Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.
This paper studies the global asymptotic stability of neural networks of neutral type with mixed delays. The mixed delays include constant delay in the leakage term (i.e. 'leakage delay'), time-varying delays and continuously distributed delays. Based on the topological degree theory, Lyapunov method and linear matrix inequality (LMI) approach, some sufficient conditions are derived ensuring the existence, uniqueness and global asymptotic stability of the equilibrium point, which are dependent on both the discrete and distributed time delays. These conditions are expressed in terms of LMI and can be easily checked by the MATLAB LMI toolbox. Even if there is no leakage delay, the obtained results are less restrictive than some recent works. It can be applied to neural networks of neutral type with activation functions without assuming their boundedness, monotonicity or differentiability. Moreover, the differentiability of the time-varying delay in the non-neutral term is removed. Finally, two numerical examples are given to show the effectiveness of the proposed method.
We consider the regularity criteria for the incompressible Navier- Stokes equations connected with one velocity component. Based on the method from Cao and Titi (2008 Indiana Univ. Math. J. 57 2643-61) we prove that the weak solution is regular, provided u(3) is an element of L-t(0, T; L-s(R-3)), 2/t + 3/s 10/3 or provided del u3 is an element of L-t (0, T; L-s(R-3)), 2/t + 3/s <= 19/12 + 1/2s if s is an element of (30/19, 3] or 2/t + 3/s <= 3/2 + 3/4s if s is an element of (3, infinity]. As a corollary, we also improve the regularity criteria expressed by the regularity of partial derivative p/partial derivative x(3) or partial derivative u(3)/partial derivative x(3).
We develop the inverse scattering transform (IST) method for the Degasperis-Procesi equation. The spectral problem is an sl(3) Zakharov-Shabat problem with constant boundary conditions and finite reduction group. The basic aspects of the IST, such as the construction of fundamental analytic solutions, the formulation of a Riemann-Hilbert problem, and the implementation of the dressing method, are presented.
In this paper, a class of neural networks with time-varying delays are investigated for the first time using a periodically intermittent control technique. First, some new and useful stabilization criteria and synchronization conditions based on p-norm are derived by introducing multi-parameters and using the Lyapunov functional technique. For infinity-norm, using the analysis technique, some novel conditions ensuring exponential stability and synchronization are also obtained. It is worth noting that the methods used in this paper are totally different from the corresponding previous works and the obtained conditions are less conservative. Particularly, the traditional assumptions on control width and time delay are removed in this paper. Finally, some numerical simulations are given to verify the theoretical results.
Recently, there has been a wide interest in the study of aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the well-posedness theory of these models. We prove local well-posedness on bounded domains for dimensions d >= 2 and in all of space for d >= 3, the uniqueness being a result previously not known for PKS with degenerate diffusion. We generalize the notion of criticality for PKS and show that subcritical problems are globally well-posed. For a fairly general class of problems, we prove the existence of a critical mass which sharply divides the possibility of finite time blow-up and global existence. Moreover, we compute the critical mass for fully general problems and show that solutions with smaller mass exists globally. For a class of supercritical problems we prove finite time blow-up is possible for initial data of arbitrary mass.
In this paper we prove the existence of ground state solutions of the modified nonlinear Schrodinger equation: -Delta u + V (x) u - 1/2u Delta u(2) = |u|(p-1)u, x is an element of R-N, N >= 3, under some hypotheses on V (x). This model has been proposed in the theory of superfluid films in plasma physics. As a main novelty with respect to some previous results, we are able to deal with exponents p is an element of (1, 3). The proof is accomplished by minimization under a convenient constraint.
We study a class of quasi-linear Schrodinger equations arising in the theory of superfluid film in plasma physics. Using gauge transforms and a derivation process we solve, under some regularity assumptions, the Cauchy problem. Then, by means of variational methods, we study the existence, the orbital stability and instability of standing waves which minimize some associated energy.
This paper is concerned with the entire solution of a diffusive and competitive Lotka-Volterra type system with nonlocal delays. The existence of the entire solution is proved by transforming the system with nonlocal delays to a four-dimensional system without delay and using the comparing argument and the sub-super-solution method. Here an entire solution means a classical solution defined for all space and time variables, which behaves as two wave fronts coming from both sides of the x-axis.
The use of linear response theory for forced dissipative stochastic dynamical systems through the fluctuation dissipation theorem is an attractive way to study climate change systematically among other applications. Here, a mathematically rigorous justification of linear response theory for forced dissipative stochastic dynamical systems is developed. The main results are formulated in an abstract setting and apply to suitable systems, in finite and infinite dimensions, that are of interest in climate change science and other applications.
Dynamical systems that describe the escape from the basins of attraction of stable invariant sets are presented and analysed. It is shown that the stable fixed points of such dynamical systems are the index-1 saddle points. Generalizations to high index saddle points are discussed. Both gradient and non-gradient systems are considered. Preliminary results on the nature of the dynamical behaviour are presented.
We give a complete analysis of low-velocity dynamics close to grazing for a generic one degree of freedom impact oscillator. This includes nondegenerate (quadratic) grazing and minimally degenerate (cubic) grazing, corresponding respectively to nondegenerate and degenerate chatter. We also describe the dynamics associated with generic one-parameter bifurcation at a more degenerate (quartic) graze, showing in particular how this gives rise to the often-observed highly convoluted structure in the stable manifolds of chattering orbits. The approach adopted is geometric, using methods from singularity theory.
We formulate and justify rigorously a numerically efficient criterion for the computation of the analyticity breakdown of quasi-periodic solutions in symplectic maps (any dimension) and 1D statistical mechanics models. Depending on the physical interpretation of the model, the analyticity breakdown may correspond to the onset of mobility of dislocations, or of spin waves (in the 1D models) and to the onset of global transport in symplectic twist maps in 2D. The criterion proposed here is based on the blow-up of Sobolev norms of the hull functions. We prove theorems that justify the criterion. These theorems are based on an abstract implicit function theorem, which unifies several results in the literature. The proofs also lead to fast algorithms, which have been implemented and used elsewhere. The method can be adapted to other contexts.
For any C-r contact Anosov flow with r >= 3, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all the C-r functions, such that the one-parameter family of transfer operators for the flow extend to them boundedly and that the extensions are quasi-compact. We also give explicit bounds on the essential spectral radii of those extensions in terms of differentiability r and the hyperbolicity exponents of the flow.
We investigate a model for the dynamics of a solid object, which moves over a randomly vibrating solid surface and is subject to a constant external force. The dry friction between the two solids is modelled phenomenologically as being proportional to the sign of the object's velocity relative to the surface, and therefore shows a discontinuity at zero velocity. Using a path integral approach, we derive analytical expressions for the transition probability of the object's velocity and the stationary distribution of the work done on the object due to the external force. From the latter distribution, we also derive a fluctuation relation for the mechanical work fluctuations, which incorporates the effect of the dry friction.
In this paper, a periodic discrete nonlinear Schrodinger equation with saturable nonlinearity is considered. Using the critical point theory in combination with periodic approximations, we establish sufficient conditions on the nonexistence and on the existence of gap solitons. Our results not only solve an open problem proposed by Pankov (2006 Nonlinearity 19 27-40) but also considerably improve some existing ones even for some special cases.
Various qualitative properties of solutions to the generalized Langevin equation (GLE) in a periodic or a confining potential are studied in this paper. We consider a class of quasi-Markovian GLEs, similar to the model that was introduced in Eckmann J-P et al 1999 Commun. Math. Phys. 201 657-97. Ergodicity, exponentially fast convergence to equilibrium, short time asymptotics, a homogenization theorem (invariance principle) and the white noise limit are studied. Our proofs are based on a careful analysis of a hypoelliptic operator which is the generator of an auxiliary Markov process. Systematic use of the recently developed theory of hypocoercivity (Villani C 2009 Mem. Am. Math. Soc. 202 iv, 141) is made.
In many semi-arid environments, vegetation cover is sparse, and is self-organized into large-scale spatial patterns. In particular, banded vegetation is typical on hillsides. Mathematical modelling is widely used to study these banded patterns, and many models are effectively extensions of a coupled reaction-diffusion-advection system proposed by Klausmeier (1999 Science 284 1826-8). However, there is currently very little mathematical theory on pattern solutions of these equations. This paper is the first in a series whose aim is a comprehensive understanding of these solutions, which can act as a springboard both for future simulation-based studies of the Klausmeier model, and for analysis of model extensions. The author focusses on a particular part of parameter space, and derives expressions for the boundaries of the parameter region in which patterns occur. The calculations are valid to leading order at large values of the 'slope parameter', which reflects a comparison of the rate of water flow downhill with the rate of vegetation dispersal. The form of the corresponding patterns is also studied, and the author shows that the leading order equations change close to one boundary of the parameter region in which there are patterns, leading to a homoclinic solution. Conclusions are drawn on the way in which changes in mean annual rainfall affect pattern properties, including overall biomass productivity.
Motivated by applications to the piston problem, to a manhole model, to blood flow and to supply chain dynamics, this paper deals with a system of conservation laws coupled with a system of ordinary differential equations. The former is defined on a domain with boundary and the coupling is provided by the boundary condition. For each of the examples considered, numerical integrations are provided.
We construct a natural extension for each of Nakada's alpha-continued fraction transformations and show the continuity as a function of alpha of both the entropy and the measure of the natural extension domain with respect to the density function (1 + xy)(-2). For 0 < alpha <= 1, we show that the product of the entropy with the measure of the domain equals pi(2)/6. We show that the interval (3 - root 5)/2 <= alpha <= (1 + root 5)/2 is a maximal interval upon which the entropy is constant. As a key step for all this, we give the explicit relationship between the a-expansion of alpha - 1 and of alpha.