Past

Coherent structures, or defects, are interfaces between wave trains with

possibly different wavenumbers: they are time-periodic in an appropriate

coordinate frame and connect two, possibly different, spatially-periodic

travelling waves. We propose a classification of defects into four

different classes which have all been observed experimentally. The

characteristic distinguishing these classes is the sign of the group

velocities of the wave trains to either side of the defect, measured

relative to the speed of the defect. Using a spatial-dynamics description

in which defects correspond to homoclinic and heteroclinic orbits, we then

relate robustness properties of defects to their spectral stability

properties. If time permits, we will also discuss how defects interact with

each other.

We describe a nice example of duality between coagulation and fragmentation associated with certain Dirichlet distributions. The fragmentation and coalescence chains we derive arise naturally in the context of the genealogy of Yule processes.

We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which don't, at present, incorporate all the competitive strategies that a population might adopt. The second is a simplification of the first in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching-annihilating random walk. For each model, using a comparison with N-dependent oriented percolation, we show that for certain parameter values both populations will coexist for all time with positive probability.

As a corollary we deduce survival for all time of branching annihilating random walk for sufficiently large branching rates.

We also present conjectures relating to the role of space in the survival probabilities for the two populations.