Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

Abstract

Reproduction-Dispersal equations, called reaction-diffusion equations in the physics literature, model the growth and spreading of biological species. IntegroDifference equations were introduced to address the shortcomings of this model, since the dispersal of invasive species is often more widespread than what the classical RD model predicts. In this paper, we extend the RD model, replacing the classical second derivative dispersal term by a fractional derivative of order 1 < α ≤ 2. Fractional derivative models are used in physics to model anomalous super-diffusion, where a cloud of particles spreads faster than the classical diffusion model predicts. This paper also establishes a connection between the new RD model and a corresponding ID equation with a heavy tail dispersal kernel. The general theory developed here accommodates a wide variety of infinitely divisible dispersal kernels that adapt to any scale. Each one corresponds to a generalised RD model with a different dispersal operator. The connection established here between RD and ID equations can also be exploited to generate convergent numerical solutions of RD equations along with explicit error bounds.

Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts

“…One of the main challenges is an implementation of the description of chemical reactions in non-Markovian transport processes governed by CTRW with non-exponential waiting time distributions. There exist several approaches and techniques to deal with this problem [2][3][4][5][6][7][8][9][10][11]. In particular, there are many efforts to incorporate the chemical reactions into subdiffusive transport.…”

Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations

“…Let u andû J be the solutions of (17)- (19) and (34)- (36), respectively. Assume that u 0 ∈ H. We have:…”

“…We note that Definitions (4) and (5) are not equivalent [17]. For the deterministic space-fractional partial differential equations where the space-fractional derivative is defined by (5), or the Riemann-Liouville space-fractional derivative, or the Caputo space-fractional derivative, many numerical methods are available, for example, finite difference methods [18][19][20][21][22][23][24][25][26][27][28][29][30], finite element methods [14,[31][32][33][34][35][36][37][38][39][40] and spectral methods [41,42]. For the deterministic space-fractional partial differential equations where the space-fractional derivative is defined by (4), some numerical methods are also available, for example the matrix transfer method (MTT) [21,22,43] and the Fourier spectral method [44].…”

A Predictor-Corrector Method for Fractional Evolution Equations

“…The equation is described as (1) ∂u(x, t) ∂t = D α x u(x, t) + f (u, x, t), (x, t) ∈ Ω × (0, T ] with an initial condition (2) u(x, 0) = u 0 (x), x ∈Ω…”

“…The analytical results on existence and uniqueness of the solution for (1) have been studied by Baeumer, Kovács and Meerschaert [2] using the semigroup theory when the source term f (u, x, t) is globally Lipschitz continuous with respect to u. They have also shown that the solution exists uniquely by introducing the cut-off function when the function f (u, x, t) is locally Lipschitz continuous.…”