In the field of Life Sciences, it is very common to deal with extremely complex systems, from both analytical and computational points of view, due to the unavoidable coupling of different interacting structures. As an example, angiogenesis has revealed to be an highly complex, and extremely interesting biomedical problem, due to the strong coupling between the kinetic parameters of the relevant branching – growth – anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. In this paper, an original revisited conceptual stochastic model of tumour-driven angiogenesis has been proposed, for which it has been shown that it is possible to reduce complexity by taking advantage of the intrinsic multiscale structure of the system; one may keep the stochasticity of the dynamics of the vessel tips at their natural microscale, whereas the dynamics of the underlying fields is given by a deterministic mean field approximation obtained by an averaging at a suitable mesoscale. While in previous papers, only an heuristic justification of this approach had been offered; in this paper, a rigorous proof is given of the so called ‘propagation of chaos’, which leads to a mean field approximation of the stochastic relevant measures associated with the vessel dynamics, and consequently of the underlying tumour angiogenic factor (TAF) field. As a side, though important result, the non-extinction of the random process of tips has been proven during any finite time interval.

Suspended fibres significantly alter fluid rheology, as exhibited in for example solutions of DNA, RNA and synthetic biological nanofibres. It is of interest to determine how this altered rheology affects flow stability. Motivated by the fact thermal gradients may occur in biomolecular analytic devices, and recent stability results, we examine the problem of Rayleigh–Bénard convection of the transversely isotropic fluid of Ericksen. A transversely isotropic fluid treats these suspensions as a continuum with an evolving preferred direction, through a modified stress tensor incorporating four viscosity-like parameters. We consider the linear stability of a stationary, passive, transversely isotropic fluid contained between two parallel boundaries, with the lower boundary at a higher temperature than the upper. To determine the marginal stability curves the Chebyshev collocation method is applied, and we consider a range of initially uniform preferred directions, from horizontal to vertical, and three orders of magnitude in the viscosity-like anisotropic parameters. Determining the critical wave and Rayleigh numbers, we find that transversely isotropic effects delay the onset of instability; this effect is felt most strongly through the incorporation of the anisotropic shear viscosity, although the anisotropic extensional viscosity also contributes. Our analysis confirms the importance of anisotropic rheology in the setting of convection.

We study a competition-diffusion model while performing simultaneous homogenization and strong competition limits. The limit problem is shown to be a Stefan-type evolution equation with effective coefficients. We also perform some numerical simulations in one and two spatial dimensions that suggest that oscillations in motilities are detrimental to invasion behaviour of a species.

This paper is concerned with a fully non-linear variant of the Allen–Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. The main purposes of this paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviours of solutions. More precisely, by deriving partial energy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solution u(x,t) converges to a solution of an elliptic obstacle problem as t → +∞.

In a prior work, the authors proved a global bifurcation theorem for spatially periodic interfacial hydroelastic travelling waves on infinite depth, and computed such travelling waves. The formulation of the travelling wave problem used both analytically and numerically allows for waves with multi-valued height. The global bifurcation theorem required a one-dimensional kernel in the linearization of the relevant mapping, but for some parameter values, the kernel is instead two-dimensional. In the present work, we study these cases with two-dimensional kernels, which occur in resonant and non-resonant variants. We apply an implicit function theorem argument to prove existence of travelling waves in both of these situations. We compute the waves numerically as well, in both the resonant and non-resonant cases.

In the singular perturbation limit ε → 0, we analyse the linear stability of multi-spot patterns on a bounded 2-D domain, with Neumann boundary conditions, as well as periodic patterns of spots centred at the lattice points of a Bravais lattice in $\mathbb{R}^2$, for the Brusselator reaction–diffusion model

$$ \begin{equation*} v_t = \epsilon^2 \Delta v + \epsilon^2 - v + fu v^2 \,, \qquad \tau u_t = D \Delta u + \frac{1}{\epsilon^2}\left(v - u v^2\right) \,, \end{equation*} $$

${\mathcal O}$

$\mathbb{R}^2$