Fractional reaction-diffusion equation for species growth and dispersal

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Boris Baeumer · Mih´ aly Kov´ acs · Mark M. Meerschaert

Fractional reaction-diffusion equation for species growth and dispersal

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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: [email protected] Mih´ aly Kov´ acs 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: [email protected] 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: [email protected]

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Baeumer, Kov´ acs, 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 ∂2u = f (u) + D 2 . ∂t ∂x

(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 ∂αu = f (u) + D α ∂t ∂x

(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], andR hydrology [4,6–8,55, 56]. If a function u(x) has Fourier transform u ˆ(λ) = e−iλx u(x)dx then the fractional derivative dα u(x)/dxα has Fourier transform (iλ)α u ˆ(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 Z ∞ u(x, t + τ ) = kτ (x, y)gτ (u(y, t))dy (3) −∞

where the dispersal kernel kτ (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 → gτ (u) to grow the population over one time step, and then we integrate against the kernel kτ (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(u0 ) is the solution to the ordinary differential equation u˙ = f (u); u(0) = u0 at time t = τ and 2 kτ (x, y) =

(x−y) − 4Dτ √ 1 e 4πDτ

(4)

is a Gaussian dispersal kernel, the fundamental solution to the classical diffusion equation ∂u ∂2u = D 2 ; u(x, t = 0) = δ(x − y) ∂t ∂x 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 kτ (x, y) is the fundamental solution to the fractional diffusion equation ∂u ∂αu = D α ; u(x, t = 0) = δ(x − y) ∂t ∂x 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.

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2 Problem outline The classical reaction-diffusion equation (1) and its fractional analogue (2) are both special cases of a general form ∂u (x, t) = (Au)(x, t) + f (u(x, t)), u(x, 0) = u0 (x) (5) ∂t 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 Z ∞ un+1 (x) = kτ (x, y)gτ (un (y)) dy (6) −∞

where un (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 Z ∞ (x−y)2 1 √ u(x, t) = e− 4Dt u0 (y) dy := [T (t)u0 ](x) 4πDt −∞ where the dispersal operator T (t) maps the initial condition at time t = 0, the function u0 (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) = ert u0 (x) := [S(t)u0 ](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)u0 ](x) = [S(t)S(s)u0 ](x) = [S(s)S(t)u0 ](x) and [T (t + s)u0 ](x) = [T (t)T (s)u0 ](x) = [T (s)T (t)u0 ](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 Z ∞ (x−y)2 − 4Dt √ 1 u(x, t) = e ert u0 (y) dy = [T (t)S(t)u0 ](x) (7) 4πDt −∞

= [S(t)T (t)u0 ](x) = ert

Z

∞

−∞

(x−y)2 − 4Dt √ 1 e 4πDt

u0 (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 un+1 (x) = u(x, (n + 1)τ ) = [T ((n + 1)τ )S((n + 1)τ )u0 ](x) = [(T (τ ))n+1 (S(τ ))n+1 u0 ](x) = [T (τ )S(τ )(T (τ ))n (S(τ ))n u0 ](x) = [T (τ )S(τ )T (nτ )S(nτ )u0 ](x) = [T (τ )S(τ )un ](x) Z ∞ Z ∞ (x−y)2 − 4Dτ rτ √ 1 e e u (y) dy := kτ (x, y)gτ (un (y)) dy. = n 4πDτ −∞

−∞


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 kτ given by (4) and gτ (un (y)) = erτ un (y). Iterations of the form un+1 = S(τ )T (τ )(un ) 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 u0 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)u0 ](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 un+1 (x) = [W (τ )un ](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 ∂u (x, t) = (Au)(x, t) ∂t

(9)

and by S(t) the solution operator to ∂u (x, t) = f (u(x, t)). ∂t

(10)

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 )u0 ](x) we compute [T (τ )S(τ )u0 ](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 un+1 (x) = [T (τ )S(τ )un ](x) with n starting value u0 , or equivalently, un+1 (x) = [(T (τ )S(τ )) u0 ] (x). Usually, un (x) 6= u(x, nτ ) (except for the commuting case), but is it true that un (x) is close to u(x, nτ ) in some sense? For example, if we fix t and choose finer and finer time steps τ = nt , then does un (x) converge to u(x, t) in some £ ¤n sense as n → ∞, in other words, does T ( nt )S( nt ) u0 converge to W (t)u0 as n → ∞? Example 2 Let us consider the fractional reaction-diffusion equation with Fisher’s growth term µ ¶ ∂u u(x, t) ∂αu (x, t) = ru(x, t) 1 − + D α (x, t); u(x, 0) = u0 (x) (11) ∂t K ∂x

6


where 1 < α ≤ 2. To apply the sequential operator splitting procedure, we first consider the fractional diffusion equation ∂αu ∂u (x, t) = D α (x, t); ∂t ∂x

u(x, 0) = u0 (x)

(12)

which methods [15,32]: Write u ˆ(λ, t) = R −iλxcan be solved by Fourier transform e u(x, t)dx and recall that ∂ α u/∂xα has Fourier transform (iλ)α u ˆ. Taking Fourier transforms on both sides of (12) yields the ordinary differential equation d u ˆ(λ, t) = D(iλ)α u ˆ(λ, t); u ˆ(λ, 0) = u ˆ0 (λ) dt whose solution is evidently of the form u ˆ(λ, t) = u ˆ0 (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 Z ∞ u(x, t) = [T (t)u0 ](x) = kt (x − y)u0 (y) dy (13) −∞

where kt (x) is an α-stable probability density. Next consider the growth equation µ ¶ ∂u u(x, t) (x, t) = ru(x, t) 1 − ; u(x, 0) = u0 (x) ∂t K with solution u(x, t) = [S(t)u0 ](x)

µ ˜ = [S(t)](u0 (x)) = K 1 −

K − u0 (x) K + u0 (x)(exp(rt) − 1)

¶

(14)

where we emphasise that the nonlinear operator S(t) maps the function u0 to ˜ is a function that maps the real another real-valued function of x, and S(t) number u0 (x) to another real number. According to the discussion above, define an iteration based on the sequential splitting by un+1 (x) = [T (τ )S(τ )un ](x) µ Z ∞ = kτ (x − y)K 1 − −∞

K − un (y) K + un (y)(exp(rτ ) − 1)

¶ dy.

(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.


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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 un+1 (x) := [S(τ )T (τ )un ](x) µZ ∞ ¶   kτ (x − y)un (y) dy K−   (16) −∞ . µZ ∞ ¶ =K 1 −  K+ kτ (x − y)un (y) dy (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 kuk, 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) = u0

(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 kT (t)uk ≤ M kuk for all u ∈ X; T (t+s) = T (t)T (s) for t, s ≥ 0; t 7→ T (t)u is continuous in the Banach space norm for all u ∈ X; and the generator Au = limh→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

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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 Z t u(t) = T (t)u0 + T (t − s)f (u(s)) ds, 0 ≤ t < δ. (18) 0

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 u0 ∈ X there is a unique global mild solution u(t) := W (t)u0 of (17) with ||W (t)u0 − W (t)v0 || ≤ MT ||u0 − v0 ||, 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)u0 . In this case T (t)u = u for all t ≥ 0, and hence it follows from (18) that Z t u(t) = u0 + f (u(s)) ds (19) 0

for all t > 0. Then u is also a strong solution, since R t if u and f are continuous, then t 7→ f (u(t)) is also continuous and t 7→ 0 f (u(s)) ds is differentiable, Rt d and 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)u0 to the abstract reaction-diffusion equation (17) can be computed by the Trotter Product Formula £ ¤n £ ¤n W (t)u0 = lim T ( nt )S( nt ) u0 = lim S( nt )T ( nt ) u0 , u0 ∈ X, (20) n→∞

n→∞

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. 2. 3. 4.

u ≤ v implies u + w ≤ v + w for all u, v, w ∈ X. u ≥ 0 implies λu ≥ 0 for all u ∈ X and λ ≥ 0. 0 ≤ u ≤ v implies ||u|| ≤ ||v|| for all u, v ∈ X. The positive cone X+ := {x ∈ X : x ≥ 0} is closed.

A typical example of an ordered Banach space is C0 (R), the space of continuous functions u : R → R such that u(x) → 0 as |x| → ∞, endowed with the supremum norm kuk = 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 Lp (R) (1 ≤ p ≤ ∞) endowed with the partial ordering u ≤ v whenever u(x) ≤ v(x)


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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)u0 ≤ S(t)T (t)u0 (21) holds for all t ∈ [0, T ] and u0 ≥ 0, then the unique mild solution W (t)u0 of the abstract reaction-diffusion equation (17) satisfies £ t ¤n £ ¤2n t t T ( n )S( nt ) u0 ≤ T ( 2n )S( 2n ) u0 ≤ W (t)u0 £ t ¤2n £ ¤n t ≤ S( 2n )T ( 2n ) u0 ≤ S( nt )T ( nt ) u0

(22)

for all u0 ≥ 0, n ∈ N and t ∈ [0, T ]. Proof It follows from our assumption that W (t)u0 is given by (20). Next we show that £ t ¤n £ ¤2n t t T ( n )S( nt ) u0 ≤ T ( 2n )S( 2n ) u0 ¤2n £ ¤n £ t t ) u0 ≤ S( nt )T ( nt ) u0 , ≤ S( 2n )T ( 2n

(23)

u0 ≥ 0, n ∈ N and t ∈ [0, T ]. For u0 ≥ 0 using (21) repeatedly we have t t t t T ( nt )S( nt )u0 = T ( 2n )T ( 2n )S( 2n )S( 2n )u0 t t t t t t t t ≤ T ( 2n )S( 2n )T ( 2n )S( 2n )u0 ≤ S( 2n )T ( 2n )T ( 2n )S( 2n )u0 t t t t t t t t ≤ S( 2n )T ( 2n )S( 2n )T ( 2n )u0 ≤ S( 2n )S( 2n )T ( 2n )T ( 2n )u0 = S( nt )T ( nt )u0 .

For any operators A, B : X → X, it is not hard to check that the inequality 0 ≤ B ≤ A implies 0 ≤ B n ≤ An (n ∈ N) which finishes the proof of (23). Fix ¤n2k £ u0 . By (23), x0 ≤ x1 ≤ ... ≤ xk ≤ ... n ∈ N and let xk := T ( n2t k )S( n2t k ) and xk → W (t)u0 as k → ∞ by (20). Therefore, it follows from the closedness of the positive cone X+ that xk ≤ W (t)u0 for all k = 0, 1, .... This shows the second inequality in (3). A similar argument yields the third inequality and the proof is complete. u t 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 R variable on R with probability density ˆ k(y) and Fourier transform k(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 Yn1 , . . . , Ynn with the same density kn such that Yn1 + · · · + Ynn 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


(see, e.g., Theorem 3.1.11 in [36]) states that if k is infinitely divisible then ˆ k(λ) = eψ(λ) where Z ³ 1 iλy ´ ψ(λ) = −iλa − λ2 b2 + e−iλy − 1 + φ(y)dy, (24) 2 1 + y2 y6=0 where a ∈ R, b ≥ 0 , and the jump intensity φ satisfies Z min{1, y 2 } φ(y)dy < ∞. y6=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 etψ(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 C0 (R) defined by Z ∞ [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 ∈ C0 (R) with u0 , u00 ∈ C0 (R) belongs to the domain of the generator A of the semigroup (25), and for such functions we have Au(x) = − au0 (x) + 21 b2 u00 (x) µ ¶ Z u0 (x)y + u(x − y) − u(x) + φ(y)dy. 1 + y2 y6=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 ∈ C0 (R) with u0 , u00 ∈ C0 (R). u t 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 etψ(λ) u ˆ(λ), and then the generator can be computed in Fourier space by t−1 [etψ(λ) − 1]ˆ u(λ) → ψ(λ)ˆ u(λ) so that ψ(λ)ˆ u(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.


11

In what follows we discuss a general abstract reaction-diffusion equation for Fisher’s equation (11) for non-negative initial data on X := C0 (R): u(t) ˙ = Au(t) + f (u(t)), u(0) = u0 ≥ 0, where f : X → X is defined via the function f˜ : R → R as µ ¶ u(x) [f (u)](x) = f˜(u(x)) := ru(x) 1 − K

(27)

(28)

We introduce a cut off function via the function fÑ : R → R as  0   ¶ if u(x) < 0 µ  u(x) if 0 ≤ u(x) ≤ N K (29) [fN (u)](x) = fÑ (u(x)) : ru(x) 1 −  K   rN K(1 − N ) if u(x) > N K. Proposition 3 Let X := C0 (R) and let f be given by (28). Then the abstract differential equation u(t) ˙ = f (u(t)), u(0) = u0 ≥ 0

(30)

has a unique strong global solution given by u(t) = S(t)u0 for each nonneg˜ ˜ ative u0 ∈ X, and in fact we have [S(t)u0 ](x) = [S(t)](u 0 (x)) where S(t) is defined by (14). For any positive integer N ≥ 2, the abstract differential equation u˙ = fN (u), u(0) = u0 ≥ 0 for the Lipschitz continuous function fN defined in (29) also has a unique strong global solution given by u(t) = SN (t)u0 for each u0 ≥ 0 in X. Furthermore, in this case, if 0 ≤ u0 (x) ≤ N K for all x ∈ R, then we also have that 0 ≤ [S(t)u0 ](x) = [SN (t)u0 ](x) ≤ N K for all x ∈ R and all t ≥ 0. Proof Consider the abstract initial value problem u(t) ˙ = fN (u(t)), u(0) = u0 ≥ 0 ∈ X which has, by the Lipschitz continuity of fN , a unique global strong solution u(t) = SN (t)u0 (see the discussion after (19)), where SN (·) is the nonlinear semigroup generated by fN . Hence, since the operator norm in this space is the supremum norm, it follows that the function ux (t) := [SN (t)u0 ](x) is for each fixed x ∈ R the unique solution of the ordinary differential equation d ux (t) = fÑ (ux (t)), ux (0) = u0 (x) ≥ 0 ∈ R. dt Then it follows easily, using the uniqueness of solutions, that SN (·) is positive, i.e., if u0 (x) ≥ 0 for all x ∈ R then ux (t) = [u(t)](x) = [SN (t)u0 ](x) ≥ 0 for all x ∈ R, and also if u0 (x) ≥ v0 (x) for all x ∈ R then ux (t) = [u(t)](x) = [SN (t)u0 ](x) ≥ vx (t) = [v(t)](x) = [SN (t)v0 ](x) for all x ∈ R. Since fÑ (y) < 0 for all y > K it also follows from uniqueness of solutions that, if u0 (x) ≤ N K for all x ∈ R, then [SN (t)u0 ](x) ≤ N K for all x ∈ R.

12


Moreover, since fÑ (u) = f˜(u) for 0 ≤ u ≤ N K it follows that SN (t)u0 also solves u(t) ˙ = f (u(t)), u(0) = u0 . (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 7→ SN (t)u0 is defined for all t ≥ 0 and hence SN (t)u0 is the unique strong global solution S(t)u0 of (31) which is necessarily given by (14) since that solves (31) evaluated pointwise. u t 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 := C0 (R). Then (27,28) has a unique mild solution u(t) = W (t)u0 for all u0 ≥ 0 in X given by the Trotter Product Formula £ ¤n £ ¤n W (t)u0 = lim T ( nt )S( nt ) u0 = lim S( nt )T ( nt ) u0 . (32) n→∞

n→∞

Moreover, for all n ∈ N, £ t ¤n £ ¤2n t t T ( n )S( nt ) u0 ≤ T ( 2n )S( 2n ) u0 ≤ W (t)u0 £ t ¤2n £ ¤n t ≤ S( 2n )T ( 2n ) u0 ≤ S( nt )T ( nt ) u0 ,

(33)

where S(·) is defined in (14). Proof Choose N ≥ 2 such that 0 ≤ u0 (x) ≤ N K for all x and consider the abstract reaction-diffusion equation u(t) ˙ = Au(t) + fN (u(t)), u(0) = u0 ≥ 0.

(34)

Since fN : X → X is globally Lipschitz continuous and A is a generator, there is a unique mild solution uN (t) = WN (t)u0 of (34) given by the Trotter Product Formula £ ¤n uN (t) = WN (t)u0 = lim T ( nt )SN ( nt ) u0 n→∞ (35) £ ¤n = lim SN ( nt )T ( nt ) u0 , n→∞

see, e.g., [12, 18, 38]. The semigroup T (·) satisfies 0 ≤ T (t)u0 ≤ T (t)v0 for 0 ≤ u0 ≤ v0 and t ≥ 0 since k t is a probability density function. If 0 ≤ u0 (x) ≤ N K for all x, then Z ∞ 0 ≤ [T (t)u0 ](x) ≤ N K k t (x − y)dy = N K (36) −∞


13

Therefore, by (36) and Proposition 3, 0 ≤ [T ( nt )SN ( nt )]n u0 (x) = [T ( nt )S( nt )]n u0 (x) ≤ N K

(37)

for all x and 0 ≤ [SN ( nt )T ( nt )]n u0 (x) = [S( nt )T ( nt )]n u0 (x) ≤ N K

(38)

for all x. This also shows that 0 ≤ [uN (t)](x) ≤ N K for all x in view of (35). Therefore uN (t) is a mild solution of (27), too, since fN (u(x)) = f (u(x)) for 0 ≤ u(x) ≤ N K. 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 uN (t) is defined for all t > 0 it follows that uN (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 7→ [S(t)](y) in (14) is concave down on y > 0 for any t > 0. Therefore, by Jensen’s inequality [21, pp. 153–154], ·Z ∞ ¸ t ˜ [S(t)T (t)u0 ](x) = S(t) k (y)u0 (x − y) dy −∞ Z ∞ ˜ [u0 (x − y)] dy ≥ k t (y)S(t) −∞ Z ∞ = k t (y) [S(t)u0 ] (x − y) dy = [T (t)S(t)u0 ](x). −∞

Thus, SN (t)T (t)u0 = S(t)T (t)u0 ≥ T (t)S(t)u0 = T (t)SN (t)u0 which finishes the proof by Proposition 1, (37) and (38).

u t

Corollary 1 Under the assumptions of Theorem 1, if u0 and its first two derivatives in x exist and and belong to C0 (R), then (27,28) has a unique strong solution u(t) = W (t)u0 for all u0 ≥ 0 in X given by the Trotter Product Formula (32). Proof In this case Proposition 2 shows that u0 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]. u t 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 un (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 u0 (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


Remark 3 Theorem 1 extends immediately to any abstract reaction-diffusion ˜ equation of the form (27) as long as u 7→ [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 ∈ Rd , 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 b2 = 2D in (24)), equation (26) shows that its generator is Au = D ∂ 2 u/∂x2 . 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 Rso if 0 < α < 2, then an easy calculation shows that the second moment r2 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 ˆ transforms k(λ) = eψ(λ) where   −iaλ − σ α |λ|α (1 − iβ(signλ) tan πα 2 ) for α 6= 1 ψ(λ) =  −iaλ − σ|λ|(1 + iβ 2 (signλ) ln λ) for α = 1 π 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


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. 0.30

β =0

0.25

0.20

β =1

β = −1

0.15

0.10

0.05

0.00 -6

-4

-2

0

2

4

6

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 ∂αu 1 + β ∂αu Au = D + 2 ∂(−x)α 2 ∂xα

¸

where ∂ α u/∂(−x)α has Fourier transform (−iλ)α u ˆ [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


0 < α < 1 the last term in the integral (26) converges, and hence we can choose a so that Z

∞

Au(x) = 0

u(x − y) − u(x) Cy −α dy. y

(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 αstable 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 Z

∞

Au(x) = −∞

u(x − y) − u(x) −c|y| e dy |y|

(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).


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: µ ¶ ∂u ∂αu u(x, t) (x, t) = D (x, t) + ru(x, t) 1 − , u(x, 0) = u0 (x). (41) ∂t ∂|x|α 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 kτ (y) is the symmetric α-stable probability density function whose α Fourier transform is e−Dτ |λ| for some D > 0.

t=32 1

α=1.8 α=2

u(x,32)

0.8 0.6 0.4 0.2 0 −100

−50

0 x

50

100

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 u0 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 )un was computed by numerically convolving un 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


t=32 1

u(x,32)

0.8 0.6 0.4 0.2 0 −100

−50

0 x

50

100

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 xc (t) by time t > 0, and since the graph of xc (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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