On the persistence of spatiotemporal oscillations generated by invasion

Many systems in biology and chemistry are oscillatory, with a stable, spatially homogeneous steady state which consists of periodic temporal oscillations in the interacting species, and such systems have been extensively studied on in"nite or semi-in"nite spatial domains. We consider the effect of a "nite domain, with zero-#ux boundary conditions, on the behaviour of solutions to oscillatory reaction–diffusion equations after invasion. We begin by considering numerical simulations of various oscillatory predator–prey systems. We conclude that when regular spatiotemporal oscillations are left in the wake of invasion, these die out, beginning with a decrease in the spatial frequency of the oscillations at one boundary, which then propagates across the domain. The long-time solution in this case is purely temporal oscillations, corresponding to the limit cycle of the kinetics. Contrastingly, when irregular spatiotemporal oscillations are left in the wake of invasion, they persist, even in very long time simulations. To study this phenomenon in more detail, we consider the λ–ω class of reaction–diffusion systems. Numerical simulations show that these systems also exhibit die-out of regular spatiotemporal oscillations and persistence of irregular spatiotemporal oscillations. Exploiting the mathematical simplicity of the λ–ω form, we derive analytically an approximation to the transition fronts in r and θx which occur during the die-out of the regular oscillations. We then use this approximation to describe how the die-out occurs, and to derive a measure of its rate, as a function of parameter values. We discuss applications of our results to ecology, calcium signalling and chemistry.