Oscillatory reaction-diffusion equations with temporally varying parameters

Periodic wave trains are the generic one-dimensional solution form for reaction-diffusion equations with a limit cycle in the kinetics. Such systems are widely used as models for oscillatory phenomena in chemistry, ecology, and cell biology. In this paper, we study the way in which periodic wave solutions of such systems are modified by periodic forcing of kinetic parameters. Such forcing will occur in many ecological applications due to seasonal variations. We study temporal forcing in detail 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 λ-ω type. Numerical simulations show that a temporal variation in the kinetic parameters causes the wave train amplitude to oscillate in time, whereas in the absence of any temporal forcing, this wave train amplitude is constant. Exploiting the mathematical simplicity of the λ-ω form, we derive analytically an approximation to the amplitude of the wave train oscillations with small forcing. We show that the amplitude of these oscillations depends crucially on the period of forcing.