This paper is concerned with a lattice dynamical system modeling the evolution of susceptible and infective individuals at discrete niches. We prove the existence of traveling waves connecting the disease-free state to non-trivial leftover concentrations. We also characterize the minimal speed of traveling waves and we prove the non-existence of waves with smaller speeds.
In this paper, the existence of a non-trivial, positive and bounded critical traveling wave solution of a diffusive disease model, whose reaction system has infinity many equilibria, is obtained for the first time. This gives an affirmative answer to an open problem left in [X. Wang, H. Wang, J. Wu, Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete Contin. Dyn. Syst. Ser. A 32 (2012) 3303-3324]. Our result shows that the critical traveling wave in this model is a mixed of front and pulse type.
In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which ou…
In the present paper, we are concerned with an SIS epidemic reaction-diffusion model governed by mass action infection mechanism and linear birth-death growth with no flux boundary condition. By performing qualitative analysis, we study the stability of the disease-free equilibrium, uniform persistence property in terms of the basic reproduction number and the global stability of the endemic equilibrium in homogeneous environment, and investigate the asymptotic profile of endemic equilibria (when exist) in heterogeneous environment as one of the movement rate of the susceptible and infected po…
The minimal wave speed of time-periodic traveling waves arising from a diffusive Kermack-McKendrick model with seasonality and nonlocal delayed interactions
The aim is to assess the combined effect of diffusion and dispersion on shocks in the moderate dispersion regime. For a diffusive dispersive approximation of the equations of one-dimensional elasticity (or p-system), we study convergence of traveling waves to shocks. The problem is recast as a Hamiltonian system with small friction, and an analysis of the length of oscillations yields convergence in the moderate dispersion regime $\eps, δ\to 0$ with $δ= o(\eps)$, under hypotheses that the limiting shock is admissible according to the Liu E-condition and is not a contact discontinuity at either…
We investigate the traveling wave solutions of a three-species system involving a single predator and a pair of strong-weak competing preys. Our results show how the predation may affect this dynamics. More precisely, we describe several situations where the environment is initially inhabited by the predator and by either one of the two preys. When the weak competing prey is an aboriginal species, we show that there exist traveling waves where the strong prey invades the environment and either replaces its weak counterpart, or more surprisingly the three species eventually co-exist. Furthermor…
This paper is concerned with the traveling wave solutions of delayed reaction-diffusion systems. By using Schauder's fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka-Volterra competition systems with distributed delays are established…
By employing the N-barrier method developed in the paper, we establish a new N-barrier maximum principle for diffusive Lotka-Volterra systems of two competing species. As an application of this maximum principle, we show under certain conditions, the existence and nonexistence of traveling waves solutions for systems of three competing species. In addition, new $(1,0,0)$-$(u^{\ast},v^{\ast},0)$ waves are given in terms of the tanh function provided that the parameters satisfy certain conditions.
We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed $ε^{-1}$ ($ε\textgreater{}0$), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter $ε$: for small $ε$ the behaviour is essentially the same as for the diffusive Fisher-KPP equation. However, for large $ε$ the traveling front with minimal speed is discontinuous and travels at the maximal speed $ε^{-1}$. The traveling front…
Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized $L^1\cap L^p\to L^p$ stability for all $p \ge 2$ and dimensions $d \ge 1$ and nonlinear $L^1\cap H^s\to L^p\cap H^s$ stability and $L^2$-asymptotic behavior for $p\ge 2$ and $d\ge 3$. The behavior can in general be rather complicated, involving both convective (i.e., wave-like) and dif…
We determine the asymptotic spreading speed of an invasive species, which invades the territory of a native competitor, governed by a diffusive competition model with a free boundary in a spherically symmetric setting. This free boundary problem was studied recently in [7] but only rough bounds of the spreading speed was obtained there. We show in this paper that the spreading speed has an asymptotic limit (as time goes to infnity), which is given by a certain traveling wave type system of one space dimension, called a semi-wave. This appears to be the first result that gives the precise asymp…
We investigate the properties of traveling wave solutions to hyperbolic conservation laws augmented with diffusion and dispersion, and review the existence and qualitative properties of the associated kinetic functions, which characterize the class of admissible shock waves selected by such regularizations.
This paper is concerned with the existence of traveling wave solutions for diffusive two-species Lotka-Volterra systems with delay in both the reaction and diffusion terms without monotonicity. We extend the partial or cross monotone iteration method to systems that satisfy the partial quasi-monotone condition via construction appropriate upper and lower solutions. This is done by using Schauder's fixed point theorem.
We study the existence and nonexistence of traveling waves of general diffusive Kermack-McKendrick SIR models with standard incidence where the total population is not constant. The three classes, susceptible $S$, infected $I$ and removed $R$, are all involved in the traveling wave solutions. We show that the minimum speed for the existence of traveling waves for this three-dimensional non-monotonic system can be derived from its linearizaion at the initial disease-free equilibrium. The proof in this paper is based on Schauder fixed point theorem and Laplace transform and provides a promising …
This work is concerned with the existence of entire solutions of the diffusive Lotka-Volterra competition system \begin{equation}\label{eq:abstract} \begin{cases} u_{t}= u_{xx} + u(1-u-av), & \qquad \ x\in\mathbb{R} \cr v_{t}= d v_{xx}+ rv(1-v-bu), & \qquad \ x\in\mathbb{R} \end{cases} \quad (1) \end{equation} where $d,r,a$, and $b$ are positive constants with $a\neq 1$ and $b\neq 1$. We prove the existence of some entire solutions $(u(t,x),v(t,x))$ of $(1)$ corresponding to $(Φ_{c}(ξ),0)$ at $t=-\infty$ (where $ξ=x-ct$ and $Φ_c$ is a traveling wave solution of the scalar Fisher-KPP defined by…
We consider a system of two reaction-diffusion-advection equations describing the one dimensional directed motion of particles with superimposed diffusion and mutual alignment. For this system we show the existence of traveling wave solutions for weak diffusion by singular perturbation techniques and provide evidence for their existence also for stronger diffusion. We discuss different types of wave fronts and their composition to more complex patterns and illustrate their emergence from generic initial data by simulations. We also investigate the dependence of the wave velocities on the model…
We prove the existence of fast traveling pulse solutions in excitable media with non-local coupling. Existence results had been known, until now, in the case of local, diffusive coupling and in the case of a discrete medium, with finite-range, non-local coupling. Our approach replaces methods from geometric singular perturbation theory, that had been crucial in previous existence proofs, by a PDE oriented approach, relying on exponential weights, Fredholm theory, and commutator estimates.
Planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction. In this paper, we investigate the stability of planar wave trains in reaction-diffusion systems. We establish nonlinear diffusive stability against perturbations that are bounded along a line in $\mathbb{R}^2$ and decay exponentially in the distance from this line. Our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation. We also consider perturbations that are fully localized and establi…
It has long been a standard practice to neglect diffusive effects in stability analyses of detonation waves. Here, with the principal aim of quantifying the impact of these oft-neglected effects on the stability characteristics of such waves, we use numerical Evans-function techniques to study the (spectral) stability of viscous strong detonation waves---particular traveling-wave solutions of the Navier-Stokes equations modeling a mixture of reacting gases. Remarkably, our results show a surprising synergy between the high-activation-energy limit typically studied in stability analyses of deto…
This paper is concerned with a diffusive Lotka-Volterra type competition system with a free boundary in one space dimension. Such a system may be used to describe the invasion of a new species into the habitat of a native competitor. We show that the longtime dynamical behavior of the system is determined by a spreading-vanishing dichotomy, and provide sharp criteria for spreading and vanishing of the invasive species. Moreover, we determine the asymptotic spreading speed of the invasive species when its spreading is successful, which involves two systems of traveling wave type equations, and …
Given a first-order nonlinear hyperbolic system of conservation laws endowed with a convex entropy-entropy flux pair, we can consider the class of weak solutions containing shock waves depending upon some small scale parameters. In this Note, we define and derive several classes of entropy-dissipating augmented models, as we call them, which involve (possibly nonlinear) second- and third-order augmentation terms. Such terms typically arise in continuum physics and model the viscosity and capillarity effects in a fluid, for instance. By introducing a new notion of positive entropy production th…
We prove the existence of traveling fronts in diffusive Rosenzweig-MacArthur and Holling-Tanner population models and investigate their relation with fronts in a scalar Fisher-KPP equation. More precisely, we prove the existence of fronts in a Rosenzweig-MacArthur predator-prey model in two situations: when the prey diffuses at the rate much smaller than that of the predator and when both the predator and the prey diffuse very slowly. Both situations are captured as singular perturbations of the associated limiting systems. In the first situation we demonstrate clear relations of the fronts wi…
We study the asymptotic stability of traveling fronts and front's velocity selection problem for the time-delayed monostable equation $(*)$ $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),\ x \in \mathbb{R},\ t >0$, considered with Lipschitz continuous reaction term $g: \mathbb{R}_+ \to \mathbb{R}_+$. We are also assuming that $g$ is $C^{1,α}$-smooth in some neighbourhood of the equilibria $0$ and $κ>0$ to $(*)$. In difference with the previous works, we do not impose any convexity or subtangency condition on the graph of $g$ so that equation $(*)$ can possess pushed traveling fronts. Our fir…
