Travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion

In this paper, we carry out a travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion. We consider two types of invasive fronts of tumour tissue into extracellular matrix (ECM), which represents healthy tissue. These types differ according to whether the density of ECM far ahead of the wave front is maximal or not. In the former case, we use a shooting argument to prove that there exists a unique travelling-wave solution for any positive propagation speed. In the latter case, we further develop this argument to prove that there exists a unique travelling-wave solution for any propagation speed greater than or equal to a strictly positive minimal wave speed. Using a combination of analytical and numerical results, we conjecture that the minimal wave speed depends monotonically on the degradation rate of ECM by tumour cells and the ECM density far ahead of the front.

Unbounded solutions of models for glycolysis

The Selkov oscillator, a simple description of glycolysis, is a system of two ordinary differential equations with mass action kinetics. In previous work the authors established several properties of the solutions of this system. In the present paper we extend this to prove that this system has solutions which diverge to infinity in an oscillatory manner at late times. This is done with the help of a Poincaré compactification of the system and a shooting argument. This system was originally derived from another system with Michaelis–Menten kinetics. A Poincaré compactification of the latter system is carried out and this is used to show that the Michaelis–Menten system, like that with mass action, has solutions which diverge to infinity in a monotone manner. It is also shown to admit subcritical Hopf bifurcations and thus unstable periodic solutions. We discuss to what extent the unbounded solutions cast doubt on the biological relevance of the Selkov oscillator and compare it with other models for the same biological system in the literature.

Boundary value problem forr2d2f/dr2+f=f3(I): existence and uniqueness

We study the equationr2d2f/dr2+f=f3with the boundary conditionsf(1)=0,f(∞)=1, andf(r)>0for1<r<∞. The existence of the solution is proved using a topological shooting argument. And the uniqueness is proved by a variation method.

Multibump periodic travelling waves in suspension bridges

We investigate different types of periodic solutions of a fourth-order, nonlinear differential equation, which has been proposed as a model for travelling waves in suspension bridges. We develop a shooting argument, which enables us to prove the existence of two families of multibump periodic solutions, each containing a countably infinite number of different solutions

TRAVELING WAVES FOR A SIMPLE DIFFUSIVE EPIDEMIC MODEL

We investigate the existence of traveling wave solutions for the infective-susceptible two-component epidemic model. The model system is described by reaction-diffusion equations with the nonlinear reaction term of the classical Kermack-McKendric type. The diffusion coefficients of infectives and susceptibles are assumed to be positive constants d1 and d2 respectively. By the shooting argument with the aid of the invariant manifold theory, we prove that there exists a positive constant c* such that the traveling wave solutions exist for any c≥c*. The minimal wave speed c* is shown to be independent of d2 and to have the same value as that for d2=0.

Static spherically symmetric solutions of a Yang–Mills field coupled to a dilaton

We give a detailed analytical study of static spherically symmetric solutions for an SU (2) Yang–Mills field coupled to a scalar graviton (or dilaton). We show by a ‘shooting’ argument that there are a countable infinity of such solutions satisfying the relevant boundary conditions, there being at least one for each given number of local maxima and minima for the Yang-Mills potential.

Instability of standing waves for non-linear Schrödinger-type equations

A theorem is proved giving a condition under which certain standing wave solutions of non-linear Schrödinger-type equations are linearly unstable. The eigenvalue equations for the linearized operator at the standing wave can be analysed by dynamical systems methods. A positive eigenvalue is then shown to exist by means of a shooting argument in the space of Lagrangian planes. The theorem is applied to a situation arising in optical waveguides.

A shooting argument with oscillation for semilinear elliptic radially symmetric equations

SynopsisA new method is developed for finding radially symmetric solutions of semilinear elliptic problems by phase space methods. The basic idea is to formulate a shooting argument from initial conditions at r = 0 which involves encoding the oscillation information about the trajectories under consideration. The main theorem is applied to a particular nonlinearity and produces a cascade of solutions wherein the multiplicity increases with the number of zeros.