This work is devoted to the study of an integro-differential system of equations modelling the genetic adaptation of a pathogen by taking into account both mutation and selection processes. First, we study the asymptotic behaviour of the system and prove that it eventually converges to a stationary state. Next, we more closely investigate the behaviour of the system in the presence of multiple EAs. Under suitable assumptions and based on a small mutation variance asymptotic, we describe the existence of a long transient regime during which the pathogen population remains far from its asymptotic behaviour and highly concentrated around some phenotypic value that is different from the one described by its asymptotic behaviour. In that setting, the time needed for the system to reach its large time configuration is very long and multiple evolutionary attractors may act as a barrier of evolution that can be very long to bypass.

If a body enters a viscous-inviscid fluid layer near a wall, then significant effects can be felt from the presence of incident vorticity, viscous forces and nonlinear forces. The focus here is on the response in the outer edge of such a wall layer. Nonlinear two-dimensional unsteady behaviour is examined through modelling, computation and analysis applied for a thin body travelling streamwise upstream or downstream or staying still relative to the wall. The wall layer with its balance between inviscid and viscous effects interacts freely with the body movement, causing relatively high magnitudes of pressure on top of the body and nonlinear responses in the gap between the body and the wall. The study finds explicit solutions for the motion of the body, separation of the flow arising near the wall and possible instabilities occurring over the length scale of any short body.

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.

Image feature points are detected as pixels which locally maximise a detector function, two commonly used examples of which are the (Euclidean) image gradient and the Harris–Stephens corner detector. A major limitation of these feature detectors is that they are only Euclidean-invariant. In this work, we demonstrate the application of a 2D equi-affine-invariant image feature point detector based on differential invariants as derived through the equivariant method of moving frames. The fundamental equi-affine differential invariants for 3D image volumes are also computed.

In this paper, we introduce a dynamical urban planning model. This leads to the study of a system of nonlinear equations coupled through multi-marginal optimal transport problems. A first example consists in solving two equations coupled through the solution to the Monge–Ampère equation. We show that theWasserstein gradient flow theory provides a very good framework to solve these highly nonlinear systems. At the end, a uniqueness result is presented in dimension one based on convexity arguments.

In lubrication problems, which concern thin film flow, cavitation has been considered as a fundamental element to correctly describe the characteristics of lubricated mechanisms. Here, the well-posedness of a cavitation model that can explain the interaction between viscous effects and micro-bubbles of gas is studied. This cavitation model consists of a coupled problem between the compressible Reynolds partial differential equation (PDE) (that describes the flow) and the Rayleigh–Plesset ordinary differential equation (that describes micro-bubbles evolution). A simplified form without bubbles convection is studied here. This coupled model seems never to be studied before from its mathematical aspects. Local times existence results are proved and stability theorems are obtained based on the continuity of the spectrum for bounded linear operators. Numerical results are presented to illustrate these theoretical results.

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.

We present a model for a class of non-local conservation laws arising in traffic flow modelling at road junctions. Instead of a single velocity function for the whole road, we consider two different road segments, which may differ for their speed law and number of lanes (hence their maximal vehicle density). We use an upwind type numerical scheme to construct a sequence of approximate solutions, and we provide uniform L∞ and total variation estimates. In particular, the solutions of the proposed model stay positive and below the maximum density of each road segment. Using a Lax–Wendroff type argument and the doubling of variables technique, we prove the well-posedness of the proposed model. Finally, some numerical simulations are provided and compared with the corresponding (discontinuous) local model.

Switched server systems are mathematical models of manufacturing, traffic and queueing systems that have being studied since the early 1990s. In particular, it is known that typically the dynamics of such systems is asymptotically periodic: each orbit of the system converges to one of its finitely many limit cycles. In this article, we provide an explicit example of a switched server system with exotic behaviour: each orbit of the system converges to the same Cantor attractor. To accomplish this goal, we bring together recent advances in the understanding of the topological dynamics of piecewise contractions and interval exchange transformations (IETs) with flips. The ultimate result is a switched server system whose Poincaré map is semiconjugate to a minimal and uniquely ergodic IET with flips.

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.

In this article, we consider diffusive transport of a reactive substance in a saturated porous medium including variable porosity. Thereby, the evolution of the microstructure is caused by precipitation of the transported substance. We are particularly interested in analysing the model when the equations degenerate due to clogging. Introducing an appropriate weighted function space, we are able to handle the degeneracy and obtain analytical results for the transport equation. Also the decay behaviour of this solution with respect to the porosity is investigated. There a restriction on the decay order is assumed, that is, besides low initial concentration also dense precipitation leads to possible high decay. We obtain nonnegativity and boundedness for the weak solution to the transport equation. Moreover, we study an ordinary differential equation (ODE) describing the change of porosity. Thereby, the control of an appropriate weighted norm of the gradient of the porosity is crucial for the analysis of the transport equation. In order to obtain global in time solutions to the overall coupled system, we apply a fixed point argument. The problem is solved for substantially degenerating hydrodynamic parameters.

We consider Fokker–Planck equations with tilted periodic potential in the subcritical regime and characterise the spatio-temporal dynamics of the partial masses in the limit of vanishing diffusion. Our convergence proof relies on suitably defined substitute masses and bounds the approximation error using the energy-dissipation relation of the underlying Wasserstein gradient structure. In the appendix, we also discuss the case of an asymmetric double-well potential and derive the corresponding limit dynamics in an elementary way.

We prove the existence of global minimisers for a class of attractive–repulsive interaction potentials that are in general not radially symmetric. The global minimisers have compact support. For potentials including degenerate power-law diffusion, the interaction potential can be unbounded from below. Further, a formal calculation indicates that for non-symmetric potentials global minimisers may neither be radial symmetric nor unique.

In this paper, we extend the work by Sato devoted to the development of economic growth models within the framework of the Lie group theory. We propose a new growth model based on the assumption of logistic growth in factors and derive the corresponding production functions, as well as a compatible notion of wage share. In the process, it is shown that the new functions compare reasonably well against relevant economic data. The corresponding problem of maximisation of profit under conditions of perfect competition is solved with the aid of one of these functions. In addition, it is explained in reasonably rigorous mathematical terms why Bowley’s law no longer holds true in the post-1960 data.

Recently we have proposed a monostable reaction-diffusion system to explain the Neolithic transition from hunter-gatherer life to farmer life in Europe. The system is described by a three-component system for the populations of hunter-gatherer (H), sedentary farmer (F1) and migratory one (F2). The conversion between F1 and F2 is specified by such a way that if the total farmers F1 + F2 are overcrowded, F1 actively changes to F2, while if it is less crowded, the situation is vice versa. In order to include this property in the system, the system incorporates a critical parameter (say F0) depending on the development of farming technology in a monotonically increasing way. It determines whether the total farmers are either over crowded (F1 + F2 >F0) or less crowded (F1 + F2 <F0) ( [9, 20]). Previous numerical studies indicate that the structure of travelling wave solutions of the system is qualitatively similar to the one of the Fisher-KPP equation, that the asymptotically expanding velocity of farmers is equal to the minimal velocity (say cm(F0)) of travelling wave solutions, and that cm(F0) is monotonically decreasing as F0 increases. The latter result suggests that the development of farming technology suppresses the expanding velocity of farmers. As a partial analytical result to this property, the purpose of this paper is to consider the two limiting cases where F0 = 0 and F0 → ∞, and to prove cm(0)>cm(∞).

Tissue engineering aims to grow artificial tissues in vitro to replace those in the body that have been damaged through age, trauma or disease. A recent approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor. A key question is to understand how the applied load is distributed throughout the construct. To address this, we employ homogenisation theory to derive equations governing the effective macroscale material properties of a periodic, elastic–poroelastic composite. We treat the fibres as a linear elastic material and the hydrogel as a poroelastic material, and exploit the disparate length scales (small inter-fibre spacing compared with construct dimensions) to derive macroscale equations governing the response of the composite to an applied load. This homogenised description reflects the orthotropic nature of the composite. To validate the model, solutions from finite element simulations of the macroscale, homogenised equations are compared to experimental data describing the unconfined compression of the fibre-reinforced hydrogels. The model is used to derive the bulk mechanical properties of a cylindrical construct of the composite material for a range of fibre spacings and to determine the local mechanical environment experienced by cells embedded within the construct.

Evolution of planar curves under a nonlocal geometric equation is investigated. It models the simultaneous contraction and growth of carbonate particles called ooids in geosciences. Using classical ODE results and a bijective mapping, we demonstrate that the steady parameters associated with the physical environment determine a unique, time-invariant, compact shape among smooth, convex curves embedded in ℝ2. It is also revealed that any time-invariant solution possesses D2 symmetry. The model predictions remarkably agree with ooid shapes observed in nature.

Two mathematical models under so-called intensity and structure frameworks to pricing a double defaultable interest rate swap are established. The default could happen or jump to a high probability in both fixed and floating parties on the predetermined boundaries. The models lead to a new and interesting mathematical problem. As the intensity approaches infinity in designated regions, the solutions of the intensity models converge to a solution of a structure-type model which is an initial value problem of a partial differential equation coupled with two obstacles problem in their restricted regions. According to the value of the fixed rate, three cases are discussed. The free boundary that determines the swap rate and the free boundaries that determine the earlier termination of the contract (due to counterparty’s default) are analysed.

Each species is subject to various biotic and abiotic factors during growth. This paper formulates a deterministic model with the consideration of various factors regulating population growth such as age-dependent birth and death rates, spatial movements, seasonal variations, intra-specific competition and time-varying maturation simultaneously. The model takes the form of two coupled reaction–diffusion equations with time-dependent delays, which bring novel challenges to the theoretical analysis. Then, the model is analysed when competition among immatures is neglected, in which situation one equation for the adult population density is decoupled. The basic reproduction number $\mathcal{R}_0$ is defined and shown to determine the global attractivity of either the zero equilibrium (when $\mathcal{R}_0\leq 1$ ) or a positive periodic solution ( $\mathcal{R}_0\gt1$ ) by using the dynamical system approach on an appropriate phase space. When the immature intra-specific competition is included and the immature diffusion rate is neglected, the model is neither cooperative nor reducible to a single equation. In this case, the threshold dynamics about the population extinction and uniform persistence are established by using the newly defined basic reproduction number $\widetilde{\mathcal{R}}_0$ as a threshold index. Furthermore, numerical simulations are implemented on the population growth of two different species for two different cases to validate the analytic results.

In a singularly perturbed limit, we analyse the existence and linear stability of steady-state hotspot solutions for an extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Jones et. al. [Math. Models. Meth. Appl. Sci., 20, Suppl., (2010)], which models urban crime with police intervention. In our extended RD model, the field variables are the attractiveness field for burglary, the criminal population density and the police population density. Our model includes a scalar parameter that determines the strength of the police drift towards maxima of the attractiveness field. For a special choice of this parameter, we recover the ‘cops-on-the-dots’ policing strategy of Jones et. al., where the police mimic the drift of the criminals towards maxima of the attractiveness field. For our extended model, the method of matched asymptotic expansions is used to construct 1-D steady-state hotspot patterns as well as to derive nonlocal eigenvalue problems (NLEPs) that characterise the linear stability of these hotspot steady states to ${\cal O}$(1) timescale instabilities. For a cops-on-the-dots policing strategy, we prove that a multi-hotspot steady state is linearly stable to synchronous perturbations of the hotspot amplitudes. Alternatively, for asynchronous perturbations of the hotspot amplitudes, a hybrid analytical–numerical method is used to construct linear stability phase diagrams in the police vs. criminal diffusivity parameter space. In one particular region of these phase diagrams, the hotspot steady states are shown to be unstable to asynchronous oscillatory instabilities in the hotspot amplitudes that arise from a Hopf bifurcation. Within the context of our model, this provides a parameter range where the effect of a cops-on-the-dots policing strategy is to only displace crime temporally between neighbouring spatial regions. Our hybrid approach to study the NLEPs combines rigorous spectral results with a numerical parameterisation of any Hopf bifurcation threshold. For the cops-on-the-dots policing strategy, our linear stability predictions for steady-state hotspot patterns are confirmed from full numerical PDE simulations of the three-component RD system.