Studies in Applied Mathematics最新文献

This paper presents the collision between two rarefaction waves for the one-dimensional blood flow model with non-constant body force. We establish the Riemann solutions using phase plane analysis, which are no longer self-similar. By employing Riemann invariants, we transform the system into a non-reducible diagonal system in Riemann invariant coordinates. This interaction problem then becomes a Goursat boundary value problem (GBVP) within the interaction region. We demonstrate that no vacuum forms within the interaction domain if the boundary data on characteristics is vacuum-free, and a vacuum only appears if the two rarefaction waves fail to penetrate each other within a finite time. Furthermore, we prove the existence and uniqueness of the $�C^1$ solution to the GBVP throughout the entire interaction region using $��\text{a priori}��C^1�$ bounds. Finally, we present the results of the interaction, showing that either the rarefaction waves completely penetrate each other or form a vacuum in the solution at a sufficiently large time during the process of penetration.

In this paper, we study the asymptotic behavior of the global solutions to a degenerate forest kinematic model, under the action of a perturbation modeling the impact of climate change. In the case where the main nonlinear term of the model is monotone, we prove that the global solutions converge to a stationary solution, by showing that the Lyapunov function derived from the system satisfies a Łojasiewicz–Simon gradient inequality. We also present an original algorithm, based on the Statistical Model Checking framework, to estimate the probability of convergence toward nonconstant equilibria. Furthermore, under suitable assumptions on the parameters, we prove the continuity of the flow and of the stationary solutions with respect to the perturbation parameter. Then, we succeed in proving the robustness of the weak attractors, by considering a weak topology phase space and establishing the existence of a family of positively invariant regions. At last, we present numerical simulations of the model and explore the behavior of the solutions under the effect of several types of perturbations. We also show that the forest kinematic model can lead to the emergence of chaotic patterns.

Microbial communities are complex ecological systems of organisms that evolve in time, with new variants created, while others disappear. Understanding how species interact within communities can help us shed light into the mechanisms that drive ecosystem processes. We studied systems with serial propagation, where the community is kept alive by taking a subsample at regular intervals and replating it in fresh medium. The data that are usually collected consist of the % of the population for each of the species, at several time points. In order to utilize this type of data, we formulated a system of equations (based on the generalized Lotka–Volterra model) and derived conditions of species noninteraction. This was possible to achieve by reformulating the problem as a problem of finding feasibility domains, which can be solved by a number of efficient algorithms. This methodology provides a cost-effective way to investigate interactions in microbial communities.

This paper is concerned with a two-species Lotka–Volterra competition patch model over a stream with better resources at downstream locations. Treating one species as the resident species and the other one as a mutant species, we first show that there exist two quantities $�\overline{d}$ and $�\underline{d}$ depending on the drift rate: if the dispersal rate of the resident species is smaller (respectively, larger) than $�\underline{d}$ (respectively, $�\overline{d}$), then a rare mutant species can invade only when its dispersal rate is faster (respectively, slower) than the resident species. Then, we show that there exists some intermediate dispersal rate, which is the unique evolutionarily stable strategy for the resident species under certain conditions. Moreover, the global dynamics of the model is obtained, and both competition exclusion and coexistence can occur. Our method for the patch model can be used for the corresponding reaction–diffusion model, and some existing results are improved.

Starting from a fairly explicit homogeneous realization of the toroidal Lie algebra $��L_{�r�+�1�}^{\mathrm{tor}}��\left(�{\mathrm{sl}}_{\ell }�\right)��$ via a lattice vertex algebra, we derive an integrable hierarchy of Hirota bilinear equations. Moreover, we represent this hierarchy in the form of Lax equations, and show that it is an extension of a certain reduction of the $�\ell$-component KP hierarchy.

We present the mathematical and numerical theory for evanescent waves in subwavelength bandgap materials. We begin in the one-dimensional case, whereby fully explicit formulas for the complex band structure, in terms of the capacitance matrix, are available. As an example, we show that the gap functions can be used to accurately predict the decay rate of the interface mode of a photonic analogue of the Su–Schrieffer–Heeger model. In two dimensions, we derive the bandgap Green's function and characterize the subwavelength gap functions via layer potential techniques. By generalizing existing lattice-summation techniques, we illustrate our results numerically by computing the complex band structure in a variety of settings.

A shortage of medical resources can arise when a multitude of patients rapidly emerge during the initial phases of an emerging infectious disease, due to limited availability of healthcare resources. Chronological age plays a pivotal role in both foreseeing and preventing infection patterns. In this investigation, we present an Susceptible-Infected-Recovered (SIR) model that integrates an age-structured and a saturated treatment function, and demonstrate its well-posedness. Our analysis reveals intricate patterns in the system, characterized by a steady-state bifurcation involving a backward bifurcation and a stable bifurcation representing a Hopf bifurcation. Notably, numerical simulations demonstrate that when , the system exemplifies a novel phenomenon wherein a disease-free equilibrium coexists harmoniously with an enduring Hopf bifurcation. We conduct a real application for model calibration and suggest that enhancing medical facilities and minimizing treatment delays may prove to be of paramount importance in curtailing the spread of the disease.