Dynamic instabilities in scalar neural field equations with space-dependent delays

In this paper we consider a class of scalar integral equations with a form of space-dependent delay. These non-local models arise naturally when modelling neural tissue with active axons and passive dendrites. Such systems are known to support a dynamic (oscillatory) Turing instability of the homogeneous steady state. In this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability. The appropriate amplitude equations are found to be the coupled mean-field Ginzburg-Landau equations describing a Turing-Hopf bifurcation with modulation group velocity of O(1). Importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio-temporal kernels defining the neural field equation of interest. Indeed our results cover not only models with axonal or dendritic delays but those which are described by a more general distribution of delayed spatio-temporal interactions. We illustrate the predictive power of this form of analysis with comparison against direct numerical simulations, paying particular attention to the competition between standing and travelling waves and the onset of Benjamin-Feir instabilities.

Coagulation equations with mass loss

We derive and solve models for coagulation with mass loss arising, for example, from industrial processes in which growing inclusions are lost from the melt by colliding with the wall of the vessel. We consider a variety of loss laws and a variety of coagulation kernels, deriving exact results where possible, and more generally reducing the equations to similarity solutions valid in the large-time limit. One notable result is the effect that mass removal has on gelation: for small loss rates, gelation is delayed, whilst above a critical threshold, gelation is completely prevented. Finally, by forming an exact explicit solution for a more general initial cluster size distribution function, we illustrate how numerical results from earlier work can be interpreted in the light of the theory presented herein.