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Interactions of Anisotropic Inclusions on a Fluid Membrane
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | 2448-2471 |
| Abstrak | : | Biological cells and membranes need to be properly shaped to fulfill many fundamental functions. This shaping is often aided by the aggregation of membrane-bound proteins that both sense the membrane curvature and shape it. Therefore, these protein inclusions interact with each other through the deformation of the membrane that they influence, and a key question is to understand the law that governs their interaction. Whereas the theoretical case of isotropic proteins is well understood, an important feature of many such proteins is their anisotropic interaction with the membrane. Here, we derive an interaction law for rigid circular membrane inclusions which impose an anisotropic contact angle on the surrounding membrane. We include the effects of both membrane bending and tension. Using asymptotic analysis, we identify two distinguished limits corresponding to weak anisotropy/weak tension and strong anisotropy/strong tension, respectively. The resulting laws exhibit a bistability in the equilibrium separation of inclusions. Inclusions with very weak anisotropy equilibrate with large separation, those with very strong anisotropy equilibrate with small separation, while there is a range of anisotropies for which both equilibria are stable. Our results provide a theoretical mechanism for the global aggregation of inclusions seen both in experiments and simulations. |
Adaptation in General Temporally Changing Environments
| Pengarang | : | Lionel Roques |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | 2420-2447 |
| Abstrak | : | We analyze a nonlocal partial differential equation (PDE) model describing the dynamics of adaptation of a phenotypically structured population, under the effects of mutation and selection, in a changing environment. Previous studies have analyzed the large-time behavior of such models, with particular forms of environmental changes---either linearly changing or periodically fluctuating. We use here a completely different mathematical approach, which allows us to consider very general forms of environmental variations and to give an analytic description of the full trajectories of adaptation, including the transient phase, before a stationary behavior is reached. The main idea behind our approach is to study a bivariate distribution of two “fitness components" that contains enough information to describe the distribution of fitness at any time. This distribution solves a degenerate parabolic equation that is dealt with by defining a multidimensional cumulant generating function associated with the distribution and solving the associated transport equation. We apply our results to several examples and check their accuracy using stochastic individual-based simulations as a benchmark. These examples illustrate the importance of being able to describe the transient dynamics of adaptation to understand the development of drug resistance in pathogens. |
A Nonlinear Hyperbolic Model for Radiative Transfer Equation in Slab Geometry
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 80 (No. 6) |
| Halaman | : | 2388-2419 |
| Abstrak | : | Linear models for the radiative transfer equation have been well developed, while nonlinear models are seldom investigated even for slab geometry due to some essential difficulties. We have proposed a moment model in [Y. Fan, R. Li, and L. Zheng, J. Comput. Phys., 404 (2020), 109128] for slab geometry, which combines the ideas of the classical ???????? and ???????? model. Though the model is far from perfect, it was demonstrated to be quite efficient in numerically approximating the solution of the radiative transfer equation, and we are motivated to improve this model further. Consequently, we propose in this paper a new model following the chartmap in [Y. Fan, R. Li, and L. Zheng, J. Comput. Phys., 404 (2020), 109128] with some significant theoretic progress. The new model is derived with global hyperbolicity, and meanwhile some necessary physical properties are preserved. We give a complete analysis of the characteristic structure and propose a numerical scheme for the new model. Numerical examples are presented to demonstrate the numerical performance of the new model. |
Asymptotic Convergence in Delay Differential Equations Arising in Epidemiology and Physiology
| Pengarang | : | Hassan A. El-Morshedy |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1781-1798 |
| Abstrak | : | In this paper we investigate the presence of oscillations and simple dynamics in ????′?(????) =−??????????(????) +?????(?????(????))??????(?????(???? −????)). Specifically, we prove that the global attraction towards a nontrivial equilibrium is reduced to the nonexistence of solutions of a certain system of inequalities. Our results cover situations in which the global attraction critically depends on the delay. From an applied point of view, this equation naturally appears in many biological situations, e.g., epidemic models with awareness and the production of platelets. In epidemiology, it is becoming clear that the different types of population awareness and the time delays of individuals' responses to available information about the disease play a critical role in its spread. Using our theoretical results, we describe qualitative properties of the behavioral responses that prevent the presence of sustainable oscillations in the number of infected individuals. Regarding physiological models, we discuss the influence of some biological parameters in certain anomalies in the production of platelets. |
Drift-Diffusion Transport in a Randomly Inhomogeneous One-Dimensional Medium
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1767-1780 |
| Abstrak | : | Organic semiconductors are intrinsic randomly inhomogeneous materials where charge transport occurs by hopping of the carriers between localized sites having a distribution of energy levels. However, the average carrier density seems to be accurately described by a simple drift-diffusion equation. We investigate the reasons for the effectiveness of the drift-diffusion model in a random material and show that the key assumption for its validity is that the correlation lengths of the randomly perturbed coefficients are much smaller than any other characteristic length of the problem. As a byproduct, we find how the effective drift and diffusion coefficients depend on the randomness. |
Distributed Delay Differential Equation Representations of Cyclic Differential Equations
| Pengarang | : | Tyler Cassidy |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1742-1766 |
| Abstrak | : | Compartmental ordinary differential equation (ODE) models are used extensively in mathematical biology. When transit between compartments occurs at a constant rate, the well-known linear chain trick can be used to show that the ODE model is equivalent to an Erlang distributed delay differential equation (DDE). Here, we demonstrate that compartmental models with nonlinear transit rates and possibly delayed arguments are also equivalent to a scalar distributed DDE. To illustrate the utility of these equivalences, we calculate the equilibria of the scalar DDE, and compute the characteristic function---without calculating a determinant. We derive the equivalent scalar DDE for two examples of models in mathematical biology and use the DDE formulation to identify physiological processes that were otherwise hidden by the compartmental structure of the ODE model. |
The Nonuniversality of Wealth Distribution Tails Near Wealth Condensation Criticality
| Pengarang | : | Sam L. Polk |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1717-1741 |
| Abstrak | : | In this work, we modify the affine wealth model of wealth distributions to examine the effects of nonconstant redistribution on the very wealthy. Previous studies of this model, restricted to flat redistribution schemes, have demonstrated the presence of a phase transition to a partially wealth-condensed state, or “partial oligarchy,” at the critical value of an order parameter. These studies have also indicated the presence of an exponential tail in wealth distribution precisely at criticality. Away from criticality, the tail was observed to be Gaussian. In this work, we generalize the flat redistribution within the affine wealth model to allow for an essentially arbitrary redistribution policy. We show that the exponential tail observed near criticality in prior work is, in fact, a special case of a much broader class of critical, slower-than-Gaussian decays that depend sensitively on the corresponding asymptotic behavior of the progressive redistribution model used. We thereby demonstrate that the functional form of the tail of the wealth distribution in a near-critical society is not universal in nature but rather entirely determined by the specifics of public policy decisions. This is significant because most major economies today are observed to be near-critical. |
Locating the Baking Isotherm in a Søderberg Electrode: Analysis of a Moving Thermistor Model
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1691-1716 |
| Abstrak | : | Søderberg electrodes feature prominently in the operation of metallurgical electrical furnaces. The electrode material must bake before entering the furnace; failure to bake will lower the efficiency of the process and may cause physical harm to the furnace itself through a soft breakage. As such, ensuring that the baking isotherm remains within the region of the electrode outside of the furnace is essential. We propose a mathematical model for a Søderberg electrode taking into account the heat, mass, and current transfer mechanisms at play, along with realistic boundary conditions on the outside of the electrode that are strongly heterogeneous in height. The resulting model describes a strongly heterogeneous cylindrical “thermistor" which moves slowly downward and is acted on by current clamps which provide Joule heating. Although it is often ignored in the literature on thermistor problems, we find that the Pe?clet number resulting from the downward motion strongly influences the position of the baking isotherm. Aside from some specific reductions leading to analytical solutions, the general form of the model is complicated enough to require numerical simulations. Still, our modeling approach provides us with a qualitative understanding of many aspects of the Søderberg electrode baking process and permits us to identify three parameters of key importance to the positioning of the baking isotherm. In particular, our results suggest desired ranges for the lowering rate of the electrode (in terms of a Pe?clet number), the radius of the electrode, and the strength of the Joule heating due to an applied current, which are the three aspects which may be controlled (to varying degrees) in industrial applications. |
Complex Resonant Ice Shelf Vibrations
| Pengarang | : | Robert H. Nazarian |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1483-1502 |
| Abstrak | : | The problem of ice shelf vibrations forced by incident waves from the open ocean is considered, using a linear hydroelastic model involving combined thin-plate and potential-flow theories. Complex resonances are shown to generate near-resonant ice shelf responses. An efficient algorithm is developed to capture the complex resonances, based on a homotopy, and initialized with the real-valued eigenfrequencies of a problem in which the ice shelf and subshelf water cavity are uncoupled from the open ocean. It is shown the complex resonances may be used to approximate the reflection coefficient in the frequency domain and the large-time ice shelf response to an incident wave packet. Shelf thickening is shown to have a major effect on complex resonances at midrange frequencies, resulting in difficulty exciting certain near resonances in transient problems. |
Erratum: Inverse Problems for Abstract Evolution Equations II: Higher Order Differentiability for Viscoelasticity
| Pengarang | : | Andreas Kirsch |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 1) |
| Halaman | : | 282-283 |
| Abstrak | : | With this erratum we correct some errors in [A. Kirsch and A. Rieder, Inverse problems for abstract evolution equations II: Higher order differentiability for viscoelasticity, SIAM J. Appl. Math. 79 (2019), pp. 2639--2662]. |