
t
s0
t
n0
If this spread rate converges to a constant, then s0 , by Eq. (40), and κ2 , a function of s0 ,
also converge to constants. In the limit of large t , it now follows that this speed is
c ≡ lim
t→∞
1 x
= ln λM(s0 )
t
s0
(53)
or equivalently,
λ=
ecs0
.
M(s0 )
(54)
In this same limit, from Eq. (40),
c=
M (s0 )
.
M(s0 )
(55)
Saddle-Point Approximations and Invasions
1801
Taking our last two equations together, we see that
λ=
es0 M (s0 )/M(s0 )
,
M(s0 )
c=
M (s0 )
.
M(s0 )
(56)
These two equations can be used to determine the speed of invasion of our expanding
population. For a given geometric growth rate λ, we can use the first equation to determine
s0 . With s0 in hand, we then use the second equation to determine the speed c. In practice,
it is often easiest to plot λ and c parametrically, with s0 as the parameter. You can then
read the speed c for a given λ from this plot.
System (56) is the standard pair of parametric equations for the asymptotic speed of
invasion for a scalar integrodifference equation found, for example, in Kot (1992), Kot et
al. (1996), and Hart and Gardner (1997). It applies to linear equations and also to nonlinear
equations that satisfy the conditions of the linear conjecture (Mollison, 1991). It is also
equivalent to the earlier formula
1 ln λM(s)
(57)
c = min
s>0 s
of Weinberger (1978, 1982). All of these equations were originally derived by other methods (but see Radcliffe and Rass, 1997).
5. Examples
We now illustrate the method of steepest descent for linear integrodifference equation (3)
by comparing our asymptotic approximation, Eq. (51), to the exact solution, Eq. (13), for
three kernels for which we derived exact solutions. The three kernels are the Gaussian,
Laplace, and the symmetric uniform distributions.
5.1. Gaussian kernel
For a Gaussian dispersal kernel with standard deviation σ ,
2
1
− x
k(x) = √ e 2σ 2 ,
σ 2π
(58)
the moment generating function is
M(s) = eσ
2 s 2 /2
,
(59)
where −∞ < Re s < ∞. For this kernel, solution (13) reduces to
n0 λt
x2
nt (x) = √
exp − 2
2σ t
σ 2πt
(see Appendix A).
(60)
1802
Kot and Neubert
Let us look at this problem using our saddle-point approximation. The cumulant generating function for our distribution is
K(s) = ln M(s) =
σ 2s2
.
2
(61)
The critical point of
f (s) = K(s) −
x
σ 2s2 x
s=
− s
t
2
t
(62)
occurs when
f (s) = σ 2 s −
x
=0
t
(63)
or at s = s0 , where
s0 =
x
.
σ 2t
(64)
Equation (51), our asymptotic approximation, now reduces to
n0 λt
x2
nt (x) = √
exp − 2 .
2σ t
σ 2πt
(65)
In this case, the method of steepest descent gives the exact solution.
Parametric formulas (56), for the asymptotic rate of expansion, reduce to
λ = eσ
2 s 2 /2
0
,
c = σ 2 s0
(66)
so that
√
c = σ 2 ln λ.
(67)
5.2. Laplace distribution
For a Laplace distribution,
1
k(x) = γ e−γ |x| ,
2
with standard deviation
√
2
σ=
,
γ
(68)
(69)
the moment generating function is
M(s) =
γ2
γ2
,
− s2
(70)
Saddle-Point Approximations and Invasions
1803
where −γ < s < γ . For this kernel, the solution may be written (see Appendix A)
1
nt (x) = n0 λt
γ t+ 2
1
|x|t− 2 Kt− 1 γ |x| ,
2
t− 12 √
2
π(t)
(71)
where Kν (x) is the modified Bessel function of the second kind of order ν (Zhang and
Jin, 1996) and (t) is the gamma function.
We proceed to our saddle-point approximation. The cumulant generating function for
the Laplace distribution is
γ2
.
(72)
K(s) = ln 2
γ − s2
Using Eq. (63), we find that critical points of
γ2
x
x
f (s) = K(s) − s = ln 2
− s
t
γ − s2
t
(73)
occur when
γ2
x
2s
− = 0.
2
−s
t
(74)
This last equation may be rewritten (for x = 0) as a quadratic equation,
t
s 2 + 2 s − γ 2 = 0.
x
(75)
By Descartes’s rule of signs, both roots are real; one root is positive and one is negative.
For each choice of x and t , we take the one root that lies in the interval −γ < s < γ . If
x > 0 and t > 0, for example, we take the root
t
t2
+ γ 2.
(76)
s0 = − +
x
x2
Our saddle-point approximation, Eq. (51), now takes the form
t
n0 λt (γ 2 − s02 )
γ2
nt (x) ∼ e−s0 x .
2
2
2 πt (γ 2 + s02 ) γ − s0
(77)
This approximation is quite close to the exact solution even for intermediate t (see Fig. 3).
Moreover, the relative error decreases with increasing t (see Fig. 4), as expected.
For the Laplace distribution, our parametric formulas for the asymptotic rate of expansion reduce to
2
2s02
2s0
γ − s02
.
(78)
exp
,
c= 2
λ=
γ2
γ 2 − s02
γ − s02
The resulting curve is shown in Fig. 5.
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Kot and Neubert
Fig. 3 Plots of exact solution (71) (solid curves)
and saddle-point approximation (77) (dotted curves) for
√
the Laplace distribution with λ = 1.2, γ = 2, n0 = 1, and t = 5, 10, 15. The approximation, asymptotic
for large t , also matches the exact solution well for intermediate values of t .
Fig. 4 The relative error
√ between saddle-point approximation (77) and exact solution (71) as a function
of x for λ = 1.2, γ = 2, for t = 10 and t = 50. The relative error decreases as time increases.
Saddle-Point Approximations and Invasions
1805
Fig. 5 The asymptotic rate of expansion for integrodifference equation (3) with Laplace distribution (68).
The speed of invasion c is an increasing function
√ of the geometric growth rate λ. To generate this curve,
parametric equations (78) were plotted for γ = 2.
5.3. Uniform distribution
For a symmetric uniform or rectangular distribution,
k(x) =
1
2γ
0,
,
|x| ≤ γ ,
|x| > γ ,
(79)
with standard deviation
γ
σ=√ ,
3
(80)
the moment generating function is
M(s) =
sinh γ s
.
γs
(81)
For this kernel, solution (13) reduces to (see Appendix A)
nt (x) =
t m̄
t (x)
t−1
n0
λ
t (−1)m
γ (t − 2m) − |x|
m
(t − 1)! 2γ
m=0
(82)
for |x| < γ t and zero otherwise, where
γ t − |x|
m̄t (x) = floor
2γ
is the largest integer less than or equal to its right-hand-side argument.
(83)
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Kot and Neubert
Fig. 6 A plot of the Langevin function, L(x). See Eq. (88).
We now consider the saddle-point approximation. The cumulant generating function
is
K(s) = ln
sinh γ s
.
γs
(84)
By Eq. (63), critical points of
f (s) = K(s) −
sinh γ s
x
x
s = ln
− s
t
γs
t
occur when
1
x
γ coth γ s −
− = 0.
γs
t
(85)
(86)
This last equation may be rewritten
L(γ s) =
1x
,
γ t
(87)
where
L(γ s) ≡ coth γ s −
1
γs
(88)
is the Langevin function (see Fig. 6).
Domb and Offenbacher (1978) solved Eq. (87) for the location of the critical point
using a linear approximation to the Langevin function. Later, Cohen (1991) showed that
the inverse Langevin function can be evaluated using the Padé approximant
L−1 (y) ≈
y(3 − y 2 )
,
(1 − y 2 )
|y| < 1.
(89)
Saddle-Point Approximations and Invasions
1807
Fig. 7 Plots of exact solution (82) (solid curves) and saddle-point
approximation (91) (dotted curves) for
√
the symmetric uniform distribution with λ = 1.2, γ = 3, n0 = 1, and t = 5, 10, 15. The approximation,
asymptotic for large t , also matches the exact solution well for intermediate values of t .
It follows that we may approximate our critical point by
s0 ≈
x(3γ 2 t 2 − x 2 )
γ 2 t (γ 2 t 2 − x 2 )
(90)
for |x| < γ t .
Our saddle-point approximation, Eq. (51), now takes the form
n0 λt
sinh γ s0 t −s0 x
e
,
nt (x) ≈ √
γ s0
2πt|κ2 |
(91)
for |x| < γ t (and zero otherwise), where
κ2 = K (s0 ) = −γ 2 csch2 γ s0 −
1
(γ s0 )2
(92)
and s0 is given by Padé approximant (90). The saddle-point approximation is close to the
exact solution even for intermediate t (see Fig. 7).
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Kot and Neubert
Fig. 8 The asymptotic rate of expansion for integrodifference equation (3) with symmetric uniform distribution (79). The speed of invasion c is an increasing function
√ of the geometric growth rate λ. To generate
this curve, parametric equations (93) were plotted for γ = 3.
For the symmetric uniform distribution, our parametric formulas for the asymptotic
rate of expansion reduce to
λ=
γ s0
eγ s0 coth γ s0 −1 ,
sinh γ s0
c = γ coth γ s0 −
1
.
s0
(93)
The resulting curve is shown in Fig. 8.
6. 2D model and analyses
The saddle-point method can also be used to approximate the solutions of linear integrodifference equations in higher spatial dimensions. In this section, we extend invasion model
(3) to two dimensions, write out the formal solution for this model, and use a bivariate
saddle-point formula to approximate the formal solution for large times. We illustrate the
method with approximations for the asymmetric Gaussian kernel and the radially symmetric Laplace kernel. In the case of radially symmetric kernels, we may also define a
unique speed of invasion. Our descriptions in this section are substantially briefer with
fewer details than those in previous sections.
6.1. 2D model and solution
We now consider the linear integrodifference equation
+∞ +∞
k(x − x , y − y )nt (x , y ) dx dy nt+1 (x, y) = λ
−∞
(94)
−∞
for a point release at the origin,
n0 (x, y) = n0 δ(x)δ(y),
(95)
Saddle-Point Approximations and Invasions
1809
of size n0 , in two dimensions. We restrict our attention to dispersal kernels k(x, y) whose
2D moment generating function, defined below, exists in some neighborhood of the origin.
Let us proceed in analogy with our previous one-dimensional analyses. We introduce
the two-dimensional exponential transform pair
+∞ +∞
¯
f (x, y)eux+vy dx dy,
(96)
f (u, v) =
−∞
f (x, y) =
−∞
1
(2πi)2
a+i∞ b+i∞
a−i∞
f¯(u, v)e−(ux+vy) du dv.
(97)
b−i∞
For a probability density function, f (x, y) = k(x, y), the two-dimensional exponential
transform is just the bivariate moment generating function
+∞ +∞
k(x, y)eux+vy dx dy.
(98)
M(u, v) =
−∞
−∞
Let us now take the exponential transform of Eq. (94). This is just
n̄t+1 (u, v) = λM(u, v)n̄t (u, v)
(99)
with the initial condition
n̄0 (u, v) = n0 .
(100)
At each time step, we multiply our transformed density by the geometric growth rate λ
and by the bivariate moment generating function M(u, v). It now follows that
n̄t (u, v) = n0 λt M t (u, v).
(101)
After applying the inverse of our exponential transform, we obtain the formal solution
a+i∞ b+i∞
1
M t (u, v)e−(ux+vy) du dv.
(102)
nt (x, y) = n0 λt
(2πi)2 a−i∞ b−i∞
6.2. 2D saddle-point approximation
We now apply the method of steepest descent to the above formal solution. We put this
solution into the standard form for the method by introducing the bivariate cumulant generating function
K(u, v) = ln M(u, v).
(103)
The solution can now be written
a+i∞ b+i∞
y
1
x
et[K(u,v)− t u− t v] du dv.
nt (x, y) = n0 λt
(2πi)2 a−i∞ b−i∞
(104)
We deform our contour through the critical point, (u, v) = (u0 , v0 ), of
f (u, v; t, x, y) ≡ K(u, v) −
x
y
u − v.
t
t
(105)
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Kot and Neubert
This critical point satisfies
∂f
∂K
x
=
− = 0,
∂u
∂u
t
∂f
∂K
y
=
− = 0.
∂v
∂v
t
(106)
We may now use the first term of the asymptotic expansion of our double integral
(Good, 1957; Renshaw, 2000; Butler, 2007; Paolella, 2007) to write the bivariate saddlepoint approximation
nt (x, y) ∼
where
n0 λt M t (u0 , v0 )e−u0 x−v0 y
,
√
2πt |K (u0 , v0 )|
2
∂ K
2
K (u0 , v0 ) = ∂u
∂ 2K
∂u∂v
∂ 2K
∂u∂v
∂ 2K
∂v 2
(107)
(108)
(u0 ,v0 )
is the Hessian determinant of the bivariate cumulant generating function evaluated at the
critical point.
6.3. Bivariate Gaussian kernel
As a simple example of approximation (107), we consider a bivariate normal distribution,
2
−1
2ρxy
y2
1
x
exp
−
+
k(x, y) =
,
(109)
2(1 − ρ 2 ) σ12
σ1 σ2
σ22
2πσ1 σ2 1 − ρ 2
with zero means, variances σ12 and σ22 , and correlation ρ (Renshaw, 2000). The cumulant
generating function for this distribution is
K(u, v) =
1 2 2
σ1 u + 2ρσ1 σ2 uv + σ22 v 2 .
2
(110)
The equations for our critical points are
σ12 u + ρσ1 σ2 v =
x
,
t
ρσ1 σ2 u + σ22 v =
y
,
t
(111)
with critical-point coordinates
u0 =
xσ22 − yρσ1 σ2
,
t (1 − ρ 2 )σ12 σ22
v0 =
yσ12 − xρσ1 σ2
.
t (1 − ρ 2 )σ12 σ22
Our saddle-point approximation now reduces to
2
n0 λt
2ρxy
y2
−1
x
nt (x, y) ∼
−
+
.
exp
2(1 − ρ 2 )t σ12
σ1 σ2
σ22
2πσ1 σ2 1 − ρ 2 t
This approximation is equal to the exact solution.
(112)
(113)
Saddle-Point Approximations and Invasions
1811
6.4. Radially symmetric kernels
2D saddle-point approximation (107) simplifies significantly if the kernel k(x, y) is radially symmetric. To see this, consider the moment generating function
+∞ +∞
M(u, v) =
−∞
k(x, y)eux+vy dx dy
(114)
−∞
for a radially symmetric kernel and let
x = r cos θ,
y = r sin θ,
(115)
u = s cos φ,
v = s sin φ.
(116)
and
It now follows that
M(s, φ) =
∞ 2π
0
esr cos(θ−φ) k(r)r dθ dr.
(117)
0
Since
2π
esr cos(θ−φ) dθ = 2πI0 (sr),
(118)
0
where I0 (r) is the modified Bessel function of the first kind of order zero, we may write
M(s) = 2π
∞
k(r)I0 (sr)r dr.
(119)
0
Using the chain rule, we now see that
∂K
∂K ∂s
M (s) u
=
=
∂u
∂s ∂u
M(s) s
(120)
∂K
∂K ∂s
M (s) v
=
=
,
∂v
∂s ∂v
M(s) s
(121)
and
so that the critical conditions,
x
∂K
= ,
∂u
t
∂K
y
= ,
∂v
t
(122)
reduce to
r
M (s)
cos φ = cos θ,
M(s)
t
M (s)
r
sin φ = sin θ.
M(s)
t
(123)
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Kot and Neubert
We thus want values of s = s0 and φ = φ0 that satisfy
M (s) r
= ,
M(s)
t
φ = θ.
(124)
Using these values of s0 and φ0 , saddle-point approximation (107) may now be rewritten
n0 λt M t (s0 )e−s0 r
,
√
2πt |K (s0 )|
(125)
1 ∂K ∂ 2 K K (s0 ) =
.
s ∂s ∂s 2 s=s0
(126)
nt (r) ∼
with
This asymptotic approximation can be used to determine the invasion speed for a population whose average spread rate converges to a positive constant. Let us imagine that a
population is first detected when it reaches the threshold density nc . We imagine that this
occurs at some r = r(t, nc ) > 0, which we, for convenience, call r. If we set nt (r) to nc
and solve for the ratio of r and t , we find that
1
1
nc
r
=
2πt K (s0 ) .
ln λM(s0 ) − ln
(127)
t
s0
t
n0
If this spread rate converges to a constant, then s0 converges to a constant and
c ≡ lim
t→∞
1 r
= ln λM(s0 )
t
s0
(128)
or, equivalently,
λ=
ecs0
.
M(s0 )
(129)
In this same limit,
c=
M (s0 )
.
M(s0 )
(130)
Taking our last two equations together, we see that
λ=
es0 M (s0 )/M(s0 )
,
M(s0 )
c=
M (s0 )
,
M(s0 )
where, you will remember,
∞
k(r)I0 (sr)r dr.
M(s) = 2π
(131)
(132)
0
Parametric Eqs. (131), with moment generating function (132), can now be used to determine the speed of invasion in two dimensions, in the same way that Eq. (56), with moment
Saddle-Point Approximations and Invasions
1813
generating function (6), was used to determine the speed of invasion in one dimension.
These two-dimensional speeds are equivalent to those reported by Lewis et al. (2006) and
Fort (2007).
6.5. Radially symmetric Laplace kernel
As a simple example of saddle-point approximation (125) we consider the radially symmetric, bivariate Laplace distribution
k(x, y) =
where σ =
γ2 2
K0 γ x + y 2 ),
2π
(133)
√
2/γ is the standard deviation of the marginal distribution,
1
k(x) = γ e−γ |x| ,
2
(134)
in the x (or y) direction. The density is unbounded at the origin. The exact density after t
iterates is
nt (r) =
n0 λt γ t+1 t−1
r Kt−1 (γ r)
2t π(t)
(135)
(Jakeman and Pusey, 1976; Hughes, 1995).
The moment generating function of our original density is just
M(u, v) =
γ2
γ2
− (u2 + v 2 )
(136)
or
M(s) =
γ2
.
γ 2 − s2
(137)
The critical point, in polar form, satisfies
M (s) r
= .
M(s)
t
(138)
This occurs when
2s
r
− = 0.
γ 2 − s2
t
(139)
This last equation may be rewritten (for r = 0) as a quadratic equation,
t
s 2 + 2 s − γ 2 = 0.
r
(140)
By Descartes’s rule of signs, both roots are real; one root is positive and one is negative.
Since r > 0 and t > 0, we take the root
t
t2
+ γ 2.
(141)
s0 = − +
r
r2
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Kot and Neubert
Fig. 9 Plots of exact solution (135) (solid curves) and saddle-point approximation
(143) (dotted curves)
√
for the radially symmetric, bivariate Laplace distribution with λ = 1.2, γ = 2, n0 = 1, and t = 15, 20, 25.
The population density is plotted on an arithmetic scale.
After some tedious calculations, one can show that
K (s0 ) = 4
(γ 2 + s02 )
.
(γ 2 − s02 )3
(142)
It now follows that
nt (x, y) ∼
n0 λt
4πt
t −s r
γ2
e 0
γ 2 −s02
,
γ 2 +s02
(143)
(γ 2 −s02 )3
where s0 is given by Eq. (141).
Figures 9 and 10 show approximate solution (143) overlaid on top of exact solution
(135) on both an arithmetic and a logarithmic scale. The agreement, as usual, is quite
good. Parametric equations (131) for the speed of invasion, in this case, are identical to
parametric equations (78) for the one-dimensional Laplace kernel.
7. Empirical saddle-point approximation
Even with good growth and dispersal data, estimates of invasion speed are sensitive to the
exact functional form chosen for the dispersal kernel (Kot et al., 1996; Clark, 1998). Clark
et al. (2001) and Lewis et al. (2006) have argued that a good way to remove this sensitivity
is to use a nonparametric estimator for the moment generating function, one that makes
no assumptions about the form of the underlying kernel, in speed formulas (56) and (57).
For one dimension, they recommend the empirical moment generating function
M̂(s) =
N
1 exp(sXi ),
N i=1
(144)
Saddle-Point Approximations and Invasions
1815
Fig. 10 Plots of exact solution (135) (solid curves) and saddle-point approximation
(143) (dotted curves)
√
for the radially symmetric, bivariate Laplace distribution with λ = 1.2, γ = 2, n0 = 1, and t = 15, 20, 25.
The population density is now plotted on a logarithmic scale.
when the Xi (the dispersal data) are N measured dispersal displacements and
M̂(s) =
N
1 cosh(sXi ),
N i=1
(145)
when the Xi are (nonnegative) distances, rather than displacements.
Predictions of the population density for large times may also be sensitive to the exact
functional form chosen for the dispersal kernel. We may, however, combine the empirical moment generating function with our earlier asymptotic approximation, Eq. (51), to
produce an empirical saddle-point approximation (Feuerverger, 1989) for the population
density. In one dimension, this empirical approximation takes the form
n̂t (x) ∼
n0 λt M̂ t (s0 )e−s0 x
,
√
2πt|κ2 |
(146)
where s0 , for each x and t , is the root of
M̂ (s0 )
M̂(s0 )
=
x
t
(147)
(cf. Eq. (40)) and
κ2 = K̂ (s0 ) =
M̂(s0 )M̂ (s0 ) − [M̂ (s0 )]2
M̂ 2 (s0 )
(148)
for that s0 . Here, K̂(s) is the empirical cumulant generating function, the natural logarithm
of the empirical moment generating function.
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Kot and Neubert
Fig. 11 Plots (thin curves) of empirical saddle-point approximation (146) for λ = 1.2, n0 = 1, t = 10 for
5 sets of simulated dispersal data.
√ Each set of data consisted of N = 100 dispersal distances drawn from a
Laplace distribution with γ = 2. The figure also shows (thick curve) saddle-point approximation (77) for
comparison. For N = 100, the empirical curves show a fair amount of scatter about the theoretical curve.
Fig. 12 Plots (thin curves) of empirical saddle-point approximation (146) for λ = 1.2, n0 = 1, t = 10 for
5 sets of simulated dispersal data.
√ Each set of data consisted of N = 500 dispersal distances drawn from
a Laplace distribution with γ = 2. The figure also shows (thick curve) saddle-point approximation (77)
for comparison. For N = 500, the empirical curves show little scatter about the theoretical curve.
We tested the empirical saddle-point approximation for the one-dimensional Gaussian,
Laplace, and uniform distributions using simulated data. For each kernel, we generated
random dispersal-distance data, used Eq. (145) for the empirical moment generating function, solved Eq. (147) numerically, using bisection, and used the resulting value of s0 in
Eq. (146). Figure 11 shows the empirical saddle-point approximations n̂10 (x) (thin curves)
for 5 sets of 100 dispersal distances (N = 100) generated from the Laplace distribution.
The figure also shows the true saddle-point approximation (thick curve), for the complete
Saddle-Point Approximations and Invasions
1817
dispersal kernel, for comparison. Figure 12 is the comparable figure for 5 sets of 500 data
points. Both sets of curves give reasonable estimates of the density, but it is clear that there
is far less scatter around the true saddle-point approximation for larger values of N , i.e.,
for better estimates of the true moment generating function. In general, the relative error
falls off with the square root of the sample size (Feuerverger, 1989). For fixed variance,
the empirical saddle-point approximation worked better for the Gaussian distribution than
for the Laplace distribution. It worked better still for the uniform distribution.
8. Censored data
The biggest potential problem with the empirical moment generating function and with
the empirical saddle-point approximation is that dispersal data are often censored. Numerous studies (Shaw, 1995; Kot et al., 1996; Lewis, 1997; Clark, 1998; Higgins and
Richardson, 1999; Brown and Hovmoller, 2002; Shigesada and Kawasaki, 2002; Caswell
et al., 2003; Lutscher, 2007) have suggested that invasion speeds are sensitive to the tails
of dispersal kernels. At the same time, long-distance dispersal data are often difficult to
collect; long-distance propagules often fall outside the range of field studies (Bullock and
Clarke, 2000; Cain et al., 2000; Nathan et al., 2003).
One theoretical approach for studying the effects of the censoring of dispersal data is
to introduce the censored dispersal kernel
k(x; r) =
k(x)
+r
,
−r k(x)dx
0,
|x| ≤ r,
|x| > r.
(149)
Here, the sampling is out to maximum distance r; we will ignore any propagule that lands
beyond that distance. This censored kernel engenders the censored moment generating
function
+r
k(x)esx dx
M(s; r) = −r
.
(150)
+r
−r k(x) dx
If one computes the invasion speed with the usual parametric formulas, but with a
censored moment generating function,
λ=
esM (s;r)/M(s;r)
,
M(s; r)
c=
M (s; r)
,
M(s; r)
(151)
one obtains a speed c that depends upon the sampling distance r. We will use the term
speed rarefaction curve for the plot of the resulting speed,
c = c(r; λ),
(152)
as a function of the distance r for a given λ. Speed rarefaction curves can be used to
understand the effects of censoring on empirical moment generating functions and on the
empirical saddle-point approximation.
1818
Kot and Neubert
As an example, consider the Laplace distribution
1
k(x) = γ e−γ |x|
2
(153)
with moment generating function
M(s) =
+∞
−∞
k(x)esx dx =
γ2
γ2
,
− s2
|s| < γ .
(154)
The censored density function is
k(x; r) =
γ e−γ |x|
,
2(1−e−γ r )
|x| ≤ r,
0,
|x| > r,
(155)
and the corresponding censored moment generating function is
M(s; r) =
1
e−r(γ −s) e−r(γ +s)
γ
2γ
−
−
.
2 (1 − e−γ r ) γ 2 − s 2
γ −s
γ +s
(156)
It now follows that
1
γ
[r(γ − s) + 1]e−r(γ −s)
4γ s
−
M (s; r) =
2 (1 − e−γ r ) (γ 2 − s 2 )2
(γ − s)2
[r(γ + s) + 1]e−r(γ +s)
+
.
(γ + s)2
(157)
Figure 13 shows the speed rarefaction curve for the Laplace distribution for λ = 2
and γ = 0.5. (The speed rarefaction curves for the Gaussian and uniform distributions
are qualitatively similar.) The speed rises quickly and saturates with increasing sampling
distance r. Overlaid on top of this curve are a series of speeds (solid dots) estimated
using a censored empirical moment generating function. In particular, we generated 1,000
dispersal distances that obeyed the appropriate Laplace distribution of displacements and
censored these data by considering only those distances that were less than the maximum
sampling distance r. The estimated speeds were computed by using the empirical moment
generating function for the censored data. These speeds, which comprise an empirical
speed rarefaction curve, closely follow the theoretical rarefaction curve, but extend along
the r-axis only as far as the maximum distance that appeared in the simulations. A field
study would need to measure large enough dispersal distances to reach the plateau of the
speed rarefaction curve in order to accurately predict potential speeds of invasion. That
is, an empirical moment generating function cannot by itself solve the problem of badly
censored data.
Censored data can also affect empirical saddle-point approximations (see Fig. 14). The
effects are the ones we would expect from the effects of censorship on speed, with more
heavily censored data typically yielding approximations with steeper fronts and narrower
tails than the true solution.
Saddle-Point Approximations and Invasions
1819
Fig. 13 A speed rarefaction curve (solid curve) for the censored Laplace distribution with γ = 0.5 and
λ = 2.0. The speed c rises and saturates with increasing sampling distance r. The curve was obtained using
parametric equations (151) and censored moment generating function (156). The solid dots are the speeds
obtained using a censored empirical moment generating function for N = 1000 dispersal distances drawn
from the Laplace distribution.
Fig. 14 Empirical saddle-point approximations (thin curves) of the population density for censored dispersal data. The curves were computed using empirical approximation (146) for λ = 2.0, n0 = 1, and
t = 10, but using a censored empirical moment generating function for 1,000 distances drawn from a
Laplace distribution with γ = 0.5. The figure shows empirical saddle-point approximations for (from top
to bottom along the vertical axis) maximum sampling distances r = 4, 6, 8, 10. The figure also shows (solid
curve) the theoretical saddle-point approximation, using the entire Laplace distribution, for comparison.
Curves for heavily censored data typically have steeper fronts and narrower tails than the theoretical curve.
9. Discussion
The method of steepest descent is commonly used in statistics (Daniels, 1954; Reid, 1988;
Goutis and Casella, 1999; Butler, 2007) and in the study of random walks (Domb and Of-
1820
Kot and Neubert
fenbacher, 1978; Hughes, 1995). Radcliffe and Rass (1997), in turn, highlighted the use
of this method for discrete-time spatial models. Despite the presence of this literature,
saddle-point approximations are rarely found in analyses of integrodifference equations
for biological invasions. In this paper, we have demonstrated that the method of steepest
descent is a powerful asymptotic method for analyzing linear integrodifference equations.
We have described the method, have applied it to a simple model of biological invasion,
and have compared the saddle-point approximation to the exact solution for five exactly
solvable kernels. Finally, we used the saddle-point method to determine the speed of invasion, much in the manner of Radcliffe and Rass (1997).
For all five examples, the saddle-point approximation provided an excellent match
to the exact solution for large times. Indeed, we found that the saddle-point method is
better than advertised. For the parameters that we studied, the method provides good
matches to the exact solution not only for large times, but for all times except (possibly)
the first few iterates. The method was straightforward to use and in using this method we
exchanged the nasty problem of calculating an inverse transform for the simpler problem
of determining a root. In our examples, we determined the root of Eq. (40) analytically or
by approximation, but for harder problems one could also solve this equation numerically.
Although we considered a simple scalar integrodifference equation, we can easily
imagine applying the same method to more complicated models with age or stage structure (Neubert and Caswell, 2000; Neubert and Parker, 2004). Even for our simple problem, there were unexpected payoffs and bonuses. One of the most exciting of these was
the empirical saddle-point approximation, which allows one to predict the density of an
invading population from empirical dispersal data without having to estimate or make assumptions about the underlying kernel. The empirical saddle-point approximation complements the use of the empirical moment generating function for predicting rates of
spread from empirical data (Clark et al., 2001; Lewis et al., 2006).
The biggest potential problem with the empirical moment generating function and with
the empirical saddle-point approximation is that dispersal data are often censored; longdistance dispersal events are often missed. An empirical speed rarefaction curve, in which
the data are artificially censored and the estimated speed is plotted as a function of the
sampling radius, can be used to assess the effect of censoring on speeds and densities.
We encourage all investigators publishing estimates of speed or density, based upon empirical moment generating functions or empirical saddle-point approximations, to include
empirical speed rarefaction curves.
Acknowledgements
This work was supported by a grant from the National Science Foundation (DEB0235692), a fellowship from the Ocean Life Institute of the Woods Hole Oceanographic
Institution, and a Mellon Independent-Study Award from the Woods Hole Oceanographic
Institution, all to MGN. Both authors thank Mark A. Lewis, Jan Medlock, and Robert E.
O’Malley, Jr. for helpful discussions or for reading and commenting on the manuscript.
We also thank the anonymous reviewers for their helpful comments.
Saddle-Point Approximations and Invasions
1821
Appendix A
In this Appendix, we evaluate solution (13),
nt (x) = n0 λt
1
2πi
a+i∞
M t (s)e−sx ds,
(A.1)
a−i∞
for the Gaussian, Laplace, and uniform distributions.
A.1 Gaussian kernel
For a Gaussian dispersal kernel with standard deviation σ ,
2
1
− x
k(x) = √ e 2σ 2 ,
σ 2π
(A.2)
the moment generating function is
M(s) = eσ
2 s 2 /2
,
where −∞ < Re s < ∞. For this kernel, solution (A.1), reduces to
a+i∞
1
2 2
nt (x) = n0 λt
eσ s t/2 e−sx ds.
2πi a−i∞
(A.3)
(A.4)
If we take our contour to be the imaginary axis (a = 0) and set s = iω, our solution,
+∞
1
2 2
nt (x) = n0 λt
e−σ ω t/2 e−iωx dω,
(A.5)
2π −∞
is in terms of a standard Fourier transform. It quickly follows that
n0 λt
x2
nt (x) = √
exp − 2 .
2σ t
σ 2πt
(A.6)
A.2 Laplace distribution
For a Laplace distribution,
1
k(x) = γ e−γ |x| ,
2
with standard deviation
√
2
,
σ=
γ
(A.7)
(A.8)
the moment generating function is
M(s) =
γ2
γ2
,
− s2
(A.9)
1822
Kot and Neubert
where −γ < s < γ . For this kernel, solution (A.1) can now be written
1
nt (x) = n0 λ
2πi
a+i∞ t
a−i∞
γ2
γ 2 − s2
t
e−sx ds.
(A.10)
The integrand has poles of order t at s = ±γ . For x < 0, we close our contour in the
left-half complex s-plane while, for x > 0, we close our contour in the right-half complex
s-plane. In both cases, we may use the calculus of residues to show that
s|x| n0 λt γ 2t
d t−1
e
lim
.
(A.11)
nt (x) =
(t − 1)! s→−γ ds t−1 (γ − s)t
It follows, using the Leibniz (generalized product) rule, that
nt (x) =
t−1
n0 λt γ −γ |x| (t − 1 + m)! (γ |x|)t−1−m
e
.
2t (t − 1)!
(t − 1 − m)!m!
2m
m=0
(A.12)
The factor
M t (s) =
γ2
γ 2 − s2
t
(A.13)
that occurs in formal solution (A.10) is the moment generating function for the generalized Laplace distribution (Kotz et al., 2001). It follows that our solution, corresponding to
a point release of size n0 at the origin, may also be written
1
γ t+ 2
1
nt (x) = n0 λ
|x|t− 2 Kt− 1 γ |x| ,
1
2
t− 2 √
2
π(t)
t
(A.14)
where Kν (x) is a modified Bessel function of the second kind of order ν and (t) is the
gamma function. The symmetric generalized Laplace distribution is thus a special case of
the Bessel function distribution (McKay, 1932).
A.3 Uniform distribution
For a symmetric uniform or rectangular distribution,
k(x) =
1
2γ
0,
,
|x| ≤ γ ,
|x| > γ ,
(A.15)
with standard deviation
γ
σ=√ ,
3
(A.16)
the moment generating function is
M(s) =
sinh γ s
.
γs
(A.17)
Saddle-Point Approximations and Invasions
1823
The solution, corresponding to a point release of size n0 at the origin, can now be written
a+i∞ 1
sinh γ s t −sx
nt (x) = n0 λt
e ds.
(A.18)
2πi a−i∞
γs
The integrand has a removable singularity at the origin. The integrand may, however,
be reformulated in a more useful way. Since
γs
e − e−γ s t
sinh γ s t
=
,
(A.19)
γs
2γ s
it follows that
t
1 sinh γ s t
t γ s t−m −γ s m
m
=
(−1)
e
e
t
m
γs
(2γ s) m=0
t
1 t
m
=
(−1)
eγ s(t−2m)
m
(2γ s)t
(A.20)
m=0
so that
nt (x) =
a+i∞
t t
n0
1 −s[x−γ (t−2m)]
λ
t
(−1)m
e
ds.
m a−i∞ s t
2πi 2γ m=0
(A.21)
We now need to calculate the residue for each term of the sum. For concreteness, let
us assume that x > 0 and that our contour lies to the right of the origin. If
x − γ (t − 2m) > 0,
(A.22)
we may close our contour off in the right-half s-plane. We will then obtain zero for our
residue. If
x − γ (t − 2m) < 0,
(A.23)
we may close our contour off in the left-half s-plane. There are
m<
γt −x
2γ
(A.24)
such terms. For each of these terms, we will pick up a contribution from the pole of order
t at the origin. It quickly follows that, for x > 0,
nt (x) =
t m̄
t (x)
t−1
n0
λ
t (−1)m
γ (t − 2m) − x
m
(t − 1)! 2γ
m=0
for 0 < x < γ t and zero for x > γ t , where
γt −x
m̄t (x) = floor
2γ
is the largest integer less than or equal to its right-hand side argument.
(A.25)
(A.26)
1824
Kot and Neubert
By symmetry and continuity, it follows that we can write the general solution, for
arbitrary x, as
nt (x) =
t m̄
t (x)
t−1
λ
n0
t (−1)m
γ (t − 2m) − |x|
m
(t − 1)! 2γ
m=0
for |x| < γ t and zero otherwise, where
γ t − |x|
m̄t (x) = floor
.
2γ
(A.27)
(A.28)
The above solution is closely related to the classical problem of deriving the probability
density for the sum of uniform random variables (Lobatschewsky, 1842; Rényi, 1970;
Feller, 1971; Lusk and Wright, 1982).
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