Periodic Travelling Wave Selection by Dirichlet Boundary Conditions in Oscillatory Reaction-Diffusion Systems

Abstract

Abstract. Periodic travelling waves are a fundamental solution form in oscillatory reactiondiffusion equations. Here I discuss the generation of periodic travelling waves in a reaction-diffusion system of the generic λ-ω form. I present numerical results suggesting that when this system is solved on a semi-infinite domain subject to Dirichlet boundary conditions in which the variables are fixed at zero, periodic travelling waves develop in the domain. The amplitude and speed of these waves are independent of the initial conditions, which I generate randomly in numerical simulations. Using a combination of numerical and analytical methods, I investigate the mechanism of periodic travelling wave selection. By looking for an appropriate similarity solution, I reduce the problem to an ODE system. Using this, I derive a formula for the selected speed and amplitude as a function of parameters. Finally, I discuss applications of this work to ecology.

A comparison of periodic travelling wave generation by Robin and Dirichlet boundary conditions in oscillatory reaction-diffusion equations

“…Analysis is much easier for the Dirichlet condition (e.g. Sherratt, 2003;Smith et al, 2008); moreover, numerical solution is much more difficult for the Robin condition, especially in two dimensions. Secondly, there are often no quantitative data on the extent of the hostility of the surrounding environment, so that although one anticipates that the Robin condition will be close to the Dirichlet limit, the actual proximity is hard to estimate.…”

“…Over the last three decades, the simple form of this periodic wave family has provided an invaluable reference point for the study of periodic travelling waves in more general reaction-diffusion systems. This work has focussed in particular on the existence and stability of periodic travelling waves (Ermentrout, 1981;Maginu, 1979Maginu, , 1981Kapitula, 1994), other cases with exact solutions (Cope, 1979;Romero et al, 2000) and the generation of periodic travelling waves by environmental heterogeneities (Auchmuty & Nicolis, 1976;Hagan, 1981;Kopell, 1981;Kay & Sherratt, 2000;Sherratt, 2003) and behind invasive wavefronts (Sherratt, 1994(Sherratt, , 1996Ermentrout et al, 1997;Petrovskii et al, 1998;Petrovskii & Malchow, 2000, 2001Webb & Sherratt, 2004;Garvie, 2007).…”

“…Note in particular that as increases from zero, the wave amplitude A gradually increases. In Sherratt (2003), I showed that in the case of = 0 (Dirichlet boundary condition), (7-9) has the exact solution…”

“…In a previous paper (Sherratt, 2003), I studied the generation of periodic travelling waves by Dirichlet boundary conditions in a 'λ-ω' system. These equations are the normal form of an oscillatory reactiondiffusion system with scalar diffusivity close to a supercritical Hopf bifurcation (Hassard et al, 1981;Guckenheimer & Holmes, 1983) and have the form ∂u/∂t = u x x + (1 − r 2 )u − (ω 0 − ω 1 r 2 )v,…”

“…The parameters ω 0 and ω 1 would be functions of ecological parameters that can be derived via the theory of normal forms (Hassard et al, 1981;Guckenheimer & Holmes, 1983;Sherratt, 2001). In Sherratt (2003), I studied the system (1) on a semi-infinite domain 0 x < ∞, subject to the boundary condition u = v = 0 at x = 0. I showed that the long-term solution has a simple analytical form (given below) and that this solution approaches a periodic travelling wave at large x.…”