We consider the impact of spatial heterogeneities on the dynamics of

localized patterns in systems of partial differential equations (in one

spatial dimension). We will mostly focus on the most simple possible

heterogeneity: a small jump-like defect that appears in models in which

some parameters change in value as the spatial variable x crosses

through a critical value -- which can be due to natural inhomogeneities,

as is typically the case in ecological models, or can be imposed on the

model for engineering purposes, as in Josephson junctions. Even such a

small, simplified heterogeneity may have a crucial impact on the

dynamics of the PDE. We will especially consider the effect of the

heterogeneity on the existence of defect solutions, which boils down to

finding heteroclinic (or homoclinic) orbits in an n-dimensional

dynamical system in `time' x, for which the vector field for x > 0

differs slightly from that for x < 0 (under the assumption that there is

such an orbit in the homogeneous problem). Both the dimension of the

problem and the nature of the linearized system near the limit points

have a remarkably rich impact on the defect solutions. We complement the

general approach by considering two explicit examples: a heterogeneous

extended Fisher–Kolmogorov equation (n = 4) and a heterogeneous

generalized FitzHugh–Nagumo system (n = 6).

# Past Industrial and Applied Mathematics Seminar

Although not all complex networks are embedded into physical spaces, it is possible to find an abstract Euclidean space in which they are embedded. This Euclidean space naturally arises from the use of the concept of network communicability. In this talk I will introduce the basic concepts of communicability, communicability distance and communicability angles. Both, analytic and computational evidences will be provided that shows that the average communicability angle represents a measure of the spatial efficiency of a network. We will see how this abstract spatial efficiency is related to the real-world efficiency with which networks uses the available physical space for classes of networks embedded into physical spaces. More interesting, we will show how this abstract concept give important insights about properties of networks not embedded in physical spaces.

I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random data particle simulations, flocks and mills, and their qualitative behavior.

Despite many years of intensive research, the modeling of contact lines moving by spreading and/or evaporation still remains a subject of debate nowadays, even for the simplest case of a pure liquid on a smooth and homogeneous horizontal substrate. In addition to the inherent complexity of the topic (singularities, micro-macro matching, intricate coupling of many physical effects, …), this also stems from the relatively limited number of studies directly comparing theoretical and experimental results, with as few fitting parameters as possible. In this presentation, I will address various related questions, focusing on the physics invoked to regularize singularities at the microscale, and discussing the impact this has at the macroscale. Two opposite “minimalist” theories will be detailed: i) a classical paradigm, based on the disjoining pressure in combination with the spreading coefficient; ii) a new approach, invoking evaporation/condensation in combination with the Kelvin effect (dependence of saturation conditions upon interfacial curvature). Most notably, the latter effect enables resolving both viscous and thermal singularities altogether, without needing any other regularizing effects such as disjoining pressure, precursor films or slip length. Experimental results are also presented about evaporation-induced contact angles, to partly validate the first approach, although it is argued that reality might often lie in between these two extreme cases.

This talk covers two topics: **(1)** Phenotype change, where we consider the steady-fitness states, in a model developed by Korobeinikov and Dempsey (2014), in which the phenotype is modelled on a continuous scale providing a structured variable to quantify the phenotype state. This enables thresholds for survival/extinction to be established in terms of fitness.

Topic **(2)** looks at the steady-size distribution of an evolving cohort of cells, such as tumour cells *in vitro, *and therein establishes thresholds for growth or decay of the cohort. This is established using a new class of non-local (but linear) singular eigenvalue problems which have point spectra, like the traditional Sturm-Liouville problems. The first eigenvalue gives the threshold required. But these problems are first order unless dispersion is added to incorporate random perturbations. But the same idea will apply here also. Current work involves binary asymmetrical division of cells, simultaneous with growth. It has implications to cancer biology, helping biologists to conceptualise non-local effects and the part they may play in cancer. This is developed in Zaidi *et al* (2015).

**Acknowledgement. **The support of Gravida (NCGD) is gratefully acknowledged.

**References**

Korobeinikov A & Dempsey C. *A continuous phenotype space model of RNA virus evolution within a host**.* Mathematical Biosciences and Engineering **11, (**2014), 919-927.

Zaidi AA, van-Brunt B, & Wake GC. *A model for asymmetrical cell division *Mathematical Biosciences and Engineering (June 2015).

It is well known that low-Reynolds-number flows ($R_e\ll1$) have unique solutions, but this statement may not be true if complex solutions are permitted.

We begin by considering Stokes series, where a general steady velocity field is expanded as a power series in the Reynolds number. At each order, a linear problem determines the coefficient functions, providing an exact closed form representation of the solution for all Reynolds numbers. However, typically the convergence of this series is limited by singularities in the complex $R_e$ plane.

We employ a generalised Pade approximant technique to continue analytically the solution outside the circle of convergence of the series. This identifies other solutions branches, some of them complex. These new solution branches can be followed as they boldly go where no flow has gone before. Sometimes these complex solution branches coalesce giving rise to real solution branches. It is shown that often, an unforced, nonlinear complex "eigensolution" exists, which implies a formal nonuniqueness, even for small and positive $R_e$.

Extensive reference will be made to Dean flow in a slowly curved pipe, but also to flows between concentric, differentially rotating spheres, and to convection in a slot. In addition, certain fundamental exact solutions are shown to possess extra complex solutions.

by Jonathan Mestel and Florencia Boshier