An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion

A class of coupled cell–bulk ODE–PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum-sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius \(\epsilon \ll 1\) are coupled through a passive bulk diffusion field. For this coupled system, the method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of the steady-state solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell–bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur, as obtained from our linear stability analysis, are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that solutions to the ODE–PDE cell–bulk system can be approximated by a finite-dimensional dynamical system. This limiting system is studied both analytically, using a linear stability analysis and, globally, using numerical bifurcation software. For one illustrative example of the theory, it is shown that when the number of cells exceeds some critical number, i.e., when a quorum is attained, the passive bulk diffusion field can trigger oscillations through a Hopf bifurcation that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a diffusion-sensing behavior is possible whereby more clustered spatial configurations of cells inside the domain lead to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell–bulk models is shown to be qualitatively rather similar to the linear stability analysis of localized spot patterns for activator–inhibitor reaction–diffusion systems in the limit of long-range inhibition and short-range activation.     

Quantum Effects on Optical Properties of a Pair of Plasmonic Particles Separated by a Subnanometer Gap

Computational Mathematics and Mathematical Physics - The discrete source method is used to study the influence exerted by nonlocal screening on the optical properties of a linear cluster of...     

Gap opening and split band edges in waveguides coupled by a periodic system of small windows

Using an example of two coupled waveguides, we construct a periodic second-order differential operator acting in a Euclidean domain and having spectral gaps whose edges are attained strictly inside the Brillouin zone. The waveguides are modeled by the Laplacian in two infinite strips of different width that have a common interior boundary. On this common boundary, we impose the Neumann boundary condition, but cut out a periodic system of small windows, while on the remaining exterior boundary we impose the Dirichlet boundary condition. It is shown that, by varying the widths of the strips and the distance between the windows, one can control the location of the extrema of the band functions as well as the number of the open gaps. We calculate the leading terms in the asymptotics for the gap lengths and the location of the extrema.     

The Contact of a Half-Space and an Uneven Base in the Presence of an Intercontact Gap Filled by an Ideal Gas

A plane contact problem of elasticity theory on the interaction of an elastic half-space and a rigid base is formulated with due regard for the presence of a gas in the gap. It is assumed that the gas is ideal and compressible and that its state can be described by the Clayperon–Mendeleyev equation. The mathematical model of the contact behavior of the system as it is loaded takes account of the transforming height and width of the intercontact gap, as well as of the changing gas pressure in the gap. The solution to the problem is presented in terms of a gap-height function, which can be obtained from a certain singular integral equation. The width of the gap is derived from an additional condition that ensures boundedness of the contact pressure. The problem is solved analytically for one form of the gap. The graphs presented illustrate the influence of the gas and its mass on the geometric parameters of the gap contact pressure.     

Exact Boundary Synchronization by Groups for a Kind of System of Wave Equations Coupled with Velocities*

This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities. These problems can not be solved directly by the usual HUM method for wave equations, however, by transforming the system into a first order hyperbolic system, the HUM method for 1-D first order hyperbolic systems, established by Li-Lu(2022) and Lu-Li(2022), can be applied to get the corresponding results.