Mean-field games (MFGs) and the best-reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system. In this paper, we present an analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in some specific modelling situations.

We investigate self-similar solutions of the thin film equation in the case of zero contact angle boundary conditions on a finite domain. We prove existence and uniqueness of such a solution and determine the asymptotic behaviour as the exponent in the equation approaches the critical value at which zero contact angle boundary conditions become untenable. Numerical and power-series solutions are also presented.

Within the online media universe, there are many underlying communities. These may be defined, for example, through politics, location, health, occupation, extracurricular interests or retail habits. Government departments, charities and commercial organisations can benefit greatly from insights about the structure of these communities; the move to customer-centred practices requires knowledge of the customer base. Motivated by this issue, we address the fundamental question of whether a sub-network looks like a collection of individuals who have effectively been picked at random from the whole, or instead forms a distinctive community with a new, discernible structure. In the former case, to spread a message to the intended user base it may be best to use traditional broadcast media (TV, billboard), whereas in the latter case a more targeted approach could be more effective. In this work, we therefore formalise a concept of testing for sub-structure and apply it to social interaction data. First, we develop a statistical test to determine whether a given sub-network (induced sub-graph) is likely to have been generated by sampling nodes from the full network uniformly at random. This tackles an interesting inverse alternative to the more widely studied “forward” problem. We then apply the test to a Twitter reciprocated mentions network where a range of brand name based sub-networks are created via tweet content. We correlate the computed results against the independent views of 16 digital marketing professionals. We conclude that there is great potential for social media based analytics to quantify, compare and interpret online brand allegiances systematically, in real time and at large scale.

A general method of placing variational bounds on the effective scalar conductivity of random heterogeneous solids is proposed. The method utilizes the Hashin-Shtrikman variational principle and trial fields which are obtained upon truncating the functional series expansion for the polarization field in the medium. It is shown how the earlier variational procedures due to Hashin and Shtrikman, Milton and Phan-Thien and Willis find natural places in the proposed general scheme. The new results reported include, for example, the fact of nonexistence of certain Gaussian-like random media and, for dispersions of spheres with volume fraction c, an exact O(c2) relation for the effective conductivity which contains absolutely convergent integrals only.

We describe the slow evolution of the wave speed and reaction temperature in a model of filtration combustion. In the counterflow configuration of the process, a porous solid matrix is converted to a porous solid product by injecting an oxidizing gas at high pressure into one end of a fresh sample of the solid while igniting it at the other end. The solid and gas react exothermically at high activation energy and, under favourable conditions, a self-sustaining combustion wave travels along the sample, converting reactants to product. Since the reaction rate depends on the gas pressure p in the pores, small gradients in p cause variations in the conditions of combustion, which, in turn, cause inhomogeneities in the physical properties of the product. We determine the slow evolution of the wave speed, the reaction temperature, and the mass flux of the gas downstream of the reaction zone. In the absence of a pressure gradient, there is a branch of steadily propagating solutions which has a fold. For planar disturbances on the slow time scale, we show that the middle part of the branch is unstable, with the change of stability occurring at the turning points of the branch. When the pressure gradient is nonzero, there are no steadily propagating solutions and the wave continually evolves. Conditions on the state of the gas at the inlet are described such that the variation in the wave speed and reaction temperature throughout the process can be minimized.

A mathematical model is presented for the washing of a band, or tow, of fibres which is pulled past a submerged obstacle. A lubrication flow between the obstacle and the tow is coupled to a flow through the tow, which is modelled as a porous medium. Coupled first-order ordinary differential equations for the fluid pressure and volume flux are solved numerically and the results are analysed. The washing of the tow induced by the reversal of the viscous boundary layers lying above and below the tow is discussed briefly. In Part 2, the model is extending to allow for fibre bending. The numerical solution of the resulting differential equation shows that at a critical value of the tow porosity, the tow touches the obstacle. An existence proof for a family of touching solutions is obtained. Using the existence of these solutions, a method for constructing a solution with a rubbing region, where the gap width is zero, is outlined. The numerical solutions for these solutions are also given.

We analyse oscillatory instabilities for a coupled partial-ordinary differential equation (PDE-ODE) system modelling the communication between localised spatially segregated dynamically active signalling compartments that are coupled through a passive extracellular bulk diffusion field in a bounded 2D domain. Each signalling compartment is assumed to secrete a chemical into the extracellular medium (bulk region), and it can also sense the concentration of this chemical in the region around its boundary. This feedback from the bulk region, resulting from the entire collection of cells, in turn modifies the intracellular dynamics within each cell. In the limit where the signalling compartments are circular discs with a small common radius ɛ ≪ 1 and where the bulk diffusivity is asymptotically large, a matched asymptotic analysis is used to reduce the dimensionless PDE–ODE system into a nonlinear ODE system with global coupling. For Sel’kov reaction kinetics, this ODE system for the intracellular dynamics and the spatial average of the bulk diffusion field are then used to investigate oscillatory instabilities in the dynamics of the cells that are triggered due to the global coupling. In particular, numerical bifurcation software on the ODEs is used to study the overall effect of coupling defective cells (cells that behave differently from the remaining cells) to a group of identical cells. Moreover, when the number of cells is large, the Kuramoto order parameter is computed to predict the degree of phase synchronisation of the intracellular dynamics. Quorum sensing behaviour, characterised by the onset of collective behaviour in the intracellular dynamics as the number of cells increases above a threshold, is also studied. Our analysis shows that the cell population density plays a dual role of triggering and then quenching synchronous oscillations in the intracellular dynamics.

We provide a detailed mathematical analysis of a model for phase separation on biological membranes which was recently proposed by Garcke, Rätz, Röger and the second author. The model is an extended Cahn–Hilliard equation which contains additional terms to account for the active transport processes. We prove results on the existence and regularity of solutions, their long-time behaviour, and on the existence of stationary solutions. Moreover, we investigate two different asymptotic regimes. We study the case of large cytosolic diffusion and investigate the effect of an infinitely large affinity between membrane components. The first case leads to the reduction of coupled bulk-surface equations in the model to a system of surface equations with non-local contributions. Subsequently, we recover a variant of the well-known Ohta–Kawasaki equation as the limit for infinitely large affinity between membrane components.

Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of ‘spurious solutions’, which led some to conclude that this was ‘the failure of MAE’. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.

This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and different particle interactions. Under general assumptions on the entropy potential, we prove the strict separation property in both two and three-space dimensions. Namely, in 2D, we notably extend the minimal assumptions on the potential adopted so far in the literature, by only requiring a mild growth condition of its first derivative near the singular points $\pm 1$, without any pointwise additional assumption on its second derivative. For all cases, we provide a compact proof using De Giorgi’s iterations. In 3D, we also extend the validity of the asymptotic strict separation property to the case of fractional Cahn-Hilliard equation, as well as show the validity of the separation when the initial datum is close to an ‘energy minimizer’. Our framework offers insights into statistical factors like particle interactions, entropy choices and correlations governing separation, with broad applicability.

We analyse a nonlinear partial differential equation system describing the motion of a microswimmer in a nematic liquid crystal environment. For the microswimmer’s motility, the squirmer model is used in which self-propulsion enters the model through the slip velocity on the microswimmer’s surface. The liquid crystal is described using the well-established Beris–Edwards formulation. In previous computational studies, it was shown that the squirmer, regardless of its initial configuration, eventually orients itself either parallel or perpendicular to the preferred orientation dictated by the liquid crystal. Furthermore, the corresponding solution of the coupled nonlinear system converges to a steady state. In this work, we rigorously establish the existence of steady state and also the finite-time existence for the time-dependent problem in a periodic domain. Finally, we will use a two-scale asymptotic expansion to derive a homogenised model for the collective swimming of squirmers as they reach their steady-state orientation and speed.

This work presents an asymptotic algorithm for the derivation of equations of thin elastic shells. The algorithm is based on the analysis of a boundary value problem for the Navier system in a thin region. The analysis covers both the membrane theory and the moment theory of elastic shells, including the eigenvalue problems.

We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviours including having two distinct classes of co-existing steady-state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveals several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite-amplitude limit cycle can occur as a singular perturbation in the nearly conserved limit.

This paper addresses a free boundary problem for a steady, uniform patch of vorticity surrounding a single flat plate of zero thickness and finite length. Exact solutions to this problem have previously been found in terms of conformal maps represented by Cauchy-type integrals. Here, however, it is demonstrated how, by considering an associated circular-arc polygon and using ideas from automorphic function theory, these maps can be expressed in a simple non-integral form.

We study plane shear waves in a continuum model for a saturated soil, assuming a hypoplastic flow rule. Following an earlier paper on velocity-controlled waves [1], we now specify a time- periodic stress disturbance on the boundary of a semi-infinite medium. As before, the stress saturates, tending at large times to a position-independent value, away from the boundary. By contrast, the dependence of this asymptotic state on the boundary data is strikingly different: in the previous case, this dependence was wildly discontinuous, while here it is very simple and smooth.

Boundary value problems for linear transport equations sometimes require the explicit construction of solutions when boundary conditions are prescribed only on parts of the boundary and thus necessitate the construction of half-range expansions. In contrast to the standard eigenfunction expansion, a half-range expansion of a given function must be reconstructed just over half the domain, using just half of the eigenfunctions. The difficulty of such expansions arises because the eigenfunctions are not orthogonal, though they are complete, over half the domain, and there is no obvious method of obtaining the expansion coefficients. Here we use complex variable techniques to find explicit formulas for the coefficients of half-range expansions for regular, negative definite Sturm–Liouville operators. We prove that the half-range expansion formula is unique, and find the corresponding half-range Green functions.

The present work is part of a series of studies conducted by the authors on analytical models of avascular tumour growth that exhibit both geometrical anisotropy and physical inhomogeneity. In particular, we consider a tumour structure formed in distinct ellipsoidal regions occupied by cell populations at a certain stage of their biological cycle. The cancer cells receive nutrient by diffusion from an inhomogeneous supply and they are subject to also an inhomogeneous pressure field imposed by the tumour microenvironment. It is proved that the lack of symmetry is strongly connected to a special condition that should hold between the data imposed by the tumour’s surrounding medium, in order for the ellipsoidal growth to be realizable, a feature already present in other non-symmetrical yet more degenerate models. The nutrient and the inhibitor concentration, as well as the pressure field, are provided in analytical fashion via closed-form series solutions in terms of ellipsoidal eigenfunctions, while their behaviour is demonstrated by indicative plots. The evolution equation of all the tumour’s ellipsoidal interfaces is postulated in ellipsoidal terms and a numerical implementation is provided in view of its solution. From the mathematical point of view, the ellipsoidal system is the most general coordinate system that the Laplace operator, which dominates the mathematical models of avascular growth, enjoys spectral decomposition. Therefore, we consider the ellipsoidal model presented in this work, as the most general analytic model describing the avascular growth in inhomogeneous environment. Additionally, due to the intrinsic degrees of freedom inherited to the model by the ellipsoidal geometry, the ellipsoidal model presented can be adapted to a very populous class of avascular tumours, varying in figure and in orientation.

Short-time existence and uniqueness results in Sobolev spaces are proved for Hele-Shaw flow with kinetic undercooling and for Stokes flow without surface tension. In both cases, the flow is driven by arbitrarily distributed sources and sinks in the interior of the liquid domain. The proofs are based on a general approach consisting of the reformulation of the problem as a Cauchy problem for a nonlinear, nonlocal evolution equation on the unit sphere, quasilinearization by equivariance, investigation of the linearization, and Galerkin approximations. In the situation discussed here, the linearized evolution operator is a first-order differential operator, and thus the evolution equation is of hyperbolic type. Finally, a brief survey of the properties of the evolution equations that arise from Hele-Shaw flow and Stokes flow with and without regularization is given.

It is proved that c-cyclical monotonicity is a sufficient condition for optimality in the multi-marginal optimal transport problem with Coulomb repulsive cost. The notion of c-splitting set and some mild regularity property are the tools. The result may be extended to Coulomb like costs.

We propose and analyse an age-structured model for within-host HIV virus dynamics which is incorporated with both virus-to-cell and cell-to-cell infection routes, and proliferations of both uninfected and infected cells in the form of logistic growth. The model turns out to be a hybrid system with two differential-integral equations and one first-order partial differential equation. We perform some rigorous analyses for the considered model. Among the interesting dynamical behaviours of the model is the occurrence of backward bifurcation in terms of the basic reproduction number R0 at R0 = 1, which raises new challenges for effective infection control. We also discuss the cause of such a backward bifurcation, based on our analytical results.