Abstract
Investigations are undertaken into simple predator-prey models with rational interaction terms in one and two spatial dimensions. Focusing on a case with linear interaction and saturation, an analysis for long domains in 1D is undertaken using ideas from spatial dynamics. In the limit that prey diffuses much more slowly than predator, the Turing bifurcation is found to be subcritical, which gives rise to localized patterns within a Pomeau pinning parameter region. Parameter regions for localized patterns and isolated spots are delineated. For a realistic range of parameters, a temporal Hopf bifurcation of the balanced equilibrium state occurs within the localized-pattern region. Detailed spectral computations and numerical simulations reveal how the Hopf bifurcation is inherited by the localized structures at nearby parameter values, giving rise to both temporally periodic and chaotic localized patterns. Simulation results in 2D confirm the onset of complex spatio-temporal patterns within the corresponding parameter regions. The generality of the results is confirmed by showing qualitatively the same bifurcation structure within a similar model with quadratic interaction and saturation. The implications for ecology are briefly discussed.