John Norbury

Mathematical modelling of capsule formation and multinodularity in benign tumour growth

Tumours that grow locally, but do not invade the surrounding tissue are called benign. Such benign tumours are characterized by the presence of a surrounding band of connective tissue called a capsule. In some cases, the tumours are also broken into a number of discrete nodules. In this paper the authors use a partial differential equation model to study the interactions of a growing tumour with the surrounding tissue. They predict mechanisms for both capsule formation and nodularity. The former has the mathematical form of bifurcation from travelling waves to aggregating waves of connective tissue, resulting in the accretion of connective tissue in a manner corresponding to capsule formation. The cause of multilobularity in tumours is currently not known. Using their model, the authors are able to predict lobulation, when tumour cell motility is retarded by aggregating connective tissue. In the final part of the paper, the authors introduce an enlarged model, and use it to demonstrate both capsule formation and the possible dissolution of the capsule following a mutation resulting in the production of proteases by the cancer cells.

The Location and Stability of Interface Solutions of an Inhomogeneous Parabolic Problem

Reaction-diffusion mechanisms have been used to explain pattern formation in developmental biology and in experimental chemical systems. Environmental effects are included through an inhomogeneous nonlinear forcing term in a parabolic equation model with a small diffusion coefficient. The small diffusivity and the form of the nonlinear forcing term yield interface solutions after a time of order one, and these interface solutions then change slowly on a longer timescale. The stability of these interface solutions is considered from both a numerical viewpoint and using asymptotic analysis, and it is found that a simple sign condition determines stability. Here the existence of a steady-state interface is determined by the zeroes of the derivative of a function H(x)equiv ln(f(x)g3 (x)), and the stability of this steady state interface (which determines a patterned solution) is controlled by the sign of the second derivative of H(x) at a zero where the interface occurs.

Dynamics of constrained differential delay equations

A class of forced first-order differential delay equations with piecewise-affine right-hand sides is introduced, as a prototype model for the speed of a motor under control. A simple pure delay form is mainly considered. When forcing is zero, an exact stable periodic solution is exhibited. For large amplitude periodic forcing, existence of stable solutions, whose period is equal to that of the forcing function, is discussed, and these solutions are constructed for square wave forcing. Traditional numerical methods are discussed briefly, and a new approach based on piecewise-polynomial structure is introduced. Simulations are then presented showing a wide range of dynamics for intermediate values of forcing amplitude, when the natural period of the homogeneous equation and the period of the forcing function compete.

Bifurcation of positive solutions for a Neumann boundary value problem

Analytical, approximate and numerical methods are used to study the Neumann boundary value problem − u xx + q 2 u = u 2 (1 + sin x ), for 0 x subject to u x (0) = 0, u x (π) = 0, for q 2 ∈ (0,∞). Asymptotic approximations to (1) are found for q 2 small and q 2 large. In the case where q 2 is large u(x) ≈ 3 qδ ( x − π /2). When q 2 = 0 we show that the only possible solution is u ≡ 0. However, there exist non-zero solutions for q 2 > 0 as well as the trivial solution u ≡ 0. To O(q 4 ) in the q 2 small case u(x) = q 2 π(π + 2) −1 , so that bifurcation occurs about the trivial solution branch u ≡ 0 at the first eigenvalue λ 0 = 0 and in the direction of the first eigenfunction ξ 0 = constant. We obtain a bifurcation diagram for (1), which confirms that there exists a positive solution for q 2 ∈ (0, 10). Symmetry-breaking bifurcations and blow-up behaviour occur on certain regions of the diagram. We show that all non-trival solutions to the problem must be positive. The formal outer solution u = q 2 u appears to satisfy u = u 2 (1 + sin x ), so that u ≡ 0 and u = (1 + sin x ) −1 are possible limit solutions. However, in the non-trivial case u x (0) = −1 and u x (π) = 1; this means that u does not satisfy the boundary conditions required for a solution of (1). This behaviour usually implies that for q 2 large a boundary layer exists near x = 0 (and one near x = π ), which corrects the slope. However, we find no evidence for such a solution structure, and only find perturbations in the direction of a delta function about u ≡ 0. We show using the monotone convergence theorem for quadratic forms that the inverse of the operator on the left-hand side of (1) is strongly convergent as q 2 → ∞. We show that strong convergence of the operator is sufficient to stop outer-layer behaviour occurring.

The bifurcation of steady gravity water waves in (R, S) parameter space

The bifurcation of steady periodic waves from irrotational inviscid streamflows is considered. Normalizing the flux Q to unity leaves two other natural quantities R (pressure head) and S (flowforce) to parameterize the wavetrain. In a well-known paper, Benjamin & Lighthill (1954) presented calculations within a cnoidal-wave theory which suggested that the corresponding values of R and S lie inside the cusped locus traced by the sub- and supercritical streamflows. This rule has been applied since to many other flow scenarios. In this paper, regular expansions for the streamfunction and profile are constructed for a wave forming on a subcritical stream and thence values for R and S are calculated. These describe, locally, how wave branches in (R, S) parameter space point inside the streamflow cusp. Accurate numerics using a boundary-integral solver show how these constant-period branches extend globally and map out parameter space. The main result is to show that the large-amplitude branches for all steady Stokes’ waves lie surprisingly close to the subcritical stream branch. This has important consequences for the feasibility of undular bores (as opposed to hydraulic jumps) in obstructed flow. Moreover, the transition from the ‘long-wave region’ towards the ‘deep-water limit’ is characterized by an extreme geometry, both of the wave branches and how they sit inside each other. It is also shown that a single (Q, R, S) triple may represent more than one wave since the global branches can overlap in (R, S) parameter space. This non-uniqueness is not that associated with the known premature maxima of wave properties as functions of wave amplitude near waves of greatest height.

Solution Structure for Nonautonomous Nonlocal Elliptic Equations with Neumann Boundary Conditions

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On the steady flow of an incompressible elastic-plastic material past thin cones: Numerical results

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A note on solitary and periodic waves of a new kind

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