Traveling waves for a reaction–diffusion–advection system with interior or boundary losses

This Note deals with the existence and qualitative properties of traveling wave solutions of a nonlinear reaction–diffusion system with losses inside the domain. In particular, we show the existence of a continuum of admissible speeds of traveling waves. Lastly, by considering losses concentrated near the boundary of the domain, these results are compared with those already known in the case of losses on the boundary.     

Traveling waves for a reaction–diffusion–advection predator–prey model

In this paper we study a reaction–diffusion–advection predator–prey model in a river. The existence of predator-invasion traveling wave solutions and prey-spread traveling wave solutions in the upstream and downstream directions is established and the corresponding minimal wave speeds are obtained. While some crucial improvements in theoretical methods have been established, the proofs of the existence and nonexistence of such traveling waves are based on Schauder’s fixed-point theorem, LaSalle’s invariance principle and Laplace transform. Based on theoretical results, we investigate the effect of the hydrological and biological factors on minimal wave speeds and hence on the spread of the prey and the invasion of the predator in the river. The linear determinacy of the predator–prey Lotka–Volterra system is compared with nonlinear determinacy of the competitive Lotka–Volterra system to investigate the mechanics of linear and nonlinear determinacy.     

Blow-up for a reaction–diffusion equation with variable coefficient

We study the blow-up behavior for positive solutions of a reaction–diffusion equation with nonnegative variable coefficient. When there is no stationary solution, we show that the solution blows up in finite time. Under certain conditions, we then show that any point with zero source cannot be a blow-up point.     

Traveling fronts for a delayed reaction–diffusion system with a quiescent stage

In this paper, we study the traveling wave fronts of a delayed reaction–diffusion system with a quiescent stage for a single species population with two separate mobile and stationary states. By transforming the corresponding wave system into a scalar delayed differential equation with an integral term, we establish the existence of the minimal wave speed c_min, and the asymptotic behavior, monotonicity and uniqueness (up to a translation) of the traveling wave fronts. In particular, the effects of the delay and transfer rates on the minimal wave speed are studied.     

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.     

Speed of wave propagation for a nonlocal reaction–diffusion equation

ABSTRACT

A nonlocal reaction–diffusion equation arising in various applications is studied. The speed of traveling waves is determined by means of a minimax representation. It is used to obtain the wave speed estimates and asymptotic values.     

Traveling wave solutions for an autocatalytic reaction–diffusion model

In this paper we consider an autocatalytic reaction–diffusion model which has many applications. We extend previous results using qualitative analysis and show the existence of an exponentially decaying traveling wave front for a minimum speed and algebraically decaying wave fronts for large speeds. Further, the wave front profiles are calculated and the minimum speed is accurately determined using different numerical methods.     

Blow-up analysis for a reaction–diffusion equation with weighted nonlocal inner absorptions under nonlinear boundary flux

A blow-up analysis for a nonlinear divergence form reaction–diffusion equation with weighted nonlocal inner absorption terms is considered under nonlinear boundary flux in a bounded star-shaped region. Based on the auxiliary function method and the modified differential inequality technique, we establish conditions on weight function and nonlinearities to guarantee the solution exists globally or blows up at finite time. Moreover, upper and lower bounds for the blow-up time are derived under appropriate measure in higher dimensional spaces. Three examples are presented to illustrate applications of our main results.     

Conservation laws for a variable coefficient nonlinear diffusion–convection–reaction equation

We find the Lie point symmetries of a class of second-order nonlinear diffusion–convection–reaction equations containing an unspecified coefficient function of the independent variable t and determine the subclasses of these equations which are nonlinearly self-adjoint. By using a general theorem on conservation laws proved recently by N.H. Ibragimov we establish conservation laws corresponding to the aforementioned Lie point symmetries, one by one, for the simultaneous system of the original equation together with its adjoint equation through a formal Lagrangian. Particularly, for the nonlinearly self-adjoint subclasses, we construct conservation laws for the corresponding equations themselves.     

Explicit periodic travelling waves for a discrete lambda–omega reaction–diffusion system

In 1973, Kopell and Howard introduced a λ–ω reaction–diffusion system and found an explicit family of periodic travelling wave solutions lying on circles with radius less than 1. Since λ–ω systems represent universal models for studying chemical processes, and onset of turbulent behaviour, etc., explicit solutions of λ–ω systems with delays or discrete λ–ω systems can be of further help when the only method for obtaining other solutions is through numerical computation. There are now much investigations of various λ–ω systems. However, it is of interest to note that none attempts to find explicit travelling wave solutions. In this paper, we investigate the existence of explicit solutions for the simplest Euler scheme of a λ–ω system with delays or advancements which is described as a coupled pair of partial difference equations. We are able to provide necessary as well as sufficient conditions for the existence of numerical periodic travelling wave solutions. Additionally, we also provide some examples to show that our explicit solutions are qualitatively different from those found by Kopell and Howard and hence they may be of interests for specialists in the area of reaction–diffusion systems.     

On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation

In this paper we study the existence and qualitative properties of traveling waves associated with a nonlinear flux limited partial differential equation coupled to a Fisher–Kolmogorov–Petrovskii–Piskunov type reaction term. We prove the existence and uniqueness of finite speed moving fronts of C²$C^2$ classical regularity, but also the existence of discontinuous entropy traveling wave solutions.     

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.     

Existence and uniqueness of forced waves in a delayed reaction–diffusion equation in a shifting environment

The existence and uniqueness of forced waves in a general reaction–diffusion equation with time delay under climate change is concerned in this paper. By using upper and lower solutions method, monotone iteration scheme combined with the strong maximum principle, we show that there exists a nondecreasing and unique wave front with the speed consistent with the habitat shifting speed. Our results indicate the propagation of both the leading and trailing edges of the comoving population wavefront lag behind the climate envelope, which drives the species to extinction. Three examples and their corresponding numerical simulations are also given to illustrate the universality of analytical conclusions.     

Time-independent reaction–diffusion equation with a discontinuous reactive term

A two-dimensional singularly perturbed elliptic equation referred to in applications as the reaction–diffusion equation is considered. The nonlinearity describing the reaction is assumed to be discontinuous on a certain closed curve. On the basis of the generalized asymptotic comparison principle, the existence of smooth solution is proven and the accuracy of the asymptotic approximation is estimated.     

Traveling waves for a periodic Lotka–Volterra predator–prey system

This paper is concerned with the time periodic traveling wave solutions for a periodic Lotka–Volterra predator–prey system, which formulates that both species synchronously invade a new habitat. We first establish the existence of periodic traveling wave solutions by combining the upper and lower solutions with contracting mapping principle and Schauder’s fixed point theorem. The asymptotic behavior of nontrivial solution is given precisely by the stability of the corresponding kinetic system that has been widely investigated. Then, the nonexistence of periodic traveling wave solutions is confirmed by applying the theory of asymptotic spreading. We show the conclusion for all positive wave speed and obtain the minimal wave speed.