Minimal Wave Speed Research Articles

Co-existence of traveling waves for a model of microbial growth and competition in a flow reactor

Consider a reaction-diffusion model for a microbial flow reactorwith two competing species. Suppose that the amount of nutrient isinput in a constant velocity at one end of the flow reactor and iswashed out at the other end of the reactor. We study the dynamicalbehavior of population growth of these two species. In particular weare interested in the problem on the coexistence of traveling wavesthat best describes the long time dynamical behavior. By developingshooting method and continuation argument with the aid of anappropriately Liapunov function, we obtain the sufficient conditionsfor the coexistence of traveling waves as well as the minimum wavespeed.

Spatial Dynamics of a Nonlocal Periodic Reaction-Diffusion Model with Stage Structure

In this paper, we investigate a nonlocal periodic reaction-diffusion population model with stage structure. In the case of unbounded spatial domain, we establish the existence of the asymptotic speed of spread and show that it coincides with the minimal wave speed for monotone periodic traveling waves. In the case of bounded spatial domain, we obtain a threshold result on the global attractivity of either zero or a positive periodic solution.

Spatial dynamics of a nonlocal and time-delayed reaction–diffusion system

In this paper, we investigate the spatial dynamics of a nonlocal and time-delayed reaction–diffusion system, which is motivated by an age-structured population model with distributed maturation delay. The spreading speed c * , the existence of traveling waves with the wave speed c ⩾ c * , and the nonexistence of traveling waves with c < c * are obtained. It turns out that the spreading speed coincides with the minimal wave speed for monotone traveling waves.

Asymptotic speed of spread and traveling fronts for a nonlocal reaction–diffusion model with distributed delay

In this paper, a nonlocal reaction–diffusion model with distributed delay is studied. The asymptotic speed of spread is established for this model, and its coincidence with the minimal wave speed for traveling wave fronts is proved. Moreover, the dependence of the asymptotic speed of spread on delay and the nonlocal effect is considered. Our main finding here is that the delay can induce slow asymptotic speed of spread while the nonlocal effect can increase fast asymptotic speed of spread.

Travelling wavefronts of Belousov–Zhabotinskii system with diffusion and delay

This paper is concerned with the existence, nonexistence and minimal wave speed of the travelling wavefronts of Belousov–Zhabotinskii system with diffusion and delay.

Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects

In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction–diffusion equation with delay $$\frac{\partial u\left(x, t\right)}{\partial t}= d\Delta u\left(x, t\right)+f\left(u\left(x, t\right),\int\limits_{-\infty }^\infty h\left(x - y\right) u\left(y, t - \tau\right) dy\right)\!.$$ Under the monostable assumption, we show that there exists a minimal wave speed c* > 0, such that the equation has no traveling wave front for 0 c*, such a traveling wave front is unique up to translation and is globally asymptotically stable. When applied to some population models, these results cover, complement and/or improve a number of existing ones. In particular, our results show that (i) if ∂2 f (0, 0) > 0, then the delay can slow the spreading speed of the wave fronts and the nonlocality can increase the spreading speed; and (ii) if ∂2 f (0, 0) = 0, then the delay and nonlocality do not affect the spreading speed.

Spreading Speeds and Traveling Waves for Nonmonotone Integrodifference Equations

The spreading speeds and traveling waves are established for a class of nonmonotone discrete-time integrodifference equation models. It is shown that the spreading speed is linearly determinate and coincides with the minimal wave speed of traveling waves.

Spreading speed and travelling waves for a spatially discrete SIS epidemic model**Supported in part by the NSERC of Canada and the MITACS of Canada.

This paper is devoted to the study of the asymptotic speed of spread and travelling waves for a spatially discrete SIS epidemic model. By appealing to the theory of spreading speeds and travelling waves for monotonic semiflows, we establish the existence of asymptotic speed of spread and show that it coincides with the minimal wave speed for monotonic travelling waves. This also gives an affirmative answer to an open problem presented by Rass and Radcliffe (2003 Spatial Deterministic Epidemics (Mathematical Surveys and Monographs vol 102) (Providence, RI: American Mathematical Society)) in the case of discrete spatial habitat.

Asymptotic speed of propagation and traveling wave solutions for a lattice integral equation

We investigate the asymptotic speed of propagation and monotone traveling wave solutions for a lattice integral equation which is an epidemic model while the population is distributed on one-dimensional lattice Z . It is proved that the asymptotic speed of propagation c ∗ coincides with the minimal wave speed.

Spreading speeds and travelling waves in a delayed population model with stage structure on a 2D spatial lattice

In this paper, we derive a lattice model for a single species in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay. The important feature of the model is the reflection of the joint effect of the diffusion dynamics, the non-local delayed effect and the direction of propagation. We study the well-posedness of the initial-value problem and establish the existence of monotone travelling waves for wave speed c ≥ c*(θ) > 0, where θ is any fixed direction of propagation. In particular, we show that the minimal wave speed c*(θ) coincides with the asymptotic speed of spread for any fixed direction θ. Moreover, we find that the asymptotic speed of spread depends on not only the maturation period and the diffusion rate of mature population monotonically but also the direction of propagation, which is different from the case when the spatial variable is continuous.

Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications

This paper is concerned with the travelling wave fronts of nonlocal reaction-diffusion systems with delays. The existence of travelling wave fronts for nonlocal reaction-diffusion systems with delays is established by using Schauder’s fixed point theorem and upper-lower solution technique. Then these results are applied to the nonlocal delayed Logistic model and the delayed Belousov-Zhabotinskii reaction-diffusion system. Our results show that the time delay can reduce the minimal wave speed while the nonlocality can increase the minimal wave speed.

Tactically-driven nonmonotone travelling waves

Models of cell invasion incorporating directed cell movement up a gradient of an external substance and carrying capacity-limited proliferation give rise to travelling wave solutions. Travelling wave profiles with various shapes, including smooth monotonically decreasing, shock-fronted monotonically decreasing and shock-fronted nonmonotone shapes, have been reported previously in the literature. The existence of tactically-driven shock-fronted nonmonotone travelling wave solutions is analysed for the first time. We develop a necessary condition for nonmonotone shock-fronted solutions. This condition shows that some of the previously reported shock-fronted nonmonotone solutions are genuine while others are a consequence of numerical error. Our results demonstrate that, for certain conditions, travelling wave solutions can be either smooth and monotone, smooth and nonmonotone or discontinuous and nonmonotone. These different shapes correspond to different invasion speeds. A necessary and sufficient condition for the travelling wave with minimum wave speed to be nonmonotone is presented. Several common forms of the tactic sensitivity function have the potential to satisfy the newly developed condition for nonmonotone shock-fronted solutions developed in this work.

On the Diffusive Nicholson’s Blowflies Equation with Nonlocal Delay

This paper is concerned with the diffusive Nicholson’s blowflies model with nonlocal (or spatiotemporal) delay. When the spatial variable is one-dimensional, we establish the existence of travelling wave-front solutions by using the approach developed by Wang, Li, and Ruan (J. Differ. Equ. 222, 185–232, 2006) on the existence of travelling front solutions of reaction–diffusion systems with nonlocal delay. Moreover, we consider the dependence of the minimal wave speed on the delay and the mobility of the population. Our main finding here is that delay can induce slow travelling wave-fronts and the mobility of the population can increase fast travelling wave-fronts. In particular, if we choose some special kernel forms, then our results include and improve some known results.

Transition radiation of elastic waves at the interface of two elastic half-planes

Radiation of elastic waves is studied that is emitted by a point load that crosses the interface of two elastic half-planes. It is assumed that the load has a constant magnitude, moves along a straight line normal to the interface, and has a constant speed that is smaller than the minimum shear wave speed in the half-planes. In this case the mechanism of excitation of elastic waves is conventionally referred to as transition radiation. The adopted model allows to obtain an analytical expression for the elastic field excited by the load in the frequency–wavenumber domain. Using this expression, the energy of transition radiation is derived in a closed form. It is shown that transition radiation of the body waves occurs at any non-zero velocity of the load. Additionally, transition radiation of interface waves may occur provided that parameters of the half-planes allow existence of Stoneley waves. A parametric analysis of the directivity diagram of radiated body waves is accomplished focusing on dependence of the diagram on the load speed, load direction, and parameters of the half-planes. Using parameters that allow radiation of interface waves, the energy of this radiation is compared to that of the body waves. It is shown that the energy of the interface waves is greater unless the load velocity is close to the lowest body wave velocity.

Travelling Waves of Attached and Detached Cells in a Wound-Healing Cell Migration Assay

During a wound-healing cell migration assay experiment, cells are observed to detach and undergo mitosis before reattaching as a pair of cells on the substrate. During experiments with mice 3T3 fibroblasts, cell detachment can be confined to the wavefront region or it can occur over the whole wave profile. A multi-species continuum approach to modelling a wound-healing assay is taken to account for this phenomenon. The first cell population is composed of attached motile cells, while the second population is composed of detached immotile cells undergoing mitosis and returning to the migrating population after successful division. The first model describes cell division occurring only in the wavefront region, while a second model describes cell division over the whole of the scrape wound. The first model reverts to the Fisher equation when the reattachment rate relative to the detachment rate is large, while the second case does not revert to the Fisher equation in any limit. The models yield travelling wave solutions. The minimum wave speed is slower than the minimum Fisher wave speed and its dependence on the cell detachment and reattachment rate parameters is analysed. Approximate travelling wave profiles of the two cell populations are determined asymptotically under certain parameter regimes.

Propagation of HBV with spatial dependence

A mathematical model is proposed to simulate the hepatitis B virus (HBV) infection with spatial dependence. The existence of traveling waves is established via the geometric singular perturbation method. Numerical simulations show that the model admits non-monotone traveling profiles. Influences of various parameters on the minimum wave speed are also discussed.

Asymptotic behaviour of a reaction–diffusion model with a quiescent stage

This paper is devoted to the investigation of the asymptotic behaviour for a reaction–diffusion model with a quiescent stage. We first establish the existence of the asymptotic speed of spread and show that it coincides with the minimal wave speed for monotone travelling waves. Then we obtain a threshold result on the global attractivity of either zero or positive steady state in the case where the spatial domain is bounded.

Some Vector Borne Diseases with Structured Host Populations: Extinction and Spatial Spread

We derive from a structured population model a system of delay differential equations describing the interaction of five subpopulations, namely susceptible and infected adult and juvenile reservoirs and infected adult vectors, for a vector borne disease with particular reference to West Nile virus, and we also incorporate spatial movements by considering the analogue reaction diffusion systems with nonlocal delayed terms. Specific conditions for the disease eradication and sharp conditions for the local stability of the disease-free equilibrium are obtained using comparison techniques coupled with the spectral theory of monotone linear semi flows. A formal calculation of the minimal wave speed for the traveling waves is given and compared with field observation data.

Pattern-generating travelling waves in a discrete multicellular system with lateral inhibition

On a one-dimensional string of cells, the juxtacrine signalling model for Delta–Notch lateral inhibition by Collier et al. [J.R. Collier, N.A.M. Monk, P.K. Maini, J.H. Lewis, Pattern formation by lateral inhibition with feedback: A mathematical model of Delta–Notch intercellular interaction, J. Theoret. Biol. 183 (1996) 429–446] exhibits a predominant alternating pattern of cells expressing either Delta or Notch, as well as many aperiodic patterns. Despite this multistationarity, in the idealised situation of no noise, travelling waves invading the unstable, homogeneous state only generate the predominant alternating pattern in their wake all over the lattice. However, this robustness is totally lost in the presence of stochastic noise because the invaded, initial state is unstable. Using linear approximations around the initial, homogeneous state and around the final, patterned state, we are able to derive analytically all essential properties of the wave: the shape of the wave front, the unique, alternating pattern generated by the wave, and the asymptotic speed of the wave front. We show that the asymptotic wave speed equals the theoretical minimum wave speed. The latter agrees extremely well with the value estimated from numerical simulations. Thus, in this system travelling waves are pulled by the leading edge of the front.

Minimal wave speed in a diffusive epidemic model with temporal delay

Taking incubation period into account, we modify Capasso et al.’s epidemic model and consider a reaction-diffusion system with time delay. By the method of upper and lower solutions to a related second-order ordinary differential equation with infinite time delay, we establish the existence of the minimal wave speed for the modified epidemic model, and show that incubation period reduces the minimal wave speed of epidemic model indeed.