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

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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.
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