Minimal Wave Speed Research Articles

Propagation Phenomena for A Reaction–Advection–Diffusion Competition Model in A Periodic Habitat

This paper is devoted to the study of propagation phenomena for a Lotka–Volterra reaction–advection–diffusion competition model in a periodic habitat. We first investigate the global attractivity of a semi-trivial steady state for the periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for spatially periodic rightward traveling waves. We also obtain a set of sufficient conditions for the rightward spreading speed to be linearly determinate. Finally, we apply the obtained results to a prototypical reaction–diffusion model.

A periodic reaction–advection–diffusion model for a stream population

This paper is devoted to the study of spatial dynamics of a periodic reaction–advection–diffusion model for a stream population. In the case of an unbounded domain, we establish the existence of the leftward and rightward spreading speeds and their coincidence with the minimal wave speeds for monotone periodic traveling waves, respectively. In the case of a bounded domain, we obtain a threshold result on the global stability of either zero or the positive periodic solution.

TRAVELING WAVEFRONTS OF A DELAYED LATTICE REACTION-DIFFUSION MODEL

We investigate a system of delayed lattice differential system which is a model of pioneer-climax species distributed on one dimensional discrete space. We show that there exists a constant $c^*>0$, such that the model has traveling wave solutions connecting a boundary equilibrium to a co-existence equilibrium for $c\geq c^*$. We also argue that $c^*$ is the minimal wave speed and the delay is harmless. The Schauder's fixed point theorem combining with upper-lower solution technique is used for showing the existence of wave solution.

Traveling bands for the Keller-Segel model with population growth.

This paper is concerned with the existence of the traveling bands to the Keller-Segel model with cell population growth in the form of a chemical uptake kinetics. We find that when the cell growth is considered, the profile of traveling bands, the minimum wave speed and the range of the chemical consumption rate for the existence of traveling wave solutions will change. Our results reveal that collective interaction of cell growth and chemical consumption rate plays an essential role in the generation of traveling bands. The research in the paper provides new insights into the mechanisms underlying the chemotactic pattern formation of wave bands.

Spreading speeds in a lattice differential equation with distributed delay

This paper studies the spreading speed for a lattice differential equation with infinite distributed delay and we find that the spreading speed coincides with the minimal wave speed of traveling waves. Here the model has been proposed to describe a single species living in a 1D patch environment with infinite number of patches connected locally by diffusion.

Propagation of delayed lattice differential equations without local quasimonotonicity

This paper is concerned with the traveling wave solutions and asymptotic spreading of delayed lattice differential equations without quasimonotonicity. The spreading speed is obtained by constructing auxiliary equations and using the theory of lattice differential equations without time delay. The minimal wave speed of invasion traveling wave solutions is established by presenting the existence and nonexistence of traveling wave solutions.

Wavefronts of a stage structured model with state--dependent delay

This paper deals with a diffusive stage structured model withstate-dependent delay which is assumed to be an increasing functionof the population density. Compared with the constant delay, thestate--dependent delay makes the dynamic behavior more complex. Forthe state--dependent delay system, the dynamic behavior is dependentof the diffusion coefficients, while the equilibrium state ofconstant delay system is not destabilized by diffusion. Throughcalculating the minimum wave speed, we find that the wave is sloweddown by the state-dependent delay. Then, the existence of travelingwaves is obtained by constructing a pair of upper--lower solutionsand using Schauder's fixed point theorem. Finally, the travelingwavefront solutions for large wave speed are also discussed, and thefronts appear to be all monotone, regardless of the state dependentdelay. This is an interesting property, since many findings arefrequently reported that delay causes a loss of monotonicity, withthe front developing a prominent hump in some other delay models.

Traveling waves in a nonlocal dispersal SIR epidemic model

This paper is concerned with traveling wave solutions of a nonlocal dispersal SIR epidemic model. The existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the minimal wave speed. This threshold dynamics are proved by Schauder’s fixed point theorem and the Laplace transform. The main difficulties are that the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity.

Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction–diffusion systems

The purpose of this work is to study the spatial dynamics of some delayed nonlocal reaction–diffusion systems in whole space. We first establish a series of comparison theorems to investigate the global attractivity of the equilibria for a delayed nonlocal reaction–diffusion system with and without quasi-monotonicity. Then we show that the spreading speed of a general system without quasi-monotone conditions is coincident with the minimal wave speed. Applying a fluctuation method, we further provide some sufficient conditions to ensure the upward convergence of the spreading speed and traveling wave solutions. Finally, we point out the effects of the delay and nonlocality on the spreading speed of the non-quasi-monotone systems.

Spatial dynamics of a lattice population model with two age classes and maturation delay

This paper is concerned with the spatial dynamics of a monostable delayed age-structured population model in a 2D lattice strip. When there exists no positive equilibrium, we prove the global attractivity of the zero equilibrium. Otherwise, we give some sufficient conditions to guarantee the global attractivity of the unique positive equilibrium by establishing a series of comparison arguments. Furthermore, when those conditions do not hold, we show that the system is uniformly persistent. Finally, the spreading speed, including the upward convergence, is established for the model without the monotonicity of the growth function. The linear determinacy of the spreading speed and its coincidence with the minimal wave speed are also proved.

Minimal wave speed and uniqueness of traveling waves for a nonlocal diffusion population model with spatio-temporal delays

A nonlocal diffusion population model with spatio-temporal delays is considered in this paper. It is shown that this nonlocal diffusion equation admits traveling wave solutions. The uniqueness and minimal wave speed of the traveling wave solutions are obtained. Furthermore, the effects of different dispersal strategies on the minimal wave speed are characterized.

Reaction, diffusion and chemotaxis in wave propagation

By constructing an invariant set in the three dimensional space, we establish the existence of traveling wave solutions to a reaction-diffusion-chemotaxis model describing biological processes such as the bacterial chemotactic movement in response to oxygen and the initiation of angiogenesis. The minimal wave speed is shown to exist and the role of each process of reaction, diffusion and chemotaxis in the wave propagation is investigated. Our results reveal three essential biological implications: (1) the cell growth increases the wave speed; (2) the chemotaxis must be strong enough to make a contribution to the increment of the wave speed; (3) the diffusion rate plays a role in increasing the wave speed only when the cell growth is present.

Traveling waves for a nonlocal dispersal SIR model with delay and external supplies

This paper is concerned with the existence, nonexistence and minimal wave speed of traveling waves of a nonlocal dispersal delayed SIR model with constant external supplies and Holling-II incidence rate. We find that the existence and nonexistence of traveling waves of the system are not only determined by the minimal wave speed c∗, but also by the so-called basic reproduction number R0 of the corresponding reaction system. That is, we establish the existence of traveling waves for R0>1 and each wave speed c⩾c∗, and the nonexistence for R0>1 and any 0<c<c∗ or R0<1. We also discuss how the latency of infection and the spatial movement of the infective individuals affect the minimal wave speed. Biologically speaking, the longer the latency of infection in a vector is, the slower the disease spreads.

Repulsion effect on superinfecting virions by infected cells.

In this paper, the repulsion effect of superinfecting virion by infected cells is studied by a reaction diffusion equation model for virus infection dynamics. In this model, the diffusion of virus depends not only on its concentration gradient but also on the concentration of infected cells. The basic reproduction number, linear stability of steady states, spreading speed and existence of traveling wave solutions for the model are discussed. It is shown that viruses spread more rapidly with the repulsion effect of infected cells on superinfecting virions, than with random diffusion only. For our model, the spreading speed of free virus is not consistent with the minimal traveling wave speed. With our general model, numerical computations of the spreading speed show that the repulsion of superinfecting virion promotes the spread of virus, which confirms, not only qualitatively but also quantitatively, the experimental result of Doceul et al. (Science 327:873-876, 2010).

Birth-jump processes and application to forest fire spotting

Birth-jump models are designed to describe population models for which growth and spatial spread cannot be decoupled. A birth-jump model is a nonlinear integro-differential equation. We present two different derivations of this equation, one based on a random walk approach and the other based on a two-compartmental reaction–diffusion model. In the case that the redistribution kernels are highly concentrated, we show that the integro-differential equation can be approximated by a reaction–diffusion equation, in which the proliferation rate contributes to both the diffusion term and the reaction term. We completely solve the corresponding critical domain size problem and the minimal wave speed problem. Birth-jump models can be applied in many areas in mathematical biology. We highlight an application of our results in the context of forest fire spread through spotting. We show that spotting increases the invasion speed of a forest fire front.

Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay

In this paper, we study the traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates of the form βS(x,t)∫0hf(τ)g(I(x,t-τ))dτ. We find that the existence of traveling waves is determined by the basic reproduction number of the corresponding spatial-homogenous delay differential equations and the minimal wave speed. The existence proof is to introduce an auxiliary system and apply Schauder’s fixed point theorem. The non-existence of traveling waves is obtained by two-sided Laplace transform.

Traveling waves for a nonlocal dispersal SIR model with standard incidence

This paper is concerned with traveling wave solutions of a nonlocal dispersal SIR epidemic model with standard incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number of the corresponding ordinary differential model and the minimal wave speed. These threshold dynamics are proved by constructing an invariant cone and applying Schauder's fixed point theorem on this cone and the Laplace transform. The main difficulties are the lack of an occurrence of a regularizing effect and the loss of the order-preserving property of this model.

Global stability and wavefronts in a cooperation model with state-dependent time delay

This article deals with a diffusive cooperative model with state-dependent delay which is assumed to be an increasing function of the population density with lower and upper bounds. For the cooperative DDE system, the positivity and boundedness of solutions are firstly given. Using the comparison principle of the state-dependent delay equations obtained, the stability criterion of model is analyzed both from local and global points of view. When the diffusion is properly introduced, the existence of traveling waves is obtained by constructing a pair of upper–lower solutions and Schauder's fixed point theorem. Calculating the minimum wave speed shows that the wave is slowed down by the state-dependent delay. Finally, the traveling wavefront solutions for large wave speed are also discussed, and the fronts appear to be all monotone, regardless of the state dependent time delay. This is an interesting property, since many findings are frequently reported that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

Minimal wave speed of a nonlocal diffusive epidemic model with temporal delay

This note is concerned with a nonlocal version of the man-environment-man epidemic model in which the dispersion of infectious agents is assumed to follow a nonlocal diffusion law modeled by a convolution operator. The purpose of this note is to show that the minimal wave speeds of properly re-scaled nonlocal diffusion equations can approximate the corresponding one of the classical diffusion equation for this model. As a byproduct, our results indicate that the temporal delay in an epidemic model can reduce the speed of epidemic spread while the nonlocal effect can increase the speed.

Traveling Waves in Spatial SIRS Models.

We study traveling wavefront solutions for two reaction–diffusion systems, which are derived respectively as diffusion approximations to two nonlocal spatial SIRS models. These solutions characterize the propagating progress and speed of the spatial spread of underlying epidemic waves. For the first diffusion system, we find a lower bound for wave speeds and prove that the traveling waves exist for all speeds bigger than this bound. For the second diffusion system, we find the minimal wave speed and show that the traveling waves exist for all speeds bigger than or equal to the minimal speed. We further prove the uniqueness (up to translation) of these solutions for sufficiently large wave speeds. The existence of these solutions are proved by a shooting argument combining with LaSalle’s invariance principle, and their uniqueness by a geometric singular perturbation argument.