Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences最新文献

Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the form$$\frac{\mathrm{d}{x}_i}{\mathrm{d}t}=x_i{\varphi }_i(x_1,\dots ,x_N),$$where $N$ represents the number of species and $x_i$, the abundance of species $i$. Among these families of coupled differential equations, Lotka–Volterra (LV) equations, corresponding to$${\varphi }_i(x_1,\dots ,x_N)=r_i-x_i+{(\Gamma \mathbf{\text{x}})}_i,$$play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here, $r_i$ is the intrinsic growth of species $i$ and $\Gamma$ stands for the interaction matrix: ${\Gamma }_{ij}$ represents the effect of species $j$ over species $i$. For large $N$, estimating matrix $\Gamma$ is often an overwhelming task and an alternative is to draw $\Gamma$ at random, parameterizing its statistical distribution by a limited number of model features. Dealing with large random matrices, we naturally rely on random matrix theory (RMT). The aim of this review article is to present an overview of the work at the junction of theoretical ecology and large RMT. It is intended to an interdisciplinary audience spanning theoretical ecology, complex systems, statistical physics and mathematical biology.

The modelling of liver tissue across multiple length scales constitutes a significant challenge, primarily due to the multiphysics coupling of mechanical response and perfusion within the complex multiscale vascularization of the organ. In this paper, we present a modelling framework that connects continuum poroelasticity and discrete vascular tree structures to model liver tissue across disparate levels of the perfusion hierarchy. The connection is achieved through a series of modelling decisions, which include source terms in the pressure equation to model inflow from the supplying tree, pressure boundary conditions to model outflow into the draining tree, and contact conditions to model surrounding tissue. We investigate the numerical behaviour of our framework and apply it to a patient-specific full-scale liver problem that demonstrates its potential to help assess surgical liver resection procedures.

The multichannel scattering problem for the stationary Schrödinger equation on a line with different thresholds at both infinities is investigated. The analytical structure of the Jost solutions and of the transition matrix relating the Jost solutions as functions of the spectral parameter is described. Unitarity of the scattering matrix is proved in the general case when some of the scattering channels can be closed and the thresholds can be different at left and right infinities on the line. The symmetry relations of the $S$-matrix are established. The condition determining the bound states is obtained. The asymptotics of the Jost functions and of the transition matrix are derived for a large spectral parameter.

Modern transducers and actuators may have functional layers with multi-field coupling and some elastic layers. This paper considers a tubular bilayer system consisting of a thin dielectric tube coated with a thick elastic layer. We study the nonlinear electromechanical response and the linear axisymmetric vibration of the system subject to different applied voltages and inner/outer pressures within the framework of the general nonlinear theory of electro-elasticity, the related linear incremental theory, and by considering the continuity conditions at the interface. We investigate instability behaviour using the same basic formulae. The state-space method provides efficient and accurate free vibration analysis, considering the dynamic response at the lowest frequencies, so we can neglect the viscous and damping effects, which is well suited to this problem. New results indicate that the bilayer system improves its frequency capability and stability compared to the monolayer dielectric tube. The thick outer elastic layer stiffens the bilayer system against axisymmetric bifurcation, bulging and necking instabilities. It also performs better in front of axisymmetric instability, increasing the system’s capability to receive or produce higher voltages, especially for long waves.This work thoroughly explains bilayer functional systems’ behaviour when exposed to extreme environments such as high voltage or pressure.

Explicit solutions to the nonlinear field equations of some gravitational theories can be obtained, by means of a Riemann–Hilbert approach, from a canonical Wiener–Hopf factorization of certain matrix functions called monodromy matrices. In this paper, we describe other types of factorization from which solutions can be constructed in a similar way. Our approach is based on an invariance problem, which does not constitute a Riemann–Hilbert problem and allows to construct solutions that could not have been obtained by Wiener–Hopf factorization of a monodromy matrix. It gives rise to a novel solution generating method based on matrix multiplications. We show, in particular, that new solutions can be obtained by multiplicative deformation of the canonical Wiener–Hopf factorization, provided the latter exists, and that one can superpose such solutions. Examples of applications include Kasner, Einstein–Rosen wave and gravitational pulse wave solutions.

The literature about the asymptotic numerical method (ANM) is reviewed in this paper as well as its application to hyperelasticity and elastoplasticity. ANM is a generic continuation method based on the computation of Taylor series for solving nonlinear partial differential equations. Modern techniques of high-order differentiation provide simple tools for computing these power series, the corresponding algorithms for finite strain elasticity and elastoplasticity being summarized here. Taylor series is not only a computation tool, but it contains also useful information about the structure of the considered solution curve. The paper ends with a short historical account about the development of this numerical method.

In biochemical networks, complex dynamical features such as superlinear growth and oscillations are classically considered a consequence of autocatalysis. For the large class of parameter-rich kinetic models, which includes generalized mass action kinetics and Michaelis–Menten kinetics, we show that certain submatrices of the stoichiometric matrix, so-called unstable cores, are sufficient for a reaction network to admit instability and potentially give rise to such complex dynamical behaviour. The determinant of the submatrix distinguishes unstable-positive feedbacks, with a single real-positive eigenvalue, and unstable-negative feedbacks without real-positive eigenvalues. Autocatalytic cores turn out to be exactly the unstable-positive feedbacks that are Metzler matrices. Thus there are sources of dynamical instability in chemical networks that are unrelated to autocatalysis. We use such intuition to design non-autocatalytic biochemical networks with superlinear growth and oscillations.