Oscillatory Reaction-Diffusion Equations with Temporally Varying Parameters

Keywords--Reaction-diffusion, Travelling waves, A-w systems. 1. I N T R O D U C T I O N Periodic wave t ra ins are the generic one-dimensional solut ion form for react ion-diffusion equat ions with a s table l imit cycle in the kinetics. Such stable l imit cycles are used widely in biological and chemical applicat ions; chemical concent ra t ion waves such as those found wi th the BelousovZhabot inski i react ion are visual ly d ramat ic examples [1], other examples inc lude the intracel lular calcium system [2], and predator-prey in teract ions [3,4]. Periodic wave t ra ins are solut ions with cons tan t shape and speed tha t oscillate in bo th space and time. T h e y were first s tudied by Kopell and Howard [5], who showed tha t all oscillatory reaction-diffusion systems have a one-parameter *Author to whom all correspondence should be addressed. This work was supported by SHEFC Research Development Grant 107 and EPSRC (earmarked studentship to S.D.W. and advanced" research fellowship to J.A.S.). S.D.W. would like to thank M. Owen (Loughborough University) for helpful discussions. 0895-7177/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by ~4.~$-TEX doi: 10.1016/S0895-7177(03)00393-5 46 S .D . WEBB AND J. A. SHERRATT family of periodic wave train solutions; here, we use the word 'oscillatory' to indicate that the reaction-diffusion kinetics have a stable limit cycle. Over the last two decades, many authors have considered the form of periodic wave trains, and their stability as solutions of the corresponding reaction-diffusion systems [6-9]. More recent work has focused on the generation of these solution forms from simple initial conditions, of the type that would arise naturally in applications [10,11]. For example, in [11], Sherratt considered the behaviour behind invasive wave fronts with initial data decaying exponentially across the domain, and showed that such initial data does indeed generate a periodic wave train. Another recent study focusing on the generation of periodic wave trains is that of Ermentrout et al. [10]. They consider wave train generation and interaction in equations that undergo a subcritical Hopf bifurcation and have a regime of bistability, leading to transition fronts between wave trains and homogeneous oscillations, and spatially localised oscillations. One important difference between real ecological systems and typical reaction-diffusion models are the temporal oscillations in parameter values due to seasonal variations. Timm and Okubo [12], and Sherratt [13] studied the effects of temporal oscillations on the ability of reactiondiffusion systems to form Turing type patterns. Timm and Okubo studied a model for a predatorprey interaction between different species of plankton, with a sinusoidal temporal variation in the dispersal rate of the predator zooplankton. They presented numerical evidence that suggested that the homogeneous steady state becomes more stable as the amplitude of the temporal variation in dispersal rate increases. More recently, Sherratt [13] extended the assumptions of Timm and Okubo to include the simple case in which the temporal variation in diffusivity has a square-tooth form, alternating between two constant values. There, analytic conditions for dispersal driven patterns were determined, which show that in some cases oscillations in the predator dispersal rate can promote pattern formation. In this paper, we consider the effects of temporal forcing on wave train propagation for systems of two reaction diffusion equations close to a supercritical Hopf bifurcation in the kinetics, with equal diffusion coefficients. In this case, the kinetics can be approximated by the Hopf normal form, giving reaction-diffusion equations of A-w form. A-w systems have been widely used in prototype studies of reaction-diffusion equations, and have proved invaluable in the study of spiral waves [14,15] and periodic plane waves [5,6]. Our objective is to understand the way in which such variations modify the generation of periodic wave trains behind invasive transition wave fronts. We do not consider the specific effects of seasonal variations in any particular ecological system; rather, we investigate the generic effect that oscillations in parameters have on oscillatory systems. Section 2 begins by introducing reaction-diffusion systems of A-w type. In Section 3, we present the results of numerical simulations for A-w systems with temporal variation. Exploiting the mathematical simplicity of the A-w form, we derive analytically (Section 4~) an approximation to the amplitude of the forced wave train oscillations. We then use this approximation to describe how the amplitude of these solutions depends on the period of forcing. We discuss the results in Section 5. 2. I N T R O D U C T I O N T O A-w S Y S T E M S We will start with an introduction to the "A-w" class of reaction-diffusion systems. These have the general form 0U 02U + (r)u ( l a ) Ot Ox 2 Ov 02v + w(r)u + A(r)v, (lb) Ot Ox 2 where r = (u 2 + v2) 1/2, and A(0) and w(0) are both strictly positive. This type of equation is a standard prototype for oscillatory reaction-diffusion systems; their form facilitates analytical Oscillatory Reaction-Diffusion Equations 47 study. In this paper, we will restrict attention to the case