Large Diffusion Rates Research Articles

Spatiotemporal dynamics in a fractional diffusive SIS epidemic model with mass action infection mechanism.

This paper is concerned with spatiotemporal dynamics of a fractional diffusive susceptible-infected-susceptible (SIS) epidemic model with mass action infection mechanism. Concretely, we first focus on the existence and stability of the disease-free and endemic equilibria. Then, we give the asymptotic profiles of the endemic equilibrium on small and large diffusion rates, which can reveal the impact of dispersal rates and fractional powers simultaneously. It is worth noting that we have some counter-intuitive findings: controlling the flow of infected individuals will not eradicate the disease, but restricting the movement of susceptible individuals will make the disease disappear.

Asymptotic profiles of a nonlocal dispersal SIS epidemic model with saturated incidence

Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (J. Differ. Equ. 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.

Asymptotic profiles of basic reproduction number for a degenerate reaction–diffusion host–pathogen model

In this paper, the basic reproduction number (BRN) is explored in terms of a degenerate reaction–diffusion host–pathogen model. The expressions of BRN and local basic reproduction number (LBRN) for the model are calculated. Some basic properties of with respect to the general diffusion rates and the asymptotic profiles of with respect to large diffusion rates and small diffusion rates are established by introducing two compact operators and . Our results reveal that is the spectral radius of a sum of the product of additive operators and strongly positive compact linear operators with spectral radii one.

A novel iron-doped mineral/carbon composite electrocatalyst synthesized by reconstituting ash phase of coal gasification fine slag for synergistically improving electrooxidation efficiency

Coal gasification fine slag is typical coal-based solid waste, and the crux to its resource utilization is the co-utilization of carbon and ash. A novel iron-doped mineral and carbon composite electrocatalyst (Fe-FSAPH) was synthesized by depolymerizing and reconstituting the ash phase for the collaborative electrocatalytic degradation of m-cresol wastewater. Exogenous iron participated in and promoted mineral reconstitution, increasing Si(-O)1/Si(-O)3 ratio (0.37→1.18) and promoting the formation of active Si-OH groups. The iron-doped minerals dispersedly developed with a sheet-like on carbon matrix, which improved intimate physical proximity and synergistic behavior of carbon/minerals. The obtained Fe-FSAPH, with a specific surface area of 246.52 m2/g, has a large diffusion rate constant KD2 (2.98 mg·min−0.5/g) in the rate-limiting step of m-cresol adsorption. Its electrochemical performances exhibit excellent, including cyclic voltammetry charge of 8.52 mC/cm3, charge-transfer resistance of 0.59 Ω, and Tafel slope of 159 mV/dec. The m-cresol with a concentration of 194.67 mg/L could be electrooxidized within 18 min due to the abundance of •OH and O2•- in the Fe-FSAPH system, which accelerated the radical attack reaction. Moreover, Si-OH groups catalyzed conversion of •OH to O2•- by generating oxygen vacancies and undergoing dehydroxylation, shortening the rapid reaction time of m-cresol removal to 14 min. The introduction of iron enhanced the synergy of carbon and minerals, with •OH generated via the Fenton-like reaction, offering a convenient environment for mineral catalysis. Hence, the study provided new ideas for the comprehensive utilization of carbon and ash and treatment of waste by waste.

Dynamics of a competition model with intra- and interspecific interference in the unstirred chemostat

This paper deals with an unstirred chemostat system with intra- and interspecific interference. We first investigate the uniqueness and asymptotic behaviors of the positive steady state solution of the single species model. The results show that the species cannot survive when the intraspecific interference is large enough. For two species system, we find that the two species can coexist when the influence of interspecific interference is relatively small, and that the system has no positive steady-state solution when the strength of interspecific interference is large enough. Furthermore, we study the influence of diffusion rate on species coexistence. The results show that for large diffusion rate, both species will be washed out, and the competition exclusion occurs at small diffusion rate. This indicates that species coexistence only occurs at intermediate diffusion rate.

Asymptotic behavior of the basic reproduction number in an age-structured SIS epidemic patch model.

In this paper we consider an age-structured SIS epidemic patch model. We define the principal eigenvalue [Formula: see text], basic reproduction number [Formula: see text] and analyze the effects of the diffusion rate d on [Formula: see text] and [Formula: see text]. More precisely, we investigate their asymptotic behavior for small and large diffusion rates. Finally we study the global behavior of this age-structured SIS epidemic patch model.

Threshold dynamics of a time-periodic nonlocal dispersal SIS epidemic model with Neumann boundary conditions

In this paper, we study a time-periodic nonlocal dispersal susceptible-infected-susceptible epidemic model with Neumann boundary conditions, where the total population number is constant. We first investigate limiting profile of the spectral bound for a time-periodic nonlocal dispersal operator, and then obtain asymptotic behavior of the basic reproduction ratio of the model as the dispersal rates go to zero and infinity, respectively. Next, we establish the existence, uniqueness and stability of steady states of the model in terms of the basic reproduction ratio. Finally, we discuss the impacts of small and large diffusion rates of the susceptible and infectious populations on the persistence and extinction of the disease.

Dynamics of a diffusion-advection Lotka-Volterra competition model with stage structure in a spatially heterogeneous environment

In this paper, we study a diffusion-advection Lotka-Volterra competition model with stage structure in a spatially heterogeneous environment. The existence and local asymptotic stability of spatially non-homogeneous semi-trivial steady-state solutions and spatially non-homogeneous positive steady-state solutions are obtained by the implicit function theorem and spectral analysis. We show that three scenarios can occur: if random diffusion rates of two species are sufficiently large, both species go extinct; if random diffusion rates of two species are relatively small, two competing species coexist; if one species has a large random diffusion rate and the other has a small random diffusion rate, the species with a large random diffusion rate are driven to extinction. An interesting finding is that a large delay does not lead to Hopf bifurcation at the spatially non-homogeneous steady-state solution, but makes this steady-state solution approach zero. We numerically demonstrate the effects of spatial heterogeneity on spatiotemporal dynamics.

Effects of diffusion and advection on predator-prey dynamics in closed environments

We investigate the dynamics and the asymptotic profiles of positive steady states of a specialist predator-prey model and a generalist predator-prey model in closed advective environments. The threshold dynamics are determined by the mortality rate of the specialist predators and the predation rate of the generalist predators, respectively. We demonstrate that, no matter how large advection and diffusion rates are, the specialist predators can successfully invade as long as the mortality rate is sufficiently small, while the generalist predators can take-over the prey if the predation rate is relatively large. Finally, we explore the impacts of diffusion and advection on the asymptotic profiles of positive steady states. Our results imply that large advection rates and small diffusion rates will lead to the concentration of the species at the downstream or the upstream. Moreover, if the diffusion rates are large, then the densities of the species will be less spatial heterogeneous.

Analysis of a diffusive host–pathogen epidemic model with two‐stage mechanism in a spatially heterogeneous environment

Due to the spatial heterogeneity present in many aspects of disease transmission, the rate of transmission of infectious diseases, human birth/mortality rate, and the mobility ability of infected humans should be different in different geographical locations. On the other hand, infected humans may exhibit distinct differences in symptoms during the different stages of the disease transmission. This paper aims to study threshold dynamics of a reaction–diffusion host–pathogen model governed by two‐stage mechanism and no flux boundary condition. In the model, we use bilinear and saturated incidence mechanism to model the interactions between host populations and pathogens and assume that (i) the parameters are spatially dependent and (ii) infected individuals in the first stage can random walk, while in the second stage, the diffusion rates are neglected. By carrying out strict analysis, the paper establishes the threshold‐type results with the basic reproduction number. Specifically, in a homogeneous case that all parameters are constants, we establish the global attractivity of the endemic equilibrium. To investigate the effects of small or large diffusion rates of susceptible individuals on the disease transmission in a heterogeneous environment, we explore the asymptotic profile of positive steady state whenever it exists. Our numerical results validate the theoretical results and indicate the importance of the infected humans in the second stage and pathogens in the environment in disease transmission, which greatly contribute to disease spread in a bounded domain and should not be ignored.

Basic reproduction ratio of a mosquito-borne disease in heterogeneous environment

To explore the influence of spatial heterogeneity on mosquito-borne diseases, we formulate a reaction-diffusion model with general incidence rates. The basic reproduction ratio [Formula: see text] for this model is introduced and the threshold dynamics in terms of [Formula: see text] are obtained. In the case where the model is spatially homogeneous, the global asymptotic stability of the endemic equilibrium is proved when [Formula: see text]. Under appropriate conditions, we establish the asymptotic profiles of [Formula: see text] in the case of small or large diffusion rates, and investigate the monotonicity of [Formula: see text] with respect to the heterogeneous diffusion coefficients. Numerically, the proposed model is applied to study the dengue fever transmission. Via performing simulations on the impacts of certain factors on [Formula: see text] and disease dynamics, we find some novel and interesting phenomena which can provide valuable information for the targeted implementation of disease control measures.

Asymptotic dynamics of a logistic SIS epidemic reaction-diffusion model with nonlinear incidence rate

A logistic SIS epidemic reaction-diffusion model with the nonlinear incidence rate SqIp in a spatially heterogeneous environment is concerned. The L∞-estimates of solutions are established under p,q>0 to explore the long-time behaviors of the model. In a special case, asymptotic dynamics and bifurcation behaviors under p=1 are analyzed in terms of the basic reproduction number. Asymptotic profiles of the endemic equilibrium for small and large diffusion rates of the susceptible and infected populations are investigated. It is found that the infectious disease is hard to control even under a policy of mobility regulation when 0<p<1 and q>1.

Dynamics and asymptotic profiles of a nonlocal dispersal SIS epidemic model with bilinear incidence and Neumann boundary conditions

This paper is concerned with a nonlocal (convolution) dispersal susceptible-infected-susceptible (SIS) epidemic model with bilinear incidence and Neumann boundary conditions. First we establish the existence and uniqueness of stationary solutions by reducing the system to a single equation. Then we study the asymptotic profiles of the endemic steady states for large and small diffusion rates to illustrate the persistence or extinction of the infectious disease. The lack of regularity of the endemic steady state makes it more difficult to obtain the limit function of the sequence of endemic steady states. We also observe the concentration phenomenon which occurs when the diffusion rate of the infected individuals tends to zero. Our analytical results demonstrate that limiting the movement of susceptible individuals is not effective in eliminating the infectious disease unless the total population size is relatively small.

The Effect of Diffusion on the Dynamics of a Predator-Prey Chemostat Model

This paper deals with a diffusive predator-prey chemostat system which describes the growth of planktonic rotifers, Brachionus calyciflorus, feeding on unicellular green algae, Chlorella vulgaris. The dynamical behavior of this system is established in terms of the diffusion rate. The results show that there exist two critical diffusion rates which classify the dynamical behavior of this system into the following three scenarios: (i) for a large diffusion rate, all species will be washed out; (ii) for an intermediate diffusion rate, the predator goes extinct and the prey survives; (iii) for a small diffusion rate, all species coexist. Finally, our numerical results show that the solution of this system may undergo a steady-state bifurcation or Hopf bifurcation for a suitably small diffusion rate, which supplements our theoretical results.

Pattern formation of a biomass–water reaction–diffusion model

In this paper, we are concerned with a biomass–water reaction–diffusion model subject to the homogeneous Neumann boundary condition. We derive several sufficient conditions on the existence and non-existence of non-constant stationary solutions with respect to large or small diffusion rate, which give the criteria for the possibility of Turing patterns in this system. Our results confirm the numerical findings of Manor and Shnerb, (2006) and also complement the theoretical results of Wang et al., (2017) for the corresponding ODE model.

Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates

Abstract This paper concerns with detailed analysis of a reaction-diffusion host-pathogen model with space-dependent parameters in a bounded domain. By considering the fact the mobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and the mobility of pathogen is ignored in the environment.We first establish the well-posedness of the model, including the global existence of solution and the existence of the global compact attractor. The basic reproduction number is identified, and also characterized by some equivalent principal spectral conditions, which establishes the threshold dynamical result for pathogen extinction and persistence. When the positive steady state is confirmed, we investigate the asymptotic profiles of positive steady state as host individuals disperse at small and large rates. Our result suggests that small and large diffusion rate of hosts have a great impacts in formulating the spatial distribution of the pathogen.

Global dynamics and travelling waves for a periodic and diffusive chemostat model with two nutrients and one microorganism

We consider a classical chemostat model with two nutrients and one microorganism, which incorporates spatial diffusion, temporal heterogeneity, and spatial heterogeneity. We study the basic reproduction number R0 and the asymptotic behaviours, which provide us some new findings in chemostat models. Global dynamics in terms of R0 is investigated in a bounded spatial domain. In the general situation where the growth rate and the loss rate of microorganisms depend on the spatiotemporal heterogeneity, we observe that microorganisms will be persistent if either the domain is of fast-growth type or there exists at least one fast-growth site and the diffusion of microorganisms is sufficiently slow. Asymptotic behaviours of the microorganism-existent steady state are discussed for the large diffusion rates, and the existence of periodic travelling wave is established by analysing a fixed point problem of a nonlinear operator in an unbounded spatial domain.

2D Si3N as a Promising Anode Material for Li/Na-Ion Batteries from First-Principles Study

The development of high-efficiency anode materials with large capacity, high stability and fast diffusion rates is a key requirement for rechargeable Li-ion and Na-ion batteries (LIBs/NIBs). In this work, the adsorption and diffusion of Li and Na atoms on two-dimensional (2D) Si3N materials is studied using first-principles calculations. The Si3N monolayers have large adsorption energies (2.74 eV for Li and 2.17 eV for Na) and a high theoretical capacity (1772.0 mAh/g for Li and 859.6 mAh/g for Na). Moreover, the low diffusion barriers (0.45 and 0.24 eV) for Li and Na atoms indicate that Si3N has an excellent high charge/discharge capability. In addition, molecular dynamics simulations showed that the structure of the 2D Si3N monolayer with adsorption of 32 Li/Na atoms has a very small change at 400 K due to the large adsorption energies for Li/Na. Owing to its good features, the Si3N monolayer is a highly promising anode material for energy storage devices.

Persistence for a Two-Stage Reaction-Diffusion System

In this article, we study how the rates of diffusion in a reaction-diffusion model for a stage structured population in a heterogeneous environment affect the model’s predictions of persistence or extinction for the population. In the case of a population without stage structure, faster diffusion is typically detrimental. In contrast to that, we find that, in a stage structured population, it can be either detrimental or helpful. If the regions where adults can reproduce are the same as those where juveniles can mature, typically slower diffusion will be favored, but if those regions are separated, then faster diffusion may be favored. Our analysis consists primarily of estimates of principal eigenvalues of the linearized system around ( 0 , 0 ) and results on their asymptotic behavior for large or small diffusion rates. The model we study is not in general a cooperative system, but if adults only compete with other adults and juveniles with other juveniles, then it is. In that case, the general theory of cooperative systems implies that, when the model predicts persistence, it has a unique positive equilibrium. We derive some results on the asymptotic behavior of the positive equilibrium for small diffusion and for large adult reproductive rates in that case.

Global dynamics of a Lotka-Volterra competition-diffusion system in advective homogeneous environments

In this paper, we mainly study a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates. One interesting feature of the model is that the boundary condition at the downstream end represents a net loss of individuals, which is tuned by a parameter b to measure the magnitude of the loss. When the upstream end has the no-flux condition, Lou and Zhou (2015) [11] have confirmed that large diffusion rate is more favorable when 0≤b<1. Here we consider the case where the upstream end has the free-flow condition, which means that the upstream end is linked to a lake. We firstly investigate the corresponding single species model. Here we establish the existence and uniqueness of positive steady states. Then for the two-species model, we find that the parameter b can be regarded as a bifurcation parameter. Precisely, when 0≤b<1, large diffusion rate is more favorable while when b>1, small diffusion rate is selected (if exists). When b=1, the system is degenerate in the sense that there is a compact global attractor consisting of a continuum of steady states.