Minimal Wave Speed Research Articles

Traveling waves for SVIR epidemic model with nonlocal dispersal.

In this paper, we studied an SVIR epidemic model with nonlocal dispersal and delay, and we find that the existence of traveling wave is determined by the basic reproduction number ℜ₀ and minimal wave speed c*. By applying Schauder's fixed point theorem and Lyapunov functional, the existence and boundary asymptotic behaviour of traveling wave solutions is investigated for ℜ₀>1 and c>c*. The existence of traveling waves is obtained for ℜ₀>1 and c=c* by employing a limiting argument. We also show that the nonexistence of traveling wave solutions by Laplace transform. Our results imply that (i) the diffusion and infection ability of infected individuals can accelerate the wave speed; (ii) the latent period and successful rate of vaccination can slow down the wave speed.

Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle

<abstract><p>This paper deals with the initial value problem of a predator-prey system with dispersal and delay, which does not admit the classical comparison principle. When the initial value has nonempty compact support, the initial value problem formulates that two species synchronously invade the same habitat in population dynamics. By constructing proper auxiliary equations and functions, we confirm the faster invasion speed of two species, which equals to the minimal wave speed of traveling wave solutions in earlier works.</p></abstract>

Spreading speeds of epidemic models with nonlocal delays.

We estimate the spreading speeds in diffusive epidemic models with nonlocal delays, nonlinear incidence rate and constant recruitment rate. The purpose is to model the process that the infective invades the habitat of the susceptible, and they coexist eventually. In order to focus on our idea, a system with a nonlinear incidence rate is firstly studied, which implies a saturation level of the infective individuals and monotone incidence rate. When the initial value of the infective has nonempty compact support, we prove the rough spreading speed that equals the minimal wave speed of traveling wave solutions in the known results. Then for a general (nonmonotone) incidence rate, we obtain the spreading speeds by constructing auxiliary systems admitting a monotone incidence rate, and prove the convergence of solutions on any compact spatial interval. Furthermore, some numerical examples are given to estimate the invasion speed and show the nontrivial effect of time delay and spatial nonlocality, which implies that the stronger spatial nonlocality leads to larger spreading speeds.

Linear and Nonlinear Speed Selection for Mono-Stable Wave Propagations

In this paper, we study the selection mechanism of the minimal wave speed for traveling waves to an abstract monotone semiflow. A necessary and sufficient condition for the nonlinear selection of the minimal wave speed is established. Based on this result, we then derive conditions under which the linear or nonlinear selection is realized by way of a comparison principle. Our results on nonlinear selection are new and novel, and they can be viewed as breakthroughs in this topic; and for the linear selection, we successfully improve previous conventional results that always require that the monotone semiflow is dominated by its linear map. The applications to various biological models are also successful. We establish a series of new results to reaction-diffusion models with delay interactions, a lattice system, a scalar integro-difference equation, and a cooperative system, which completely solve some open problems and conjectures in the related references.

Chemotaxis Effect on Algae by Inorganic Polymer Flocculants: Backward Bifurcations and Traveling Wave Solutions

In recent years, inorganic polymer flocculants have been developed into a new type of water treatment reagents, which are more efficient than traditional inorganic flocculants and much cheaper than organic polymer flocculants. Based on the mechanism of inorganic polymer flocculants, a diffusive model is proposed to study the chemotaxis effect on algae. The chemotaxis flux of algae depends on not only its own density, but also the density of flocculants and the density gradient of flocculants. For the spatially heterogeneous model in the absence of chemotaxis, threshold dynamics can be expressed by the basic reproduction number [Formula: see text] which describes the average number of new population generated by initial fertile algae individuals. Further, the phenomenon of backward and forward bifurcations, local asymptotic stability properties and the existence of traveling wave solutions are studied for the spatially homogeneous model in the presence of chemotaxis. Our results suggest that reducing [Formula: see text] to be smaller than one may not be sufficient to eradicate the algae. Numerical analysis reveals that the minimal wave speed may be linearly deterministic in the absence of chemotaxis, while it is not linearly deterministic in the presence of chemotaxis.

Mathematical analysis of a tumour-immune interaction model: A moving boundary problem

A spatio-temporal mathematical model, in the form of a moving boundary problem, to explain cancer dormancy is developed. Analysis of the model is carried out for both temporal and spatio-temporal cases. Stability analysis and numerical simulations of the temporal model replicate experimental observations of immune-induced tumour dormancy. Travelling wave solutions of the spatio-temporal model are determined using the hyperbolic tangent method and minimum wave speeds of invasion are calculated. Travelling wave analysis depicts that cell invasion dynamics are mainly driven by their motion and growth rates. A stability analysis of the spatio-temporal model shows a possibility of dynamical stabilization of the tumour-free steady state. Simulation results reveal that the tumour swells to a dormant level.

Wave propagation in a diffusive SEIR epidemic model with nonlocal reaction and standard incidence rate

We study the existence and nonexistence of traveling waves of the reaction‐diffusion equations that describes a diffusive SEIR model with nonlocal reaction between the infected subpopulation and the susceptible subpopulation, where the total population is not constant. The existence of traveling waves depends on the basic reproduction number R0 of the corresponding ordinary differential equations and the minimal wave speed c∗. The main difficulty is the lack of order‐preserving property of our general system. Its proof is showed by constructing a closed and convex set and applying Schauder fixed point theorem. The proof of nonexistence result is obtained by two‐side Laplace transform. Finally, we present some numerical results of the minimal wave speed.

Spreading Speeds and Traveling Wave Solutions for a Delayed Periodic Equation without Quasimonotonicity

This paper is concerned with the propagation theory of a delayed periodic equation without quasimonotonicity. Because of time delay, it does not generate a monotone semiflow. A threshold is obtained, which is the spreading speed as well as the minimal wave speed. To estimate the spreading speed, some auxiliary equations are given, which also implies the nonexistence of traveling wave solutions. The existence of traveling wave solutions is investigated by Schauder’s fixed point and regularity of analytic semigroup, and is confirmed by showing the existence of generalized upper and lower solutions.

Spatial invasion dynamics for a time period predator‐prey system

This paper is concerned with the propagation theory of a predator‐prey system in a periodic environment. We consider the invasion process that the prey is the aboriginal, whereas the predator is the invader. If the initial habitat size of the predator is finite, then the asymptotic speed of spreading of the predator is given. Furthermore, the existence and nonexistence of traveling wave solutions are obtained, which defined a minimal wave speed of traveling wave solutions. Our results imply that the minimal wave speed of traveling wave solutions is equal to the asymptotic speed of spreading of the predator.

Wave propagation in a diffusive SIR epidemic model with spatiotemporal delay

In this paper, we propose a SIR epidemic model incorporating Laplacian diffusion and the spatiotemporal delay to model the transmission of communicable diseases. The existence and nonexistence of traveling wave solutions for the model are investigated. It is found that the threshold dynamics are determined by the basic reproduction number and the minimum wave speed c∗. By introducing an auxiliary system and Schauder's fixed point theorem, we establish the existence of traveling wave solutions for the model if and c > c∗. Employing Fubini theorem and the two‐sided Laplace transform, we obtain the nonexistence of traveling wave solutions for the model if or 0 < c < c∗. Our results cover and improve some results in the literature.

Spatial propagation for a parabolic system with multiple species competing for single resource

A model of \begin{document}$m$\end{document} species competing for a single growth-limiting resource is considered. We aim to use the dynamics of such a problem to describe the invasion and spread of \begin{document}$m$\end{document} species which are introduced localized in space \begin{document}$\mathbb{R}^N$\end{document} . The existence, uniqueness and uniform boundedness of the Cauchy problem are investigated by semigroup theory and local \begin{document}$L^p$\end{document} -estimates. The asymptotic speed of spread is achieved by uniform persistence ideas. The existence of traveling wave is obtained by upper-lower solutions and sliding techniques. Our result shows that the asymptotic speed of spread for \begin{document}$m$\end{document} species is characterized by the minimum wave speed of the positive traveling wave solutions associated with this system.

Minimal wave speed of competitive lattice dynamical systems with delays

This paper deals with the minimal wave speed of delayed lattice dynamical systems. We obtain the minimal wave speed by presenting the existence and nonexistence of traveling wave solutions, which completes the earlier results. In particular, when the wave speed equals the minimal wave speed, traveling wave solutions do not exponentially decay.

Wave propagation in a nonlocal diffusion epidemic model with nonlocal delayed effects

A nonlocal diffusion epidemic model with nonlocal delayed effects is investigated. The existence and non-existence of the non-trivial and non-negative traveling wave solutions for the model are obtained, respectively. It is found that the threshold dynamics of the model is determined by the basic reproduction number of the corresponding reaction system and minimal wave speed.

Traveling waves in an SEIR epidemic model with a general nonlinear incidence rate

ABSTRACTWe study the traveling waves of reaction-diffusion equations for a diffusive SEIR model with a general nonlinear incidence. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed. Its proof is showed by introducing an auxiliary system, applying Schauder fixed point theorem and then a limiting argument. The non-existence proof is obtained by two-sided Laplace transform when the speed is less than the critical velocity. Finally, we present some examples to support our theoretical results.

Some Spreading Properties of a Monotone Non-autonomous Parabolic Equation in a Periodic Cylinder

This paper is concerned with the asymptotic spreading speed and the periodic traveling wave fronts for a monotone non-autonomous parabolic equation in a cylinder. Main results concerning the two topics are obtained by virtue of some autonomous auxiliary equations. It turns out that the asymptotic spreading speed is the minimal wave speed of such periodic traveling wave front.

Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal

<p style='text-indent:20px;'>This paper is concerned with the traveling waves for a three-species competitive system with nonlocal dispersal. It has been shown by Dong, Li and Wang (DCDS 37 (2017) 6291-6318) that there exists a minimal wave speed such that a traveling wave exists if and only if the wave speed is above this minimal wave speed. In this paper, we first investigate the asymptotic behavior of traveling waves at negative infinity by a modified version of Ikehara's Theorem. Then we prove the uniqueness of traveling waves by applying the stronger comparison principle and the sliding method. Finally, we establish the exponential stability of traveling waves with large speeds by the weighted-energy method and the comparison principle, when the initial perturbation around the traveling wavefront decays exponentially as <i>x</i> → -∞, but can be arbitrarily large in other locations.

Traveling waves for time-delayed reaction diffusion equations with degenerate diffusion

This paper is concerned with time-delayed reaction–diffusion equations with degenerate diffusion. When the term for birth rate is a nonlocal integral with a heat kernel, the family of minimum wave speeds corresponding to all the degenerate diffusion coefficients is proved to admit a uniform positive infimum. However, when the term for birth rate is local, there is no positive infimum of all the minimum wave speeds. This difference indicates that the nonlocal effect plays a role as Laplacian such that a positive lower bound independent of the degenerate diffusion exists for the minimum wave speeds. The approach adopted for the proof is the monotone technique with the viscosity vanishing method. The degeneracy of diffusion for the equation causes us essential difficulty in the proof. A number of numerical simulations are also carried out at the end of the paper, which further numerically confirm our theoretical results.

Minimal Wave Speed in a Predator-Prey System with Distributed Time Delay

This paper is concerned with the minimal wave speed of traveling wave solutions in a predator-prey system with distributed time delay, which does not satisfy comparison principle due to delayed intraspecific terms. By constructing upper and lower solutions, we obtain the existence of traveling wave solutions when the wave speed is the minimal wave speed. Our results complete the known conclusions and show the precisely asymptotic behavior of traveling wave solutions.

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

ABSTRACTThis 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.

Minimal wave speed in a dispersal predator\u2013prey system with delays

This paper is concerned with the minimal wave speed in a nonlocal dispersal predator–prey system with delays. We define a threshold. By presenting the existence and nonexistence of traveling wave solutions, we confirm that the threshold is the minimal wave speed, which completes the known results.