Even understanding the way we perceive very simple images presents a major challenge. In this talk I will discuss two visual effects. In one, random dots are superimposed on themselves following a linear transformation. In the second, a rotating disk with radial spokes is viewed under stroboscopic illumination, where the frequency and duration of the stroboscopic flash are varied. Though these phenomena are very different, in both correlation plays a major role in defining the structure of the image. In this talk, I will give demonstrations of these phenomena and discuss related experimental and theoretical work by ourselves and others. In particular, I focus on a theory that uses the theory of forced nonlinear oscillations to predict the percept of rotating disks during stroboscopic illumination over a wide range of disk rotation speeds and strobe frequencies. Finally, I suggest that the anatomical structure of the human visual system plays a major role in enabling the amazingly rapid and accurate computation of spatial and time dependent correlation functions carried out by the visual system.
One of the most salient spatio-temporal patterns in population ecology is the synchronization of fluctuating local populations across vast spatial extent. Synchronization of abundance has been widely observed across a range of spatial scales in relation to rate of dispersal among discrete populations. However, the dependence of synchrony on patterns of among-patch movement across heterogeneous landscapes has been largely ignored. Here we consider the duration of movement between two predator-prey communities connected by weak dispersal, and its effect on population synchrony. More specifically, we introduce time delayed dispersal to incorporate the finite transmission time between discrete populations across a continuous landscape. Reducing the system to a phase model using weakly connected network theory, it is found that the time delay is an important factor determining the nature and stability of phase-locked states. Our analysis predicts enhanced convergence to stable synchronous fluctuations in general, and a decreased ability of systems to produce in-phase synchronization dynamics in the presence of delayed dispersal. These results introduce delayed dispersal as a tool for understanding the importance of dispersal time across a landscape matrix in affecting metacommunity dynamics. They further highlight the importance of landscape and dispersal patterns for predicting the onset of synchrony between weakly-coupled populations.
Pattern formation in self-organised biological aggregation is a phenomenon that has been studied intensively over the past twenty years. I will present a class of models of animal aggregation in the form of two first-order hyperbolic partial differential equations on a one-dimensional domain with periodic boundary conditions, describing the motion of left and right moving individuals. The nonlinear terms appear using nonlocal social interaction terms for attraction, repulsion and alignment. This class of models has been introduced in the Ph.D thesis of R. Eftimie. In this talk, I will show that the equations are O(2) equivariant where the group O(2) is generated by space-translations and a reflection which interchanges left-moving individuals with right-moving individuals across the middle of the interval. I will show the existence of codimension two steady-state/steady-state, steady-state/Hopf and Hopf/Hopf bifurcation points with O(2) symmetry. Using the existing symmetry-breaking bifurcation theory, one can study the neighborhood of those bifurcation points and classify the patterns obtained. This is joint work with R. Eftimie (U. Dundee, Scotland).
A recent nonlocal animal aggregation model published by Dr. Eftimie et al is introduced. An overview of the numerical methods involved in time stepping and continuing solutions of the model is given; including the pseudo-spectral method involved, as well as the use of a matrix-free continuation method by Dr. Sanchez et al. Highlights are given to the use of the flow operator in the Newton update step, which serves to better condition the system; and to interesting problems arising from symmetries and the non-locality of the model.