Calcium sparks in cardiac muscle cells occur when a cluster of Ca2+ channels open and release Ca2+ from an internal store. A simplified model of Ca2+ sparks has been developed to describe the dynamics of a cluster of channels, which is of the form of a continuous time Markov chain with nearest neighbour transitions and slowly varying jump functions. The chain displays metastability, whereby the probability distribution of the state of the system evolves exponentially slowly, with one of the metastable states occurring at the boundary. An asymptotic technique for analysing the Master equation (a differential-difference equation) associated with these Markov chains is developed using the WKB and projection methods. The method is used to re-derive a known result for a standard class of Markov chains displaying metastability, before being applied to the new class of Markov chains associated with the spark model. The mean first passage time between metastable states is calculated and an expression for the frequency of calcium sparks is derived. All asymptotic results are compared with Monte Carlo simulations.

A vectorial extension of the Keller–Rubinow method of computing asymptotic approximations of eigenvalues in bounded domains is presented. The method is applied to the problem of a multimode step-profile cylindrical optical fibre, including the effects of polarisation. A comparison of the asymptotic results with the exact eigenvalues is made when these are available, and the agreement is shown to be good.

We consider the activator-inhibitor Gierer–Meinhardt reaction-diffusion system of biological pattern formation in a closed bounded domain. The existence and stability of a boundary apike-layer solution to the Gierer–Meinhardt model, and it, so-called shadow limit, is analysed. In the limit of small activator diffusivity, together with a large inhibitor diffusivity, an equilibrium boundary spike-layer solution is constructed that concentrates at a non-degenerate critical point P of the boundary. By non-degenerate we mean that every principal curvature of the boundary has a local maximum at P, and hence the mean curvature at the boundary has a local maximum at P. Rigorous results for the stability of such a boundary spike-layer solution are given.

Non-local degenerate parabolic systems arise in three-phase capillary flows in porous media under a pressure control at the inflow- and outflow-boundaries. A mathematical study of such systems is performed for a class of capillarity pressure functions corresponding to triangular capillarity-diffusion tensors. To this end a theory of non-degenerate parabolic approximations is developed: the unique global solvability of initial boundary-value problems is proved.

Thin-film models for the flow of a low reduced-Reynolds-number poroviscous droplet over a planar substrate are developed. One of the formulations is used to develop a minimal model for active animal cell motion in which the microscopic mechanisms of polymerisation and depolymerisation near the outer cell periphery are modelled by specifying the rate of mass transfer between the phases at the contact-line in terms of the velocity of the latter. An asymptotic analysis in the limit corresponding to strong cell-substrate adhesion is shown to lead to a novel class of multi-valued contact-line laws, a qualitative analysis of which leads in two dimensions to some intriguing behaviour, including (i) periodic contraction and expansion (pulsation), (ii) steady propagation at a contant speed, (iii) an unsteady combination of pulsation and propagation, and (iv) a bistable regime in which both non-motile and motile solutions are admissible, each of them being stable to sufficiently small perturbations, but with sufficiently large perturbations being able to ‘prod’ a stationary cell into motion or halt a moving one; these qualitative predictions are where possible compared with experiment. The contact-line behaviour is likely to be highly sensitive to environmental signals; the formulation may, therefore, provide a useful ‘minimal’ modelling framework for investigation of chemotactic effects at the cell scale. The corresponding extensional flow formulations are also noted.