Unsteady state mathematical model of chemically reactive pollutants from an instantaneous line source into a stable atmospheric boundary layer

A time dependent atmospheric model represented for chemically reactive primary pollutants emitted from an elevated line source into a stable atmospheric boundary layer over a surface terrain. The model obtained from an analytical solution of the atmospheric diffusion equation with the quadratic diffusion coefficient (exchange coefficient) and the variable wind velocity taken to be of three different types’ viz. constant, constant shear and parabolic functions of vertical height. The pollutants considered to be of chemically reactive primary pollutants emitted from a time-dependent line source of Instantaneous type. In order to facilitate the application of the model the results for the general situation that includes chemical reaction rate & time dependent source incorporated in the model.     

On a Lotka-Volterra Competition Diffusion Model with Advection

In this paper, the author focuses on the joint effects of diffusion and advection on the dynamics of a classical two species Lotka-Volterra competition-diffusion-advection system, where the ratio of diffusion and advection rates are supposed to be a positive constant. For comparison purposes, the two species are assumed to have identical competition abilities throughout this paper. The results explore the condition on the diffusion and advection rates for the stability of former species. Meanwhile, an asymptotic behavior of the stable coexistence steady states is obtained.     

Buoyancy and chemical reaction effects on MHD mixed convection heat and mass transfer in a porous medium with thermal radiation and Ohmic heating

The combined effect of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conducting fluid past a vertical permeable surface embedded in a porous medium is analyzed. The heat equation includes the terms involving the radiative heat flux, Ohmic dissipation, viscous dissipation and the internal absorption whereas the mass transfer equation includes the effects of chemically reactive species of first-order. The non-linear coupled differential equations are solved analytically by perturbation technique. The results obtained show that the velocity, temperature and concentration fields are appreciably influenced by the presence of chemical reaction, thermal stratification and magnetic field. It is observed that the effect of thermal radiation and magnetic field is to decrease the velocity, temperature and concentration profiles in the boundary layer. There is also considerable effect of magnetic field and chemical reaction on skin-friction coefficient and Nusselt number.     

A fast precipitation and dissolution reaction for a reaction–diffusion system arising in a porous medium

This paper is devoted to the study of a fast reaction–diffusion system arising in reactive transport. It extends the articles [R. Eymard, T. Gallouët, R. Herbin, D. Hilhorst, M. Mainguy, Instantaneous and noninstantaneous dissolution: Approximation by the finite volume method, ESAIM Proc. (1998); J. Pousin, Infinitely fast kinetics for dissolution and diffusion in open reactive systems, Nonlinear Anal. 39 (2000) 261–279] since a precipitation and dissolution reaction is considered so that the reaction term is not sign-definite and is moreover discontinuous. Energy type methods allow us to prove uniform estimates and then to study the limiting behavior of the solution as the kinetic rate tends to infinity in the special situation of one aqueous species and one solid species.     

Competitive exclusion in a nonlocal reaction–diffusion–advection model of phytoplankton populations

We continue our study on the global dynamics of a nonlocal reaction–diffusion–advection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where both populations depend solely on light for their metabolism. In our previous work, we proved that system (1.1) is a strongly monotone dynamical system with respect to a non-standard cone related to the cumulative distribution functions, and further determined the global dynamics when the species have either identical diffusion rate or identical advection rate. In this paper, we study the trade-off of diffusion and advection and their joint influence on the outcome of competition. Two critical curves for the local stability of two semi-trivial equilibria are analyzed, and some new competitive exclusion results are obtained. Our main tools, besides the theory of monotone dynamical system, include some new monotonicity results for the principal eigenvalues of elliptic operators in one-dimensional domains.     

Modeling the reactive processes within a catalytic porous medium

A one-dimensional modelling approach to the reactive processes within a heated homogeneously premixed fuel–air mixture in its passage through a non-adiabatic catalytically reactive porous medium is described. The main focus of this contribution was comparison of the results obtained while using different modeling approaches that include mass diffusion to solid pores versus neglecting it; single step reaction versus detailed kinetic simulation; adiabatic versus non-adiabatic reactor operation; two different approaches accounting for radiation heat transfer. This model was tailored to our experimental results so as to obtain original kinetic data for corresponding global reactions for different types of catalysts and validate at the same time the predictive approaches.     

Optimal harvesting for a population dynamics problem with age-structure and diffusion

In this work, optimal harvesting policy for the predator-prey system of three species with age-dependent and diffusion is discussed. Existence and uniqueness of non-negative solution to the system are investigated by using the fixed point theorem. The existence of optimal control strategy is discussed and optimality conditions are obtained. Our results extend some known criteria.     

一个特殊的三维捕食链非自治扩散系统的持续生存

对一个特殊的三种群非自治捕食链扩散系统进行讨论.其特殊之处在于扩散现象发生在构成捕食链的中间种群间.通过利用比较原理进行微分不等式的比较,得到了该系统持续生存的条件;分析了扩散运动对该系统种群动力学行为的影响.     

Bifurcation analysis of a diffusive predator–prey system with a herd behavior and quadratic mortality

In this paper, a diffusive predator–prey system, in which the prey species exhibits herd behavior and the predator species with quadratic mortality, has been studied. The stability of positive constant equilibrium, Hopf bifurcations, and diffusion‐driven Turing instability are investigated under the Neumann boundary condition. The explicit condition for the occurrence of the diffusion‐driven Turing instability is derived, which is determined by the relationship of the diffusion rates of two species. The formulas determining the direction and the stability of Hopf bifurcations depending on the parameters of the system are derived. Finally, numerical simulations are carried out to verify and extend the theoretical results and show the existence of spatially homogeneous periodic solutions and nonconstant steady states. Copyright © 2014 John Wiley & Sons, Ltd.     

Application of the finite-element method to the diffusion and reaction of chemical species in multilayered polymeric bodies

Application of the finite-element method (FEM) to chemical species diffusion and reaction in polymers by Fickian mass transport is described. The method is developed by analogy to heat conduction and is extended to include multiple, reactive chemical species dissolved in multilayered polymeric materials. Because of the analogy to conductive heat transfer, existing FEM thermal codes can be readily adapted to solve chemical diffusion problems. The method described is limited to Fickian diffusion at a constant material temperature.     

Balance laws for liquid crystal mixtures

Equations of balance of mass, linear momentum, angular momentum and energy are obtained for each of the species in a chemically reacting mixture of liquid crystals. The forms of these equations for the mixture as a whole are then arrived at.     

Global attractor of a coupled finite difference reaction diffusion system with delays

In the study of asymptotic behavior of solutions for reaction diffusion systems, an important concern is to determine whether and when the system has a global attractor which attracts all positive time-dependent solutions. The aim of this paper is to investigate the global attraction problem for a finite difference system which is a discrete approximation of a coupled system of two reaction diffusion equations with time delays. Sufficient conditions are obtained to ensure the existence and global attraction of a positive solution of the corresponding steady-state system. Applications are given to three types of Lotka-Volterra reaction diffusion models, where time-delays may appear in the opposing species.     

Spatial Dynamics of a Lattice Lotka-Volterra Competition Model with a Shifting Habitat

In this paper, we concern with the spatial dynamics of the lattice Lotka-Volterra competition system in a shifting habitat. We study the impact of the environmental deterioration rate on the population density under the strong competition condition. Our results show that if the environment deteriorates rapidly, both species will become extinct. However, when the environmental degradation rate is not so fast, the species with slow diffusion will go extinct, while those with fast diffusion will survive. The extinction of species with slow diffusion can be divided into two situations: one is the extinction caused by environmental deterioration faster than its own diffusion speed, the other is the extinction caused by slow diffusion speed under the influence of strong competition.     

The Global Existence of Solutions for a Cross-diffusion System

This paper is concerned with the global existence of solutions for a class of quasilinear cross-diffusion system describing two species competition under self and cross population pressure. By establishing and using more detailed interpolation results between several different Banach spaces, the global existence of solutions are proved when the self and cross diffusion rates for the first species are positive and there is no self or cross-diffusion for the second species.     

Effect of chemical reaction on the dispersion of a solute in a porous medium

A mathematical model is presented in this paper which describes the dispersion of a chemically active solute in the laminar flow in a sparsely packed porous medium. The validity of time-dependent dispersion coefficient is widened by using a generalized dispersion coefficient. The effect of porous parameter and chemical reaction on the dispersion coefficient is studied. The exact solution for the mean concentration distribution of a chemically active solute is obtained as a function of downwind distance and time. Results are also obtained for pure convection.     

On a nonlinear nonautonomous predator–prey model with diffusion and distributed delay

In this paper, a nonlinear nonautonomous predator–prey model with diffusion and continuous distributed delay is studied, where all the parameters are time-dependent. The system, which is composed of two patches, has two species: the prey can diffuse between two patches, but the predator is confined to one patch. We first discuss the uniform persistence and global asymptotic stability of the model; after that, by constructing a suitable Lyapunov functional, some sufficient conditions for the existence of a unique almost periodic solution of the system are obtained. An example shows the feasibility of our main results.     

基于比率的三种群混合扩散模型的动力学行为

讨论了一类基于比率的非自治三种群混合扩散模型,三种群间既有捕食关系又有竞争关系.我们研究了该模型的动力学行为:包括一致持久性,全局渐近稳定性,周期解,概周期解的存在唯一性.表明即使食饵种群在某些孤立的斑块中可能绝灭,也可以通过适当选取扩散率来保证系统持续生存.     

Analysis of a Prey-predator Model with Disease in Prey

In this paper, a system of reaction-diffusion equations arising in ecoepidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. Local stability of the constant positive solution is considered. A number of existence and non-existence results about the nonconstant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the prey with disease is treated as a bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steadystate solution under some conditions.     

Effect of prey-taxis on predator’s invasion in a spatially heterogeneous environment

In this study, we consider a diffusive predator–prey system with prey-taxis and ratio-dependent functional responses in a spatially heterogeneous environment. Prey-taxis implies that the predator exhibits directed movement in the presence of prey. We claim that when the predator diffuses uniformly without prey-taxis, only the diffusion of the species plays a role in the predator’s invasion in a spatially homogeneous environment. However, when the predator disperses through uniform diffusion together with prey-taxis, we observe that both the diffusion of species and prey-taxis affect the invasion of the predator, which is not the case in a spatially homogeneous environment. The results are obtained by investigating the local stability of a semi-trivial solution, using an eigenvalue analysis.     

A fully adaptive numerical approximation for a two-dimensional epidemic model with nonlinear cross-diffusion

An epidemic model is formulated by a reaction–diffusion system where the spatial pattern formation is driven by cross-diffusion. The reaction terms describe the local dynamics of susceptible and infected species, whereas the diffusion terms account for the spatial distribution dynamics. For both self-diffusion and cross-diffusion, nonlinear constitutive assumptions are suggested. To simulate the pattern formation two finite volume formulations are proposed, which employ a conservative and a non-conservative discretization, respectively. An efficient simulation is obtained by a fully adaptive multiresolution strategy. Numerical examples illustrate the impact of the cross-diffusion on the pattern formation.