Fractional reaction-diffusion equation for species ... - ResearchGate

Noname manuscript No.
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Boris Baeumer · Mihály Kovács · Mark
M. Meerschaert
Fractional reaction-diffusion equation
for species growth and dispersal
Received: date / Revised: date
Abstract Reaction-diffusion equations are commonly used to model the
growth and spreading of biological species. The classical diffusion term implies
a Gaussian dispersal kernel in the corresponding integro-difference equation,
which is often unrealistic in practice. In this paper, we propose a fractional
reaction-diffusion equation where the classical second derivative diffusion
term is replaced by a fractional derivative of order less than two. The resulting
model captures the faster spreading rates and power law invasion profiles
observed in many applications, and is strongly motivated by a generalised
central limit theorem for random movements with power-law probability tails.
We then develop practical numerical methods to solve the fractional reactiondiffusion
equation by time discretization and operator splitting, along with
Partially supported by NSF grants DMS-0139927 and DMS-0417869 and the Marsden
Fund administered by the Royal Society of New Zealand.
Boris Baeumer
Department of Mathematics and Statistics, University of Otago,
P.O. Box 56, Dunedin 9001, New Zealand
Tel.: +643-479-7763
Fax: +643-479-8427
E-mail: bbaeumer@maths.otago.ac.nz
Mihály Kovács
Department of Mathematics and Statistics, University of Otago,
P.O. Box 56, Dunedin 9001, New Zealand
Tel.: +643-479-7889
Fax: +643-479-8427
E-mail: mkovacs@maths.otago.ac.nz
Mark M. Meerschaert
Department of Mathematics and Statistics, University of Otago,
P.O. Box 56, Dunedin 9001, New Zealand
Tel.: +643-479-7889
Fax: +643-479-8427
E-mail: mcubed@maths.otago.ac.nz

2 Baeumer, Kovács, Meerschaert
some existing methods from the literature on anomalous super-diffusion. In
the process, we establish the mathematical relationship between the discrete
time integro-difference and continuous time reaction-diffusion analogues of
the model, along with error bounds. Our general approach also applies to
other alternative non-Gaussian dispersal kernels, and it identifies the analogous
continuous time evolution equations for those models.
Keywords reaction-diffusion equation · growth and dispersal · fractional
derivative · anomalous diffusion · operator splitting
Mathematics Subject Classification (2000) 35K57 · 26A33 · 47D03
1 Introduction
Classical reaction-diffusion equations are useful to model the spread of invasive
species [40,41]. In this model, the population density u(x, t) at location
x and time t is the solution of the partial differential equation
∂u
∂t = f(u) + D ∂2 u
∂x 2 . (1)
The first term on the right is the reaction term that models population
growth; a typical choice is Fisher’s equation f(u) = ru(1 − u/K) where r is
the intrinsic growth rate and K is the carrying capacity. The second term
is the diffusion term; it models spreading/dispersion. Solutions to (1) spread
at a rate proportional to t with exponential leading edges. The main failure
of this model in real applications is the unrealistically slow spreading, since
typical invasive species have population densities that spread faster than t,
with power law leading edges [13,16,17,26,28,43]. The power law exhibits as
a straight line on a log-log plot of dispersive distance, a phenomenon often
seen in field data [58–60]. In this paper, we propose an alternative fractional
reaction-diffusion equation
∂u
∂t = f(u) + D ∂α u
∂x α (2)
with 0 < α ≤ 2, which reduces to the classical equation if α = 2. Solutions to
(2) spread faster than t with power law leading edges [19] and hence provide
a more realistic model for invasive species.
Fractional derivatives are almost as old as their more familiar integerorder
counterparts [37,48]. Fractional derivatives have recently been applied
to many problems in physics [5,10,11,27,32–35,47,61], finance [22,45,46,51–
53], and hydrology [4,6–8,55,56]. If a function u(x) has Fourier transform
û(λ) = ∫ e −iλx u(x)dx then the fractional derivative d α u(x)/dx α has Fourier
transform (iλ) α û(λ), extending the familiar formula for integer α. Fractional
derivatives are used to model anomalous diffusion or dispersion, where a
particle plume spreads at a different rate than the classical diffusion model
predicts [15,34]. Just as the fundamental solution to the classical diffusion
equation is provided by the normal or Gaussian probability density functions
of a Brownian motion, the α-stable probability density functions [21,49] of

Fractional reaction-diffusion equation 3
a Lévy motion [9,50] solve the fractional diffusion equation [32,57]. This is
because fractional derivatives arise from sums of random movements with
power law probability tails [56,31], for which the usual central limit theorem
is replaced by its heavy tail analogue [21,36].
An alternative discrete time model for population growth and dispersal
uses an integro-difference equation
u(x, t + τ) =
∫ ∞
−∞
k τ (x, y)g τ (u(y, t))dy (3)
where the dispersal kernel k τ (x, y) is the probability density of the location
x to which an individual disperses at time t + τ, given that the individual
is located at spatial coordinate y at time t [30,41]. Equation (3) can be
understood as a two-step process: First we apply the iteration equation u →
g τ (u) to grow the population over one time step, and then we integrate
against the kernel k τ (x, y) to disperse the population over the same time step.
The classical reaction-diffusion model (1) can be approximated (in a way we
will make precise later in this paper) with the discrete time model (3) where
g(u 0 ) is the solution to the ordinary differential equation ˙u = f(u); u(0) = u 0
at time t = τ and
k τ (x, y) = √ 1
4πDτ
e − (x−y)2
4Dτ (4)
is a Gaussian dispersal kernel, the fundamental solution to the classical diffusion
equation
∂u
∂t = D ∂2 u
; u(x, t = 0) = δ(x − y)
∂x2 at time t = τ. In the same manner, we will show in this paper that solutions to
the fractional reaction-diffusion equation (2) can be usefully approximated by
the same iteration equation (3) with the only change being that the dispersal
kernel k τ (x, y) is the fundamental solution to the fractional diffusion equation
∂u
∂t = D ∂α u
; u(x, t = 0) = δ(x − y)
∂xα at time t = τ. This dispersal kernel can be explicitly computed [14,42,49].
Ecologists have also experimented with other alternative kernels incorporating
skewness and heavier probability tails, such as the Gamma, Laplace,
Weibull, and Student-t distribution [13,16,17,30,39]. This paper will also
show how those models relate to a variant of the fractional reaction-diffusion
equation (2), and the exact manner in which the discrete time evolution
equation (3) can be used to approximate the solution of the continuous time
reaction-diffusion equation in this case. Our methods also provide explicit
error bounds on the approximation, which are useful in numerical simulations.
These methods also extend to the case where the reaction term f(u, x)
explicitly depends on the space variable x. This includes, for example, the
Fisher equation with space-variable intrinsic growth rate r(x) and carrying
capacity K(x). Finally we note that, since reaction-diffusion equations are
also important in other areas where anomalous diffusion is widely observed,
including cell biology, chemistry, physics, geophysics, and finance, it seems
likely that the results reported here will also be useful in a broader context.

4 Baeumer, Kovács, Meerschaert
2 Problem outline
The classical reaction-diffusion equation (1) and its fractional analogue (2)
are both special cases of a general form
∂u
∂t (x, t) = (Au)(x, t) + f(u(x, t)), u(x, 0) = u 0(x) (5)
where A is a pseudo-differential operator [25] that appears as the generator
of some continuous convolution semigroup [1,44]. Our goal in this section is
to explore the connection between this continuous time evolution equation
and its discrete time analogue, the integro-difference equation
u n+1 (x) =
∫ ∞
−∞
k τ (x, y)g τ (u n (y)) dy (6)
where u n (x) = u(x, nτ) with some τ > 0 fixed. Our general approach is based
operator theory for abstract differential equations [1,20,44] and infinitely
divisible probability distributions [21,36].
Example 1 To demonstrate the basic idea let us consider the classical reactiondiffusion
equation (1) with a growth term f(u) = ru for some real r ≥ 0. It
is well know that if r = 0 then the solution is given by
u(x, t) =
∫ ∞
−∞
1
√ e − (x−y)2
4Dt u 0 (y) dy := [T (t)u 0 ](x)
4πDt
where the dispersal operator T (t) maps the initial condition at time t = 0,
the function u 0 (x), to the solution of the diffusion equation at time t > 0.
Likewise, if D = 0 then the solution is given by
u(x, t) = e rt u 0 (x) := [S(t)u 0 ](x).
where the growth operator S(t) maps the initial condition to the solution of
the growth equation at time t > 0. Clearly, [S(t+s)u 0 ](x) = [S(t)S(s)u 0 ](x) =
[S(s)S(t)u 0 ](x) and [T (t + s)u 0 ](x) = [T (t)T (s)u 0 ](x) = [T (s)T (t)u 0 ](x) for
all t, s ≥ 0 (the semigroup property). Using the product rule its is easy to see
that the solution to (1) is of the form
u(x, t) =
∫ ∞
−∞
√ 1
4πDt
e − (x−y)2
4Dt e rt u 0 (y) dy = [T (t)S(t)u 0 ](x) (7)
= [S(t)T (t)u 0 ](x) = e rt ∫ ∞
−∞
√ 1
4πDt
e − (x−y)2
4Dt u 0 (y) dy, (8)
that is; the solution of the complex problem (1) can be written using the
solution of the simpler sub-problems. Now if we fix τ > 0, then by (7) and
(8) we have
u n+1 (x) = u(x, (n + 1)τ) = [T ((n + 1)τ)S((n + 1)τ)u 0 ](x)
= [(T (τ)) n+1 (S(τ)) n+1 u 0 ](x) = [T (τ)S(τ)(T (τ)) n (S(τ)) n u 0 ](x)
= [T (τ)S(τ)T (nτ)S(nτ)u 0 ](x) = [T (τ)S(τ)u n ](x)
=
∫ ∞
−∞
√ 1
4πDτ
e − (x−y)2
4Dτ e rτ u n (y) dy :=
∫ ∞
−∞
k τ (x, y)g τ (u n (y)) dy.

Fractional reaction-diffusion equation 5
This means that the exact solution at time t = nτ, for any positive integer
n = 1, 2, . . . satisfies an iteration of the form (6) with k τ given by (4) and
g τ (u n (y)) = e rτ u n (y). Iterations of the form u n+1 = S(τ)T (τ)(u n ) are called
sequential splitting.
We now try to follow the same basic idea in the case when both the differential
operator and the growth term are more complicated. Assume that
the continuous model is well posed in the sense that given u 0 we obtain a
unique solution on R × [0, T ]. In this case, at least formally, the solution can
be written as u(x, t) = [W (t)u 0 ](x) and we call W (t) the solution operator
of (5). In Example 1 the solution operator is given by (7) and since T (t)
and S(t) commute W (t) = T (t)S(t). If we would be able to construct W (t)
as in Example 1, then the solution of (5) would satisfy the natural iteration
u n+1 (x) = [W (τ)u n ](x). Unfortunately if the function f is not just a multiplication
by a constant, then it is usually impossible to obtain a closed form
of W (t). Therefore, we might want to follow the idea of sequential splitting
here, too, that is; instead of looking at (5) we look at two separate differential
equations which are easier to solve and construct an iteration based on
their solutions. Of course we then have to investigate what this has to do
with W (t), the solution operator of (5). Let us denote by T (t) the solution
operator to the pseudo-differential equation
and by S(t) the solution operator to
∂u
(x, t) = (Au)(x, t) (9)
∂t
∂u
(x, t) = f(u(x, t)). (10)
∂t
The idea behind sequential splitting is that for τ small, we solve first (10)
on [0, τ] and then using the obtained solution at time t = τ as initial data
for (9) we solve that on [0, τ]. Then hope that this will be close to the result
if we would do both at the same time. Mathematically, instead of computing
[W (τ)u 0 ](x) we compute [T (τ)S(τ)u 0 ](x). Now if S(t) and T (t) commute,
then this is the same thing. If not, then it is usually different. Assume
that we are interested in the solution of (5) at time t = nτ. Then, by the
above described idea, we set up an iteration u n+1 (x) = [T (τ)S(τ)u n ](x) with
starting value u 0 , or equivalently, u n+1 (x) = [(T (τ)S(τ)) n u 0 ] (x). Usually,
u n (x) ≠ u(x, nτ) (except for the commuting case), but is it true that u n (x)
is close to u(x, nτ) in some sense For example, if we fix t and choose finer
and finer time steps τ = t n , then does u n(x) converge to u(x, t) in some
sense as n → ∞, in other words, does [ T ( t n )S( t n )] n
u0 converge to W (t)u 0
as n → ∞
Example 2 Let us consider the fractional reaction-diffusion equation with
Fisher’s growth term
( )
∂u
∂t (x, t) = ru(x, t) u(x, t)
1 − + D ∂α u
K ∂x α (x, t); u(x, 0) = u 0(x) (11)

6 Baeumer, Kovács, Meerschaert
where 1 < α ≤ 2. To apply the sequential operator splitting procedure, we
first consider the fractional diffusion equation
∂u
∂t (x, t) = D ∂α u
∂x α (x, t); u(x, 0) = u 0(x) (12)
which ∫ can be solved by Fourier transform methods [15,32]: Write û(λ, t) =
e −iλx u(x, t)dx and recall that ∂ α u/∂x α has Fourier transform (iλ) α û. Taking
Fourier transforms on both sides of (12) yields the ordinary differential
equation
d
dtû(λ, t) = D(iλ)α û(λ, t); û(λ, 0) = û 0 (λ)
whose solution is evidently of the form
û(λ, t) = û 0 (λ) exp(tD(iλ) α )
and now we use the fact from probability theory that exp(C(iλ) α ) is known
to be the Fourier transform of an α-stable probability density, which appears
in the generalised central limit theorem [21,36,49]. Then we may write
u(x, t) = [T (t)u 0 ](x) =
∫ ∞
−∞
k t (x − y)u 0 (y) dy (13)
where k t (x) is an α-stable probability density. Next consider the growth
equation
( )
∂u
∂t (x, t) = ru(x, t) u(x, t)
1 − ; u(x, 0) = u 0 (x)
K
with solution
u(x, t) = [S(t)u 0 ](x)
= [ ˜S(t)](u 0 (x)) = K
(
)
K − u 0 (x)
1 −
K + u 0 (x)(exp(rt) − 1)
(14)
where we emphasise that the nonlinear operator S(t) maps the function u 0 to
another real-valued function of x, and ˜S(t) is a function that maps the real
number u 0 (x) to another real number. According to the discussion above,
define an iteration based on the sequential splitting by
u n+1 (x) = [T (τ)S(τ)u n ](x)
=
∫ ∞
−∞
k τ (x − y)K
(
)
K − u n (y)
1 −
dy.
K + u n (y)(exp(rτ) − 1)
(15)
This is a discrete time evolution equation of the form (6), an approximate
solution u(x, nτ) of the fractional reaction-diffusion equation (11). Our goal
is to show that this approximate solution converges to the actual solution as
the time step τ tends to zero.

Fractional reaction-diffusion equation 7
Remark 1 The splitting method (15) has a useful biological interpretation.
Given a typical time scale τ, the formula (15) expresses that the population
first increases via an application of the growth operator S(τ), and then
spreads out via an application of the dispersal operator T (τ). Similarly, if we
model first dispersion and then growth, we obtain the alternative sequential
splitting
u n+1 (x) := [S(τ)T (τ)u n ](x)
⎛
(∫ ∞
K −
= K ⎜
⎝ 1 −
−∞
)
k τ (x − y)u n (y) dy
)
(∫ ∞
K + k τ (x − y)u n (y) dy
−∞
⎞
⎟
⎠ . (16)
(exp(rτ) − 1)
In this paper, we will show that both approaches lead to the same continuous
time limit.
In the next section, we will prove that solutions to the discrete time
integro-difference equation with an α-stable dispersal kernel converge to solutions
of the fractional reaction-diffusion equation as the time step tends to
zero. We will do this by proving a more general result, that also applies to
other useful dispersal kernels. If the growth term f(u) = ru(1 − u/K) as in
Example 2, we will also show that these splitting procedures provide lower
and upper bounds for solutions to the continuous time model. These bounds
will be useful in the numerical methods discussed in Section 4.
3 Mathematical details
Let X be a Banach space of functions u : R → R with associated norm
‖u‖, and consider the abstract reaction-diffusion equation (5) which we will
rewrite here in operator theory notation as
˙u(t) = Au(t) + f(u(t)), t > 0, u(0) = u 0 (17)
to emphasise our point of view that u : [0, ∞) → X and f : X → X. Here
A is the generator of a strongly continuous semigroup {T (t)} t≥0 on X, a one
parameter family of linear operators T (t) : X → X such that: T (0) = I the
identity operator (Iu = u); each T (t) is bounded, meaning that there exists
a real number M > 0 depending on t > 0 such that ‖T (t)u‖ ≤ M‖u‖ for all
u ∈ X; T (t+s) = T (t)T (s) for t, s ≥ 0; t ↦→ T (t)u is continuous in the Banach
space norm for all u ∈ X; and the generator Au = lim h→0+ h −1 (T (h)u − u)
exists for at least some nonzero u ∈ X. We call the set D(A) ⊂ X for
which this limit exists the domain of the linear operator A, and we say that
the semigroup T (t) is generated by A. We say that u : [0, δ) → X is a
local classical/strong solution of (17) if u is continuous on [0, δ), continuously
differentiable on (0, δ), u(t) ∈ D(A) for t ∈ (0, δ) and u satisfies (17) on
(0, δ). If δ can be chosen arbitrarily large then u is a global classical/strong

8 Baeumer, Kovács, Meerschaert
solution of (17). A function u : [0, δ) → X is a local mild solution of (17) if u
is continuous and satisfies the corresponding integral equation
u(t) = T (t)u 0 +
∫ t
0
T (t − s)f(u(s)) ds, 0 ≤ t < δ. (18)
We note that the integral in (18) is a Bochner integral [1,24,44], an extension
of the Lebesgue integral to the Banach space setting which coincides with a
Riemann integral if the integrand is continuous in the Banach space norm.
If δ can be chosen arbitrarily large then u is a global mild solution of (17).
If f : X → X is Lipschitz continuous, that is; ||f(u) − f(v)|| ≤ M||u − v||
for all u, v ∈ X, then for all u 0 ∈ X there is a unique global mild solution
u(t) := W (t)u 0 of (17) with ||W (t)u 0 − W (t)v 0 || ≤ M T ||u 0 − v 0 ||, t ∈ [0, T ]
(see, for example, [44, Section 6.1]). Denote the unique mild solution of (17)
in the special case A ≡ 0 by u(t) := S(t)u 0 . In this case T (t)u = u for all
t ≥ 0, and hence it follows from (18) that
u(t) = u 0 +
∫ t
0
f(u(s)) ds (19)
for all t > 0. Then u is also a strong solution, since if u and f are continuous,
then t ↦→ f(u(t)) is also continuous and t ↦→ ∫ t
f(u(s)) ds is differentiable,
∫
0
and d t
dt
f(u(s)) ds = f(u(t)) (see, e.g., [24, page 67]). Therefore t → u(t) is
0
differentiable in view of (19), and ˙u(t) = f(u(t)). It also follows from (19) that
the collection of nonlinear operators {S(t)} t≥0 forms a semigroup, sometimes
called the flow of the abstract differential equation ˙u = f(u). We say that the
collection S(·) is generated by f. It is well known that the solution operator
W (t)u 0 to the abstract reaction-diffusion equation (17) can be computed by
the Trotter Product Formula
[
W (t)u 0 = lim T (
t
n→∞ n )S( t n )] n [
u0 = lim S(
t
n→∞ n )T ( t n )] n
u0 , u 0 ∈ X, (20)
see, for example, [12,18,38]. This result means that the mild solution to the
abstract reaction-diffusion equation (17) can be computed as an approximation
using the mild solutions of the two sub-problems, the abstract reaction
equation ˙u = f(u) and the abstract diffusion equation ˙u = Au.
The following proposition is a simplified variant of [18, Theorem 15]. We
call a Banach space X an ordered Banach space if it is a real Banach space
endowed with a partial ordering ≤ such that
1. u ≤ v implies u + w ≤ v + w for all u, v, w ∈ X.
2. u ≥ 0 implies λu ≥ 0 for all u ∈ X and λ ≥ 0.
3. 0 ≤ u ≤ v implies ||u|| ≤ ||v|| for all u, v ∈ X.
4. The positive cone X + := {x ∈ X : x ≥ 0} is closed.
A typical example of an ordered Banach space is C 0 (R), the space of continuous
functions u : R → R such that u(x) → 0 as |x| → ∞, endowed with the
supremum norm ‖u‖ = sup{|u(x)| : x ∈ R}, and endowed with the partial
ordering u ≤ v whenever u(x) ≤ v(x) for all x ∈ R. Another example is L p (R)
(1 ≤ p ≤ ∞) endowed with the partial ordering u ≤ v whenever u(x) ≤ v(x)

Fractional reaction-diffusion equation 9
for x ∈ R almost everywhere. An operator A on an ordered Banach space is
called positive if 0 ≤ u ≤ v implies 0 ≤ Au ≤ Av. We also write B ≤ A if
0 ≤ Bu ≤ Au for any u ≥ 0.
Proposition 1 Let X be an ordered Banach space, and assume that the
strongly continuous semigroup T (·) generated by the linear operator A and
the nonlinear semigroup S(·) generated by the Lipschitz continuous function
f are positive. If
T (t)S(t)u 0 ≤ S(t)T (t)u 0 (21)
holds for all t ∈ [0, T ] and u 0 ≥ 0, then the unique mild solution W (t)u 0 of
the abstract reaction-diffusion equation (17) satisfies
[
T (
t
n )S( t n )] n
u0 ≤ [ T ( t
2n )S( t
2n )] 2n
u0 ≤ W (t)u 0
for all u 0 ≥ 0, n ∈ N and t ∈ [0, T ].
≤ [ S( t
2n )T ( t
2n )] 2n
u0 ≤ [ S( t n )T ( t n )] n
u0 (22)
Proof It follows from our assumption that W (t)u 0 is given by (20). Next we
show that
[
T (
t
n )S( t n )] n
u0 ≤ [ T ( t
2n )S( t
2n )] 2n
u0
≤ [ S( t
2n )T ( t
2n )] 2n
u0 ≤ [ S( t n )T ( t n )] n
u0 , (23)
u 0 ≥ 0, n ∈ N and t ∈ [0, T ]. For u 0 ≥ 0 using (21) repeatedly we have
T ( t n )S( t n )u 0 = T ( t
2n )T ( t
2n )S( t
2n )S( t
2n )u 0
≤ T ( t
2n )S( t
2n )T ( t
2n )S( t
2n )u 0 ≤ S( t
2n )T ( t
2n )T ( t
2n )S( t
2n )u 0
≤ S( t
2n )T ( t
2n )S( t
2n )T ( t
2n )u 0 ≤ S( t
2n )S( t
2n )T ( t
2n )T ( t
2n )u 0 = S( t n )T ( t n )u 0.
For any operators A, B : X → X, it is not hard to check that the inequality
0 ≤ B ≤ A implies 0 ≤ B n ≤ A n (n ∈ N) which finishes the proof of (23). Fix
n ∈ N and let x k := [ T ( t )S( t ) ] n2 k u
n2 k n2 k 0 . By (23), x 0 ≤ x 1 ≤ ... ≤ x k ≤ ...
and x k → W (t)u 0 as k → ∞ by (20). Therefore, it follows from the closedness
of the positive cone X + that x k ≤ W (t)u 0 for all k = 0, 1, .... This shows the
second inequality in (3). A similar argument yields the third inequality and
the proof is complete.
⊓⊔
One useful class of strongly continuous semigroups are the infinitely divisible
semigroups, which are associated with certain families of probability
densities. Suppose that Y is a random variable on R with probability density
k(y) and Fourier transform ˆk(λ) = ∫ e −iλy k(y)dy. Let k n = k ∗ · · · ∗ k denote
the n−fold convolution of k with itself. We say that Y (or k) is infinitely
divisible if for each n = 1, 2, 3, . . . there exist independent random variables
Y n1 , . . . , Y nn with the same density k n such that Y n1 + · · · + Y nn is identically
distributed with Y . The normal, Cauchy, double-Gamma, Laplace, α-stable,
and Student-t densities are all infinitely divisible. The Lévy representation

10 Baeumer, Kovács, Meerschaert
(see, e.g., Theorem 3.1.11 in [36]) states that if k is infinitely divisible then
ˆk(λ) = e ψ(λ) where
ψ(λ) = −iλa − 1 ∫ (
2 λ2 b 2 + e −iλy − 1 +
iλy )
y≠0
1 + y 2 φ(y)dy, (24)
where a ∈ R, b ≥ 0 , and the jump intensity φ satisfies
∫
min{1, y 2 } φ(y)dy < ∞.
y≠0
The triple [a, b, φ] is unique, and we call this the Lévy representation of
the infinitely divisible density k. It follows that we can define the convolution
power k t to be the infinitely divisible density with Lévy representation
[ta, tb, tφ], so that k t has Fourier transform e tψ(k) for any t ≥ 0. It is well
known (e.g., see [25, Example 4.1.3]) that every infinitely divisible density is
associated with a strongly continuous semigroup on C 0 (R) defined by
[T (t)u](x) :=
∫ ∞
−∞
k t (x − y) u(y)dy. (25)
The following result allows us to compute the generator of this semigroup
from the Lévy representation.
Proposition 2 Every function u ∈ C 0 (R) with u ′ , u ′′ ∈ C 0 (R) belongs to the
domain of the generator A of the semigroup (25), and for such functions we
have
Au(x) = − au ′ (x) + 1 2 b2 u ′′ (x)
∫ (
)
+ u(x − y) − u(x) + u′ (x)y
1 + y 2 φ(y)dy.
y≠0
(26)
Proof Example 4.1.12 in [25] shows that, for any u ∈ Cc
∞ (R), the space of
infinitely differentiable functions u : R → R that vanish off a compact set, u
belongs to the domain of A and (26) holds. Then it follows from Corollary
2.4 of [54] that the same holds for all u ∈ C 0 (R) with u ′ , u ′′ ∈ C 0 (R). ⊓⊔
Remark 2 The integral formula in the Lévy representation (24) is a slight
generalisation of the formula for the Fourier transform of a compound Poisson
density, and indeed any infinitely divisible density is in some sense the
limiting case of a compound Poisson [36, Remark 3.1.18], a random sum with
a Poisson distributed number of summands where the relative frequency of
jumps of size y is given by the jump intensity φ(y). The generator formula
comes from the fact that T (t)u is a convolution k t ∗ u so that its Fourier
transform is e tψ(λ) û(λ), and then the generator can be computed in Fourier
space by t −1 [e tψ(λ) − 1]û(λ) → ψ(λ)û(λ) so that ψ(λ)û(λ) is the Fourier
transform of Au(x). Then (26) comes from inverting this Fourier transform
using (24) and the fact that multiplication by iλ, e −iλy corresponds with
taking the derivative or shifting in real space, respectively.

Fractional reaction-diffusion equation 11
In what follows we discuss a general abstract reaction-diffusion equation
for Fisher’s equation (11) for non-negative initial data on X := C 0 (R):
˙u(t) = Au(t) + f(u(t)), u(0) = u 0 ≥ 0, (27)
where f : X → X is defined via the function ˜f : R → R as
(
[f(u)](x) = ˜f(u(x)) := ru(x) 1 − u(x) )
K
We introduce a cut off function via the function f ˜ N : R → R as
⎧
0
⎪⎨ (
if u(x) < 0
[f N (u)](x) = ˜f N (u(x)) : ru(x) 1 − u(x) )
if 0 ≤ u(x) ≤ NK
K
⎪⎩
rNK(1 − N) if u(x) > NK.
(28)
(29)
Proposition 3 Let X := C 0 (R) and let f be given by (28). Then the abstract
differential equation
˙u(t) = f(u(t)), u(0) = u 0 ≥ 0 (30)
has a unique strong global solution given by u(t) = S(t)u 0 for each nonnegative
u 0 ∈ X, and in fact we have [S(t)u 0 ](x) = [ ˜S(t)](u 0 (x)) where ˜S(t) is
defined by (14). For any positive integer N ≥ 2, the abstract differential equation
˙u = f N (u), u(0) = u 0 ≥ 0 for the Lipschitz continuous function f N defined
in (29) also has a unique strong global solution given by u(t) = S N (t)u 0
for each u 0 ≥ 0 in X. Furthermore, in this case, if 0 ≤ u 0 (x) ≤ NK for all
x ∈ R, then we also have that 0 ≤ [S(t)u 0 ](x) = [S N (t)u 0 ](x) ≤ NK for all
x ∈ R and all t ≥ 0.
Proof Consider the abstract initial value problem
˙u(t) = f N (u(t)), u(0) = u 0 ≥ 0 ∈ X
which has, by the Lipschitz continuity of f N , a unique global strong solution
u(t) = S N (t)u 0 (see the discussion after (19)), where S N (·) is the nonlinear
semigroup generated by f N . Hence, since the operator norm in this space is
the supremum norm, it follows that the function u x (t) := [S N (t)u 0 ](x) is for
each fixed x ∈ R the unique solution of the ordinary differential equation
d
dt u x(t) = ˜f N (u x (t)), u x (0) = u 0 (x) ≥ 0 ∈ R.
Then it follows easily, using the uniqueness of solutions, that S N (·) is positive,
i.e., if u 0 (x) ≥ 0 for all x ∈ R then u x (t) = [u(t)](x) = [S N (t)u 0 ](x) ≥
0 for all x ∈ R, and also if u 0 (x) ≥ v 0 (x) for all x ∈ R then u x (t) =
[u(t)](x) = [S N (t)u 0 ](x) ≥ v x (t) = [v(t)](x) = [S N (t)v 0 ](x) for all x ∈ R.
Since ˜f N (y) < 0 for all y > K it also follows from uniqueness of solutions
that, if u 0 (x) ≤ NK for all x ∈ R, then [S N (t)u 0 ](x) ≤ NK for all x ∈ R.

12 Baeumer, Kovács, Meerschaert
Moreover, since ˜f N (u) = ˜f(u) for 0 ≤ u ≤ NK it follows that S N (t)u 0 also
solves
˙u(t) = f(u(t)), u(0) = u 0 . (31)
Since f is locally Lipschitz, Theorem [24, Chapter 3, Theorem 3.4.1] also implies
that (31) has a unique strong local solution. The function t ↦→ S N (t)u 0
is defined for all t ≥ 0 and hence S N (t)u 0 is the unique strong global solution
S(t)u 0 of (31) which is necessarily given by (14) since that solves (31)
evaluated pointwise.
⊓⊔
Now we come to the main result of this paper. It connects an abstract
reaction-diffusion equation (5) in continuous time with its discrete time counterpart
(6), and shows that in the case of a Fisher’s equation growth term,
the discrete integro-difference equation yields bounds on the solution of the
continuous reaction-diffusion equation that converge to the unique solution
of the continuous equation as the time step tends to zero.
Theorem 1 Let k be infinitely divisible and let A denote the generator of the
associated strongly continuous convolution semigroup T (t) defined by (25) on
X := C 0 (R). Then (27,28) has a unique mild solution u(t) = W (t)u 0 for all
u 0 ≥ 0 in X given by the Trotter Product Formula
W (t)u 0 = lim
n→∞
Moreover, for all n ∈ N,
[
T (
t
n )S( t n )] n
u0 = lim
n→∞
[
T (
t
n )S( t n )] n
u0 ≤ [ T ( t
2n )S( t
2n )] 2n
u0 ≤ W (t)u 0
where S(·) is defined in (14).
[
S(
t
n )T ( t n )] n
u0 . (32)
≤ [ S( t
2n )T ( t
2n )] 2n
u0 ≤ [ S( t n )T ( t n )] n
u0 , (33)
Proof Choose N ≥ 2 such that 0 ≤ u 0 (x) ≤ NK for all x and consider the
abstract reaction-diffusion equation
˙u(t) = Au(t) + f N (u(t)), u(0) = u 0 ≥ 0. (34)
Since f N : X → X is globally Lipschitz continuous and A is a generator,
there is a unique mild solution u N (t) = W N (t)u 0 of (34) given by the Trotter
Product Formula
[
u N (t) = W N (t)u 0 = lim T (
t
n→∞ n )S N ( t n )] n
u0
[
= lim SN ( t
n→∞ n )T ( t n )] n (35)
u0 ,
see, e.g., [12,18,38]. The semigroup T (·) satisfies 0 ≤ T (t)u 0 ≤ T (t)v 0 for
0 ≤ u 0 ≤ v 0 and t ≥ 0 since k t is a probability density function. If 0 ≤
u 0 (x) ≤ NK for all x, then
0 ≤ [T (t)u 0 ](x) ≤ NK
∫ ∞
−∞
k t (x − y)dy = NK (36)

Fractional reaction-diffusion equation 13
Therefore, by (36) and Proposition 3,
for all x and
0 ≤ [T ( t n )S N ( t n )]n u 0 (x) = [T ( t n )S( t n )]n u 0 (x) ≤ NK (37)
0 ≤ [S N ( t n )T ( t n )]n u 0 (x) = [S( t n )T ( t n )]n u 0 (x) ≤ NK (38)
for all x. This also shows that 0 ≤ [u N (t)](x) ≤ NK for all x in view of (35).
Therefore u N (t) is a mild solution of (27), too, since f N (u(x)) = f(u(x)) for
0 ≤ u(x) ≤ NK. Since f is locally Lipschitz, a well known result [44, Chapter
6, Theorem 1.4] implies that (27) has a unique local mild solution and since
u N (t) is defined for all t > 0 it follows that u N (t) is the unique global mild
solution of (27) and is given by the Trotter product formula (32) in view of
(35), (37) and (38). Finally, an easy computation shows that the function
y ↦→ [ ˜S(t)](y) in (14) is concave down on y > 0 for any t > 0. Therefore, by
Jensen’s inequality [21, pp. 153–154],
[∫ ∞
]
[S(t)T (t)u 0 ](x) = ˜S(t) k t (y)u 0 (x − y) dy
−∞
Thus,
≥
=
∫ ∞
−∞
∫ ∞
−∞
k t (y) ˜S(t) [u 0 (x − y)] dy
k t (y) [S(t)u 0 ] (x − y) dy = [T (t)S(t)u 0 ](x).
S N (t)T (t)u 0 = S(t)T (t)u 0 ≥ T (t)S(t)u 0 = T (t)S N (t)u 0
which finishes the proof by Proposition 1, (37) and (38).
Corollary 1 Under the assumptions of Theorem 1, if u 0 and its first two
derivatives in x exist and and belong to C 0 (R), then (27,28) has a unique
strong solution u(t) = W (t)u 0 for all u 0 ≥ 0 in X given by the Trotter
Product Formula (32).
Proof In this case Proposition 2 shows that u 0 is in the domain of the operator
A, and since f : X → X is continuously differentiable, u is also a strong
solution by [44, Chapter 6, Theorem 1.5].
⊓⊔
Theorem 1 shows that the fractional reaction-diffusion equation (11) in
continuous time can be solved numerically by computing solutions to one of
its discrete time counterparts (15) or (16) with τ = t/n. The approximate
solutions u n (x, t) converge to the unique solution u(x, t) of the fractional
reaction-diffusion equation at any time t > 0 for any smooth initial population
density u 0 (x) as n → ∞. This result links the continuous time partial
differential equation model with the corresponding discrete time integrodifference
equation model. Furthermore, the approximation (15) gives a lower
bound to the exact solution while the approximation (16) gives an upper
bound. Hence these two approximation can be compared to yield exact error
bounds. See Section 4 for an illustration.
⊓⊔

14 Baeumer, Kovács, Meerschaert
Remark 3 Theorem 1 extends immediately to any abstract reaction-diffusion
equation of the form (27) as long as u ↦→ [ ˜S(t)](u) is concave down on some
interval u ∈ [0, M] for each t > 0, and solutions to the differential equation
˙u = f(u) remain in this interval for all time whenever u(0) ∈ [0, M]. It also
extends to the case where f(u, x) depends on the point x in space, and/or
the multidimensional case x ∈ R d , since all the proofs are point-wise. Hence
the results of this paper can also be adapted to model patchy populations in
d-dimensional space, where the growth rate and carrying capacity vary with
spatial location.
The most familiar choice for the infinitely divisible semigroup T (t) in
Theorem 1 is the Gaussian semigroup, where k t is the normal density with
mean zero and variance 2Dt for some D > 0. This is the semigroup introduced
in Example 1, and since the Lévy representation of k is [0, 2D, 0] (i.e., we have
a = 0, φ ≡ 0, and b 2 = 2D in (24)), equation (26) shows that its generator
is Au = D ∂ 2 u/∂x 2 . In applications to biology, the probability density k t is
the dispersal kernel that describes the dispersion of population over a time
interval of length t. The choice of a Gaussian dispersal kernel is justified by
the central limit theorem [21,36] which states that the accumulation of a
series of independent random movements will converge to a Gaussian form
as the number of movements increases to infinity. This is the same argument
that justifies the Brownian motion model in cell and molecular biology.
There are several reasons why the Gaussian dispersal kernel might not
apply as a model for the spread of a population density. First of all, the time
interval may be too short for the limit theorem to apply. Second, correlation
in movements can invalidate the Gaussian limit. Finally, and most important
for the purposes of this paper, heavy tailed movements can also invalidate
the Gaussian limit. Recent empirical research [13,16,17,26,28,43,58–60] has
produced a body of evidence indicating a power law distributions of movements,
so that if R is the magnitude (radius) of the movement of a randomly
selected member of the population, then we have P (R > r) ≈ Cr −α for r
large. A log-log plot of such movements will exhibit as a straight line with
slope −α (power law tail), see for example Figure 2 in [60] or Figure 3 in
[58] or Figure 2 in [13]. This leads to a density function k(r) ≈ Cαr −α−1
for r large, and so if 0 < α < 2, then an easy calculation shows that the
second moment ∫ r 2 k(r)dr = ∞. The non-existence of the second moment
invalidates the Gaussian central limit theorem, and in this case an alternative
limit theorem applies [21,36] in which the Gaussian limit distribution is
replaced by the more general Lévy stable distribution. The stable probability
density function cannot be written in closed form, except in a few special
cases, so it is common to describe these distributions in terms of their Fourier
transforms ˆk(λ) = e ψ(λ) where
⎧
⎨ −iaλ − σ α |λ| α (1 − iβ(signλ) tan πα 2 ) for α ≠ 1
ψ(λ) =
⎩
−iaλ − σ|λ|(1 + iβ 2 π
(signλ) ln λ) for α = 1
where the stable index 0 < α ≤ 2, the skewness −1 ≤ β ≤ 1, the center
a ∈ R, and the scale σ ≥ 0 [49]. The Gaussian distribution is the special

Fractional reaction-diffusion equation 15
case α = 2, and in that case the skewness is superfluous. Stable distributions
represent the only possible limits of normalised sums of independent random
variables, so they are in some sense universal models for diffusion processes.
Stable densities generalise the more familiar Gaussian densities to include
the possibility of skewness and heavy power law tails, since for α < 2 the
stable density k(y) ≈ Cqα|y| −α−1 as y → −∞ and k(y) ≈ Cpαy −α−1 as
y → ∞ where β = p − q. The weights p, q represent the relative chance of
large negative versus positive movements, respectively. In the symmetric case
one has β = 0 since p = q. Figure 1 shows the standard σ = 1, µ = 0 stable
densities in the case α = 1.6 to illustrate the skewness. The symmetric stable
density is similar to a Gaussian density but with power-law tails.
β = 0
0.30
0.25
β = −1
β =1
0.20
0.15
0.10
0.05
-6 -4 -2 0 2 4 6
0.00
Fig. 1 Standard stable dispersal kernels with α = 1.6 illustrating the bell shape
and skewness.
For stable semigroups the generator formula is
[ 1 − β ∂ α u
Au = D
2 ∂(−x) α + 1 + β ∂ α ]
u
2 ∂x α
where ∂ α u/∂(−x) α has Fourier transform (−iλ) α û [3]. This formula is obtained
from (26) by computing the integral in the last term, using the α-stable
jump intensity φ(y) = Cq|y| −α−1 for y < 0 and φ(y) = Cpy −α−1 for y > 0.
In the symmetric case β = 0, one often writes Au = D∂ α u/∂|x| α the Riesz
fractional derivative. This form also coincides with the classical fractional
power of the Laplacian or second derivative operator [1,24].
The fractional derivative can be computed in real space from the generator
formula (26), see [3]. For example, in the simplest case p = 1 − q = 1 and

16 Baeumer, Kovács, Meerschaert
0 < α < 1 the last term in the integral (26) converges, and hence we can
choose a so that
Au(x) =
∫ ∞
0
u(x − y) − u(x)
y
Cy −α dy. (39)
Then we see that the fractional derivative is a weighted average of difference
quotients, with power law weights deriving from the underlying jump
intensity. Using the fractional derivative generator in the abstract reactiondiffusion
equation (5) corresponds to an α-stable dispersal kernel in the
corresponding discrete-time integro-difference equation (6). Along with the
Gaussian kernel (the special case α = 2), these dispersal kernels are distinguished
by the stability property: k t (y) = t −1/α k(yt −1/α ) for all y ∈ R and
all t > 0. Hence they are strongly indicated when the shape of the dispersal
kernel is time-independent. Furthermore, since they derive from central
limit theorems, they are in some sense universal. Finally, although these α-
stable probability density functions cannot usually be written in closed form,
probabilists have developed fast numerical methods for computing them [42].
Combining this with Theorem 1 allows the efficient solution of the fractional
reaction-diffusion equation, as well as monotone error bounds. We will illustrate
the application of these methods in the next section.
Remark 4 A wide variety of alternative models can be obtained by considering
different infinitely divisible densities k as the dispersal kernel. The infinitely
divisible densities are convenient because one can define the integrodifference
equation (6) at any time scale τ based on the convolution power k τ .
The results of this paper also yield the corresponding reaction-diffusion model
with a diffusion operator A exhibiting as the generator of the corresponding
infinitely divisible semigroup. Lockwood et al. [30] employ a Laplace (double
exponential), or more generally, a double Gamma family of dispersion
kernels, which are infinitely divisible with Lévy representation [ta, 0, tφ] with
jump intensity φ(y) = |y| −1 e −c|y| . In this case the last term in the integral
(26) converges, and hence we can choose a so that
Au(x) =
∫ ∞
−∞
u(x − y) − u(x)
|y|
e −c|y| dy (40)
an exponentially weighted derivative, which is similar to the fractional derivative
formula (39) but with different weights [29]. For an exponential or
Gamma dispersal kernel, the generator formula is the same as (40) except
that the integral is taken over y > 0. Clarke et al. [16,17] employ the Student-t
dispersal kernel, which is infinitely divisible with Lévy representation as specified
in [23, Remark 2.3] in terms of Bessel functions. Substituting into (26)
yields the corresponding generator, connecting the integro-difference equation
(6) with a Student-t dispersal kernel to the analogous reaction-diffusion
equation (5).

Fractional reaction-diffusion equation 17
4 Numerical Experiments
Consider a fractional reaction-diffusion equation using Fisher’s growth equation
and a symmetric (Riesz) fractional diffusion term of order 1 < α ≤ 2:
( )
∂u
∂t (x, t) = D ∂α u
∂|x| α (x, t) + ru(x, t) u(x, t)
1 − , u(x, 0) = u 0 (x). (41)
K
Solutions to this equation can be computed using the results of Theorem 1,
using the integro-difference model (6) as an approximation where the dispersal
kernel k τ (y) is the symmetric α-stable probability density function whose
Fourier transform is e −Dτ|λ|α for some D > 0.
u(x,32)
1
0.8
0.6
0.4
0.2
t=32
α=1.8
α=2
0
−100 −50 0 50 100
x
Fig. 2 Solution of the fractional Fisher’s equation (41) with α = 1.8 versus α = 2
at t = 32 with K = 1, r = 0.25 and D = 0.1 showing heavier tails and faster
spreading in the fractional case α < 2.
Numerical simulations were performed with K = 1, c = 0.25 and D = 0.1
and a smooth step-like initial function u 0 which takes the constant value
u = 0.8 around the origin and rapidly decays to 0 away from the origin. S(τ)
was computed analytically at each time step using the explicit analytical solution
(14) and T (τ)u n was computed by numerically convolving u n against
the symmetric α-stable dispersal kernel k τ , which was computed numerically
using the method of Nolan [42]. Note that the dispersal kernel can be computed
for any time scale τ, and that it only has to be computed once for
each time scale. Figure 2 shows that the use of the conventional diffusion
term α = 2 in (41) produces a rapidly decaying solution away from the origin.
However, if one replaces the diffusion term by a fractional one, even with
an order α which is close to 2, then the solution picks up heavy tails and is
spreading faster, similar to the results reported in del-Castillo-Negrete et al.
[19] for the one-sided fractional reaction-diffusion equation (2).
Figure 3 visualises the conclusions of Theorem 1; that is, the sequential
splitting approximations (15) and (16) converge in a pointwise monotone
increasing and decreasing fashion, respectively, to the solution of (41).

18 Baeumer, Kovács, Meerschaert
1
t=32
0.8
u(x,32)
0.6
0.4
0.2
0
−100 −50 0 50 100
x
Fig. 3 Sequential splitting approximations of the solution of the fractional Fisher’s
equation (41) with α = 1.8, K = 1, r = 0.25 and D = 0.1. Dashed lines are
computed from (16) with time step τ = 32, 8, 2 and dotted lines are computed
from (15) at the same time steps to show monotone convergence to the exact (solid
line) solution.
Figure 4 shows evidence of an accelerating front. The selected population
density level u = c has spread a distance x c (t) by time t > 0, and since the
graph of x c (t) versus t for various c closely resembles a straight line on the
semi-log plot, we conclude that the front expands exponentially with time,
in agreement with results reported by del-Castillo-Negrete et al. [19] for the
one-sided model (2).
Acknowledgements We would like to thank Shona Lamoureaux of AgResearch
New Zealand, Jon Pitchford at the University of Leeds, and Alex James at the
University of Canterbury, New Zealand for stimulating discussions.
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Fractional reaction-diffusion equation 19
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