Time-dependent localized patterns in a predator-prey model

Numerical continuation is used to compute solution branches in a two-component reaction-diffusion model of Leslie--Gower type. %in the vicinity of a Turing-Hopf interaction. Two regimes are studied in detail. In the first, the homogeneous state loses stability to supercritical spatially uniform oscillations, followed by a subcritical steady state bifurcation of Turing type. The latter leads to spatially localized states embedded in an oscillating background that bifurcate from snaking branches of localized steady states. Using two-parameter continuation we uncover a novel mechanism whereby disconnected segments of oscillatory states zip up into a continuous snaking branch of time-periodic localized states, some of which are stable. In the second, the homogeneous state loses stability to supercritical Turing patterns, but steady spatially localized states embedded either in the homogeneous state or in a small amplitude Turing state are nevertheless present. We show that such behavior is possible when sideband Turing states are strongly subcritical and explain why this is so in the present model. In both cases the observed behavior differs significantly from that expected on the basis of a supercritical primary bifurcation.