Concentration profile of endemic equilibrium of a reaction–diffusion–advection SIS epidemic model

Cui and Lou (J Differ Equ 261:3305–3343, 2016) proposed a reaction–diffusion–advection SIS epidemic model in heterogeneous environments, and derived interesting results on the stability of the DFE (disease-free equilibrium) and the existence of EE (endemic equilibrium) under various conditions. In this paper, we are interested in the asymptotic profile of the EE (when it exists) in the three cases: (i) large advection; (ii) small diffusion of the susceptible population; (iii) small diffusion of the infected population. We prove that in case (i), the density of both the susceptible and infected populations concentrates only at the downstream behaving like a delta function; in case (ii), the density of the susceptible concentrates only at the downstream behaving like a delta function and the density of the infected vanishes on the entire habitat, and in case (iii), the density of the susceptible is positive while the density of the infected vanishes on the entire habitat. Our results show that in case (ii) and case (iii), the asymptotic profile is essentially different from that in the situation where no advection is present. As a consequence, we can conclude that the impact of advection on the spatial distribution of population densities is significant.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.     

The spatial behavior of a competition–diffusion–advection system with strong competition

We study a competition–diffusion–advection system for two competitive species inhabiting a spatially heterogeneous environment. We show that they spatially segregate as the interspecific competition rate tends to infinity. Besides, by using a blow up method, we obtain the uniform Hölder bounds for solutions of the system.     

Spatio-temporal dynamics of a reaction–diffusion–advection food-limited system with nonlocal delayed competition and Dirichlet boundary condition

Spatio-temporal dynamics of a reaction–diffusion–advection food-limited population model with nonlocal delayed competition and Dirichlet boundary condition are considered. Existence and stability of the positive spatially nonhomogeneous steady state solution are shown. Existence and direction of the spatially nonhomogeneous steady-state-Hopf bifurcation are proved. Stable spatio-temporal patterns near the steady-state-Hopf bifurcation point are numerically obtained. We also investigate the joint influences of some important parameters including advection rate, food-limited parameter and nonlocal delayed competition on the dynamics. It is found that the effect of advection on Hopf bifurcation is opposite with the corresponding no-flux system. The theoretical results provide some interesting highlights in ecological protection in streams or rivers.     

Existence and stability of periodic contrast structures in the reaction–advection–diffusion problem in the case of a balanced nonlinearity

We study a singularly perturbed periodic problem for the parabolic reaction–advection–diffusion equation with small advection. We consider the case in which there exists an internal transition layer under the conditions of balanced nonlinearity. An asymptotic expansion of the solution is constructed. To substantiate this asymptotics, we use the asymptotic method of differential inequalities. The Lyapunov asymptotic stability of the periodic solution is analyzed.     

Solution with an inner transition layer of a two-dimensional boundary value reaction–diffusion–advection problem with discontinuous reaction and advection terms

Theoretical and Mathematical Physics - We study the problem of the existence and asymptotic stability of a stationary solution of an initial boundary value problem for the...     

On a reaction–advection–diffusion equation with Robin and free boundary conditions

ABSTRACT

A reaction–advection–diffusion equation with variable intrinsic growth rate, Robin and free boundary conditions is investigated in this paper. Firstly, we present a spreading–vanishing dichotomy for the asymptotic behavior of the solutions of the equation. Then, we obtain criteria for spreading and vanishing, and get an estimate for the asymptotic spreading speed of the spreading front. Moreover, numerical simulation is also given to illustrate the impact of the expansion capacity on the free boundary.     

Bifurcation for a reaction–diffusion system with unilateral and Neumann boundary conditions

We consider a reaction–diffusion system of activator–inhibitor or substrate-depletion type which is subject to diffusion-driven instability if supplemented by pure Neumann boundary conditions. We show by a degree-theoretic approach that an obstacle (e.g. a unilateral membrane) modeled in terms of inequalities, introduces new bifurcation of spatial patterns in a parameter domain where the trivial solution of the problem without the obstacle is stable. Moreover, this parameter domain is rather different from the known case when also Dirichlet conditions are assumed. In particular, bifurcation arises for fast diffusion of activator and slow diffusion of inhibitor which is the difference from all situations which we know.     

The global existence and asymptotic stability of solutions for a reaction–diffusion system

This paper studies the solutions of a reaction–diffusion system with nonlinearities that generalize the Lengyel–Epstein and FitzHugh–Nagumo nonlinearities. Sufficient conditions are derived for the global asymptotic stability of the solutions. Furthermore, we present some numerical examples.     

Global asymptotic stability of a Lotka–Volterra competition model with stochasticity in inter-specific competition

In the paper we first propose a two-species Lotka–Volterra competition model with the stochastic terms related to the inter-specific competition rates and the coexistence equilibrium of the deterministic model. Then we establish the global asymptotic stability of the coexistence equilibrium. Finally, we provide some discussions and numerical examples to illustrate our mathematical results.     

Boundedness and global stability of the predator–prey model with prey-taxis and competition

In this paper, we consider a reaction–diffusion system of the population dynamics of two predators and one prey with prey-taxis and competition. We prove the global existence and uniform boundedness of the positive classical solutions for the fully parabolic system over a bounded domain with Neumann boundary conditions. Furthermore, we establish the asymptotic behavior of solutions by constructing some appropriate Lyapunov functionals. Our results not only generalize the previously known one, but also present some new conclusions.     

The stability of a diffusion model of plankton allelopathy with spatio–temporal delays

A reaction–diffusion system with non-local delay is proposed to describe two competitive planktonic growths in aquatic ecology. The local and global stability of the axial equilibria as well as the positive equilibrium are discussed. Our results show that the delay has no effect on the stability of the axial equilibria; on the other hand, the positive equilibrium can be induced to be locally unstable by the delay. Finally, the corresponding numerical simulations are also demonstrated.     

Global asymptotic stability of Lotka–Volterra competition systems with diffusion and time delays

In the Lotka–Volterra competition system with N-competing species if the effect of dispersion and time-delays are both taken into consideration, then the densities of the competing species are governed by a coupled system of reaction–diffusion equations with time-delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to the competing rate constants to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the competing system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays, and the conclusions for the reaction–diffusion system are directly applicable to the corresponding ordinary differential system.     

Bifurcation for a reaction–diffusion system with unilateral obstacles with pointwise and integral conditions

A reaction–diffusion system of activator–inhibitor or substrate-depletion type is considered which is subject to diffusion driven instability. It is shown that obstacles (e.g. a unilateral membrane) for one or both quantities introduce a new bifurcation of spatially non-homogeneous steady states in a parameter domain where the trivial branch is exponentially stable without obstacles. The obstacles are modeled in terms of inclusions. Moreover, simultaneously some of the obstacles can be modeled also using nonlocal integral conditions.     

A note on a nonlocal nonlinear reaction–diffusion model

We give an application of the Crandall–Rabinowitz theorem on local bifurcation to a system of nonlinear parabolic equations with nonlocal reaction and cross-diffusion terms as well as nonlocal initial conditions. The system arises as steady-state equations of two interacting age-structured populations.     

Hopf bifurcation in a reaction–diffusion model with Degn–Harrison reaction scheme

In this paper, we consider a chemical reaction–diffusion model with Degn–Harrison reaction scheme under homogeneous Neumann boundary conditions. The existence of Hopf bifurcation to ordinary differential equation (ODE) and partial differential equation (PDE) models are derived, respectively. Furthermore, by using the center manifold theory and the normal form method, we establish the bifurcation direction and stability of periodic solutions. Finally, some numerical simulations are shown to support the analytical results, and to reveal new phenomenon on the Hopf bifurcation.