Nonlocal delay gives rise to vegetation patterns in a vegetation-sand model

The vegetation pattern generated by aeolian sand movements is a typical type of vegetation patterns in arid and semi-arid areas. This paper presents a vegetation-sand model with nonlocal interaction characterized by an integral term with a kernel function. The instability of the Turing pattern was analyzed and the conditions of stable pattern occurrence were obtained. At the same time, the multiple scales method was applied to obtain the amplitude equations at the critical value of Turing bifurcation. The spatial distributions of vegetation under different delays were obtained by numerical simulation. The results revealed that the vegetation biomass increased as the interaction intensity decreased or as the nonlocal interaction distance increased. We demonstrated that the nonlocal interaction between vegetation and sand is a crucial mechanism for forming vegetation patterns, which provides a theoretical basis for preserving and restoring vegetation

Coexistence and stability of an unstirred chemostat model with Beddington–DeAngelis function

AbstractThis paper deals with an unstirred chemostat model with the Beddington–DeAngelis functional response. First, a sufficient condition to the existence of positive steady state solutions is established. Second, the effect of the parameter β1 in the Beddington–DeAngelis functional response which models mutual interference between species u is considered. The result shows that if β1 is sufficiently large, the solution of this model is determined by a limiting equation. The main tool used here includes the fixed point index theory, the perturbation technique and the bifurcation theory

Qualitative Analysis of a Three-Species Reaction-Diffusion Model with Modified Leslie-Gower Scheme

The qualitative analysis of a three-species reaction-diffusion model with a modified Leslie-Gower scheme under the Neumann boundary condition is obtained. The existence and the stability of the constant solutions for the ODE system and PDE system are discussed, respectively. And then, the priori estimates of positive steady states are given by the maximum principle and Harnack inequality. Moreover, the nonexistence of nonconstant positive steady states is derived by using Poincaré inequality. Finally, the existence of nonconstant positive steady states is established based on the Leray-Schauder degree theory

Calculation of Mechanical Properties, Electronic Structure and Optical Properties of CsPbX₃ (X = F, Cl, Br, I)

We utilized a first-principle density functional theory for a comprehensive analysis of CsPbX3 (X = F, Cl, Br, I) to explore its physical and chemical properties, including its mechanical behavior, electronic structure and optical properties. Calculations show that all four materials have good stability, modulus of elasticity, hardness and wear resistance. Additionally, CsPbX3 demonstrates a vertical electron leap and serves as a semiconductor material with direct band gaps of 3.600 eV, 3.111 eV, 2.538 eV and 2.085 eV. In examining its optical properties, we observed that the real and imaginary components of the dielectric function exhibit peaks within the low-energy range. Furthermore, the dielectric function gradually decreases as the photon energy increases. The absorption spectrum reveals that the CsPbX3 material exhibits the highest UV light absorption, and as X changes (with the increase in atomic radius within the halogen group of elements), the light absorption undergoes a red shift, becoming stronger and enhancing light utilization. These properties underscore the material’s potential for application in microelectronic and optoelectronic device production. Moreover, they provide a theoretical reference for future investigations into CsPbX3 materials