Minimal Wave Speed Research Articles

Mathematical Models for Cell Migration with Real-Time Cell Cycle Dynamics

The fluorescent ubiquitination-based cell cycle indicator, also known as FUCCI, allows the visualization of the G1and S/G2/M cell cycle phases of individual cells. FUCCI consists of two fluorescent probes, so that cells in the G1 phase fluoresce red and cells in the S/G2/M phase fluoresce green. FUCCI reveals real-time information about cell cycle dynamics of individual cells, and can be used to explore how the cell cycle relates to the location of individual cells, local cell density, and different cellular microenvironments. In particular, FUCCI is used in experimental studies examining cell migration, such as malignant invasion and wound healing. Here we present, to our knowledge, new mathematical models that can describe cell migration and cell cycle dynamics as indicated by FUCCI. The fundamental model describes the two cell cycle phases, G1 and S/G2/M, which FUCCI directly labels. The extended model includes a third phase, early S, which FUCCI indirectly labels. We present experimental data from scratch assays using FUCCI-transduced melanoma cells, and show that the predictions of spatial and temporal patterns of cell density in the experiments can be described by the fundamental model. We obtain numerical solutions of both the fundamental and extended models, which can take the form of traveling waves. These solutions are mathematically interesting because they are a combination of moving wavefronts and moving pulses. We derive and confirm a simple analytical expression for the minimum wave speed, as well as exploring how the wave speed depends on the spatial decay rate of the initial condition.

Minimal wave speed of a competition integrodifference system

In this paper, we study the minimal wave speed of a competitive system. By constructing upper and lower solutions, we confirm the existence of travelling wave solution at the critical wave speed. This completes earlier results found in the literature. Our conclusion implies that the asymptotic decay behaviour of solutions at the critical wave speed is different from that of solutions at larger wave speeds.

Propagation dynamics for a spatially periodic integrodifference competition model

In this paper, we study the propagation dynamics for a class of integrodifference competition models in a periodic habitat. An interesting feature of such a system is that multiple spreading speeds can be observed, which biologically means different species may have different spreading speeds. We show that the model system admits a single spreading speed, and it coincides with the minimal wave speed of the spatially periodic traveling waves. A set of sufficient conditions for linear determinacy of the spreading speed is also given.

A reaction-diffusion within-host HIV model with cell-to-cell transmission.

In this paper, a reaction-diffusion within-host HIV model is proposed. It incorporates cell mobility, spatial heterogeneity and cell-to-cell transmission, which depends on the diffusion ability of the infected cells. In the case of a bounded domain, the basic reproduction number [Formula: see text] is established and shown as a threshold: the virus-free steady state is globally asymptotically stable if [Formula: see text] and the virus is uniformly persistent if [Formula: see text]. The explicit formula for [Formula: see text] and the global asymptotic stability of the constant positive steady state are obtained for the case of homogeneous space. In the case of an unbounded domain and [Formula: see text], the existence of the traveling wave solutions is proved and the minimum wave speed [Formula: see text] is obtained, providing the mobility of infected cells does not exceed that of the virus. These results are obtained by using Schauder fixed point theorem, limiting argument, LaSalle's invariance principle and one-side Laplace transform. It is found that the asymptotic spreading speed may be larger than the minimum wave speed via numerical simulations. However, our simulations show that it is possible either to underestimate or overestimate the spread risk [Formula: see text] if the spatial averaged system is used rather than one that is spatially explicit. The spread risk may also be overestimated if we ignore the mobility of the cells. It turns out that the minimum wave speed could be either underestimated or overestimated as long as the mobility of infected cells is ignored.

具有时空时滞的非局部扩散SIR模型的行波解

In view of the infected individuals with the ability to move freely and spread disease, the traveling wave solutions for nonlocal dispersal SIR models with spatiotemporal delays were investigated. The threshold dynamics was determined by means of the basic reproduction number and the minimal wave speed. Firstly, based on Schauder’s fixed point theorem, the existence of fixed points on the cone was proved through construction of an invariant cone of the initial function on a bounded region. Then, the nonexistence of traveling wave solutions was verified through the twosided Laplace transform. Since the minimum propagation velocity of the traveling wave solution had important practical significance to control the disease transmission, the influences of the nonlocal diffusion term and the delay on the minimum wave velocity of the disease were discussed.

Propagation phenomena for CNNs with asymmetric templates and distributed delays

The aim of this work is to study propagation phenomena for monotone and nonmonotone cellular neural networks with the asymmetric templates and distributed delays. More precisely, for the monotone case, we establish the existence of the leftward ($c_{-}^*$) and rightward ($c_{+}^*$) spreading speeds for CNNs by appealing to the theory developed in [26,27], and $c_{-}^*+c_{+}^*>0$. Especially, if cells possess the symmetric templates and the same delayed interactions, then $c_{-}^*=c_{+}^*>0$. Moreover, if the effect of the self-feedback interaction $α f'(0)$ is not less than 1, then both $c_{-}^*>0$ and $c_{+}^*>0$. For the non-monotone case, the leftward and rightward spreading speeds are investigated by using the results of the spreading speed for the monotone case and squeezing the given output function between two appropriate nondecreasing functions. It turns out that the leftward and rightward spreading speeds are linearly determinate in these two cases. We further obtain the existence and nonexistence of travelling wave solutions under the weaker conditions than those in [46, 47] and show that the spreading speed coincides with the minimal wave speed.

Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations

This paper is concerned with the stability of noncritical/critical traveling waves for nonlocal time-delayed reaction-diffusion equation. When the birth rate function is non-monotone, the solution of the delayed equation is proved to converge time-exponentially to some (monotone or non-monotone) traveling wave profile with wave speed $c>c_*$, where $c_*>0$ is the minimum wave speed, when the initial data is a small perturbation around the wave. However, for the critical traveling waves ($c = c_*$), the time-asymptotical stability is only obtained, and the decay rate is not gotten due to some technical restrictions. The proof approach is based on the combination of the anti-weighted method and the nonlinear Halanay inequality but with some new development.

Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model.

yme disease is transmitted via blacklegged ticks, the spatial spread of which is believed to be primarily via transport on white-tailed deer. In this paper, we develop a mathematical model to describe the spatial spread of blacklegged ticks due to deer dispersal. The model turns out to be a system of differential equations with a spatially non-local term accounting for the phenomenon that a questing female adult tick that attaches to a deer at one location may later drop to the ground, fully fed, at another location having been transported by the deer. We first justify the well-posedness of the model and analyze the stability of its steady states. We then explore the existence of traveling wave fronts connecting the extinction equilibrium with the positive equilibrium for the system. We derive an algebraic equation that determines a critical value c* which is at least a lower bound for the wave speed in the sense that, if c<c*, there is no traveling wave front of speed c connecting the extinction steady state to the positive steady state. Numerical simulations of the wave equations suggest that c* is the minimum wave speed. We also carry out some numerical simulations for the original spatial model system and the results seem to confirm that the actual spread rate of the tick population coincides with c*. We also explore the dependence of c* on the dispersion rate of the white tailed deer, by which one may evaluate the role of the deer's dispersion in the geographical spread of the ticks.

Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity

This work is devoted to study the spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. In the case of the spatial domain is bounded and heterogeneous, we assume some key parameters in the model explicitly depend on spatial location. We first define the basic reproduction number $\mathcal{R}_0$ for the disease transmission, which generalizes the existing definition of $\mathcal{R}_0$ for the system in spatially homogeneous environment. Then we establish a threshold type result for the disease eradication ($\mathcal{R}_0 &lt;1$) or uniform persistence ($\mathcal{R}_0&gt;1)$. In the case of the domain is linear, unbounded, and spatially homogenerous, we further establish the existence of traveling wave solutions and the minimum wave speed $c^*$ for the disease transmission. At the end of this work, we characteristic the minimum wave speed $c^*$ and provide a method for the calculation of $c^*$.

Analysis of the Minimal Traveling Wave Speed via the Methods of Young Measures

In this paper, we consider the equation $u_t = u_{xx} + h(b(x)) (1-u)u, x \in \mathbb{R}$, where $\gamma \mapsto h(\gamma)$ is a nonnegative continuous function and $b(x)$ is a periodic function. In some sense, $b(x)$ represents a controllable parameter and the function $\gamma \mapsto h(\gamma)$ describes how the intrinsic growth rate of $u$ depends on the parameter $b$. It is known that there exists a nonnegative number $c^{*}(h(b))$ such that the traveling wave with average speed $c$ exists if and only if $c \geq c^{*}(h(b))$. We study minimizing problems and maximizing problems of the minimal speed $c^{*}(h(b))$ by varying $b(x)$ under the constraint $\frac{1}{L} \int_{0}^{L} b(x) dx = \alpha$, where $\alpha$ is a given positive constant. We prove the existence of minimizers and maximizers of the minimal speed, but it turns out that the minimizers for certain choices of $h$ do not exist in the class of functions but do exist in the space of Young measures. This means that the minimizing sequence exhi...

A new estimate on the minimal wave speed for travelling fronts in reaction–diffusion–convection equations

A new estimate on the minimal wave speed for travelling fronts in reaction–diffusion–convection equations

Propagation of monostable traveling fronts in discrete periodic media with delay

This paper is devoted to study the front propagation for a class of discrete periodic monostable equations with delay and nonlocal interaction. We first establish the existence of rightward and leftward spreading speeds and prove their coincidence with the minimal wave speeds of the pulsating traveling fronts in the right and left directions, respectively. The dependency of the speeds of propagation on the heterogeneity of the medium and the delay term is also investigated. We find that the periodicity of the medium increases the invasion speed, in comparison with a homogeneous medium; while the delay decreases the invasion speed. Further, we prove the uniqueness of all noncritical pulsating traveling fronts. Finally, we show that all noncritical pulsating traveling fronts are globally exponentially stable, as long as the initial perturbations around them are uniformly bounded in a weight space.

Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity

A non-local delayed reaction-diffusion model with a quiescent stage is investigated. It is shown that the spreading speed of this model without quasi-monotonicity is linearly determinate and coincides with the minimal wave speed of traveling waves.

Traveling wave phenomena of [formula omitted]-dimensional diffusive predator–prey systems

In this work, we deal with the n-dimensional reaction-diffusive predator–prey systems with the discrete time delay, which have numerous applications in biology and ecology. By applying an asymptotic analysis, the Schauder’s fixed point theorem as well as the upper and lower solution method, we investigate the existence, non-existence, exponentially asymptotic stability and asymptotic behaviors of traveling wave solutions with the appropriate small delay when the wave speed c varies. Moreover, we derive the minimal wave speed c∗ based on the well-established properties of the Fisher equation, and prove the existence of traveling wave solutions when the wave speed c≥c∗, which provides a sharp contrast to the case of c<c∗. The obtained results extend the existing ones in the literature to arbitrary finite dimensional systems with the discrete time delay and demonstrate asymptotic behaviors of traveling wave solutions.

Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate

A nonlocal dispersal SIR epidemic model with nonlinear incidence rate is introduced. It is shown that the existence and non-existence of nontrivial and nonnegative traveling wave solutions of this model are fully determined by the threshold values, that is, the basic reproduction number R0 and the minimal wave speed c∗. For R0>1 and c≥c∗, the existence theorem is obtained by the method of auxiliary system, Schauder’s fixed point theorem and three limiting arguments. For R0>1 and 0<c<c∗, the non-existence theorem is derived by applying the two-sided Laplace transform and making full use of the structure of the model. For R0=1 with c>0 and R0<1 with c>0, the non-existence theorems are also established.

Meandering Rivers: How Important is Lateral Variability for Species Persistence?

Models for population dynamics in rivers and streams have highlighted the importance of spatial and temporal variations for population persistence. We present a novel model that considers the longitudinal variation as introduced by the sinuosity of a meandering river where a main channel is laterally extended to point bars in bends. These regions offer different habitat conditions for aquatic populations and therefore may enhance population persistence. Our model is a nonstandard reaction-advection-diffusion model where the domain of definition consists of the real line (representing the main channel) with periodically added intervals (representing the point bars). We give an existence and uniqueness proof for solutions of the equations. We then study population persistence as the (in-) stability of the trivial solution and population spread as the minimal wave speed of traveling periodic waves. We conduct a sensitivity analysis to highlight the importance of each parameter on the model outcome. We find that sinuosity can enhance species persistence.

Minimal wave speed of competitive diffusive systems with time delays

This paper is concerned with the minimal wave speed of competitive diffusive systems, of which the corresponding wave system is of infinite dimensional. By constructing upper and lower solutions, the existence of traveling wave solutions is confirmed, which shows the minimal wave speed. Therefore, it completes the known results.

Monotonicity, uniqueness, and stability of traveling waves in a nonlocal reaction‐diffusion system with delay

The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction‐diffusion system with delay arising from the spread of an epidemic by oral‐faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c⩾c∗. Then we show that the traveling wave fronts with speed c&gt;c∗ are exponentially asymptotically stable.

Spatial Dynamics of Multilayer Cellular Neural Networks

The purpose of this work is to study the spatial dynamics of one-dimensional multilayer cellular neural networks. We first establish the existence of rightward and leftward spreading speeds of the model. Then we show that the spreading speeds coincide with the minimum wave speeds of the traveling wave fronts in the right and left directions. Moreover, we obtain the asymptotic behavior of the traveling wave fronts when the wave speeds are positive and greater than the spreading speeds. According to the asymptotic behavior and using various kinds of comparison theorems, some front-like entire solutions are constructed by combining the rightward and leftward traveling wave fronts with different speeds and a spatially homogeneous solution of the model. Finally, various qualitative features of such entire solutions are investigated.

Traveling waves for a diffusive SEIR epidemic model with non-local reaction

• This paper is the first study of the traveling waves of diffusive SEIR model with non-local reaction. • We give a more precise estimation on the traveling wave speed of infectious diseases. • Our results are of great practical value in the control of the propagation of infectious diseases. This paper is devoted to the existence of the traveling waves of the equations describing a diffusive SEIR model with non-local reaction between the infected and the susceptible. The existence of traveling waves depends on the basic reproduction rate S − ∞ β / γ and the minimal speed. We use the Schauder fixed point theorem and the Laplace transform to get the existence and non-existence of traveling waves in our paper. We also give some numerical results of the minimal wave speed.