Minimal Wave Speed Research Articles

Invasion dynamics of a diffusive pioneer-climax model: Monotone and non-monotone cases

<p style='text-indent:20px;'>In this paper, we study the invasion dynamics of a diffusive pioneer-climax model in monotone and non-monotone cases. For parameter ranges in which the system admits monotone properties, we establish the existence of spreading speeds and their coincidence with the minimum wave speeds by monotone dynamical system theories. The linear determinacy of the minimum wave speeds is also studied by constructing suitable upper solutions. For parameter ranges in which the system is non-monotone, we further determine the existence of spreading speeds and traveling waves by the sandwich technique and upper-lower solution method. Our results generalize the existing results established under monotone assumptions to more general cases.

Existence of traveling wave solutions for a diffusive tuberculosis model with fast and slow progression

A diffusive tuberculosis model with fast and slow progression is investigated. By using sub–super solution method, Schauder’s fixed point theorem and Lyapunov functional, we obtain that there exists a traveling wave solutions (TWS) for the model if the threshold number ℜ0>1 and c>c∗, where c∗ is the minimum wave speed.

Spontaneous initiation and development of hydrogen–oxygen detonation with ozone sensitization

This work studies numerically the spontaneous initiation and sustenance of a detonation wave from a hot spot with a nonuniform initial temperature embedded within an H2O2 mixture with and without O3 addition. For the case with either no or just a small amount of O3 addition, a weak reaction wave is auto-ignited at the hot spot, accelerates and then transitions to a pulsating detonation, which propagates along the temperature gradient and quenches as it runs into the cold fresh mixture. However, with increasing O3 addition, the possibility of sustenance of a developing detonation within the gradient is significantly enhanced as it enters the cold mixture. Furthermore, the reduced induction time by O3 addition leads to earlier appearance of the spontaneous reaction wave and detonation formation in the cold mixture, demonstrating that quenching of the detonation is largely related to the instability property of the mixture because the shortened induction time reduces substantially the instability. It is also noted that, for 5%O3 addition, a low-temperature flame produced by the O3 reactions is present in front of the spontaneous reaction wave, inducing a local pressure wave, which facilitates spontaneous initiation and sustains the detonation entering the cold mixture. Moreover, O3 addition renders the critical temperature to induce the minimum spontaneous wave speed higher than the crossover temperature, while they are very close for the case without O3.

Traveling waves of a diffusive SIR epidemic model with general nonlinear incidence and infinitely distributed latency but without demography

In this paper, we are concerned with existence/non-existence of traveling waves of a diffusive SIR epidemic model with general incidence rate of the form of f(S)g(I) and infinitely distributed latency but without demography. We show that the existence of traveling waves only depends on the basic reproduction number of the corresponding spatial-homogeneous system of delay differential equations, which is determined by the recovery rate, the local properties of f and g and a minimal wave speed c∗ that is affected by the distributed delay. The proof of existence of traveling waves is by employing Schauder’s fixed point theorem, and the proof of nonexistence is completed with the aid of the bilateral Laplace transform.

Traveling wave solutions for a three-species predator–prey model with two aborigine preys

In this paper, we study the invading phenomenon of an alien predator to the habitat of two aborigine preys by traveling waves connecting the predator-free state to the co-existence state. We characterize the minimal wave speed of this invading process based on an application of Schauder’s fixed point theorem with the help of (generalized) upper-lower-solutions. New form of upper-lower-solutions are constructed to derive the existence of traveling waves for all admissible speeds.

Invasion waves for a diffusive predator–prey model with two preys and one predator

In this paper, we study the existence of invasion waves of a diffusive predator–prey model with two preys and one predator. The existence of traveling semi-fronts connecting invasion-free equilibrium with wave speed [Formula: see text] is obtained by Schauder’s fixed-point theorem, where [Formula: see text] is the minimal wave speed. The boundedness of such waves is shown by rescaling method and such waves are proved to connect coexistence equilibrium by LaSalle’s invariance principle. The existence of traveling front with wave speed [Formula: see text] is got by rescaling method and limit arguments. The non-existence of traveling fronts with speed [Formula: see text] is shown by Laplace transform.

Non-monotone waves of a stage-structured SLIRM epidemic model with latent period

We propose and investigate a stage-structured SLIRM epidemic model with latent period in a spatially continuous habitat. We first show the existence of semi-travelling waves that connect the unstable disease-free equilibrium as the wave coordinate goes to − ∞, provided that the basic reproduction number$\mathcal {R}_0 &gt; 1$and$c &gt; c_*$for some positive number$c_*$. We then use a combination of asymptotic estimates, Laplace transform and Cauchy's integral theorem to show the persistence of semi-travelling waves. Based on the persistent property, we construct a Lyapunov functional to prove the convergence of the semi-travelling wave to an endemic (positive) equilibrium as the wave coordinate goes to + ∞. In addition, by the Laplace transform technique, the non-existence of bounded semi-travelling wave is also proved when$\mathcal {R}_0 &gt; 1$and$0 &lt; c &lt; c_*$. This indicates that$c_*$is indeed the minimum wave speed. Finally simulations are given to illustrate the evolution of profiles.

Spreading speed for a KPP type reaction-diffusion system with heat losses and fast decaying initial data

In this work we consider a two-components reaction-diffusion system of KPP type with heat losses posed in a straight cylinder and equipped with fast decaying initial data. We derive a criteria condition – expressed in term of the sign of a suitable elliptic eigenvalue – for extinction and for propagation of the solutions. In the case of propagation we derive a spreading speed property and we obtain an asymptotic expansion in the large times for the location of the front, that is strongly related with the minimal wave speed of the travelling waves. We also obtain decay estimates for the solutions ahead the front.

Minimum wave speeds in monostable reaction-diffusion equations: sharp bounds by polynomial optimization.

Many monostable reaction-diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. We present methods for finding upper and lower bounds on minimum wave speed. They rely on constructing trapping boundaries for dynamical systems whose heteroclinic connections correspond to the travelling waves. Simple versions of this approach can be carried out analytically but often give overly conservative bounds on minimum wave speed. When the reaction-diffusion equations being studied have polynomial nonlinearities, our approach can be implemented computationally using polynomial optimization. For scalar reaction-diffusion equations, we present a general method and then apply it to examples from the literature where minimum wave speeds were unknown. The extension of our approach to multi-component reaction-diffusion systems is then illustrated using a cubic autocatalysis model from the literature. In all three examples and with many different parameter values, polynomial optimization computations give upper and lower bounds that are within 0.1% of each other and thus nearly sharp. Upper bounds are derived analytically as well for the scalar reaction-diffusion equations.

Traveling Waves and Estimation of Minimal Wave Speed for a Diffusive Influenza Model with Multiple Strains.

Antiviral treatment remains one of the key pharmacological interventions against influenza pandemic. However, widespread use of antiviral drugs brings with it the danger of drug resistance evolution. To assess the risk of the emergence and diffusion of resistance, in this paper, we develop a diffusive influenza model where influenza infection involves both drug-sensitive and drug-resistant strains. We first analyze its corresponding reaction model, whose reproduction numbers and equilibria are derived. The results show that the sensitive strains can be eliminated by treatment. Then, we establish the existence of the three kinds of traveling waves starting from the disease-free equilibrium, i.e., semi-traveling waves, strong traveling waves and persistent traveling waves, from which we can get some useful information (such as whether influenza will spread, asymptotic speed of propagation, the final state of the wavefront). On the other hand, we discuss three situations in which semi-traveling waves do not exist. When the control reproduction number R_{C} is larger than 1, the conditions for the existence and nonexistence of traveling waves are determined completely by the reproduction numbers R_{SC}, R_{RC} and the wave speed c. Meanwhile, we give an interval estimation of minimal wave speed for influenza transmission, which has important guiding significance for the control of influenza in reality. Our findings demonstrate that the control of influenza depends not only on the rates of resistance emergence and transmission during treatment, but also on the diffusion rates of influenza strains, which have been overlooked in previous modeling studies. This suggests that antiviral treatment should be implemented appropriately, and infected individuals (especially with the resistant strain) should be tested and controlled effectively. Finally, we outline some future directions that deserve further investigation.

A Non-local Cross-Diffusion Model of Population Dynamics II: Exact, Approximate, and Numerical Traveling Waves in Single- and Multi-species Populations.

We study traveling waves in a non-local cross-diffusion-type model, where organisms move along gradients in population densities. Such models are valuable for understanding waves of migration and invasion and how directed motion can impact such scenarios. In this paper, we demonstrate the emergence of traveling wave solutions, studying properties of both planar and radial wave fronts in one- and two-species variants of the model. We compute exact traveling wave solutions in the purely diffusive case and then perturb these solutions to analytically capture the influence directed motion has on these exact solutions. Using linear stability analysis, we find that the minimum wavespeeds correspond to the purely diffusive case, but numerical simulations suggest that advection can in general increase or decrease the observed wavespeed substantially, which allows a single species to more rapidly move into unoccupied resource-rich spatial regions or modify the speed of an invasion for two populations. We also find interesting effects from the non-local interactions in the model, suggesting that single species invasions can be enhanced with stronger non-locality, but that invasion of a competitive species may be slowed due to this non-local effect. Finally, we simulate pattern formation behind waves of invasion, showing that directed motion can have substantial impacts not only on wavespeed but also on the existence and structure of emergent patterns, as predicted in the first part of our study (Taylor et al. in Bull Math Biol, 2020).

A sharp-front moving boundary model for malignant invasion

We analyse a novel mathematical model of malignant invasion which takes the form of a two-phase moving boundary problem describing the invasion of a population of malignant cells into a population of background tissue, such as skin. Cells in both populations undergo diffusive migration and logistic proliferation. The interface between the two populations moves according to a two-phase Stefan condition. Unlike many reaction–diffusion models of malignant invasion, the moving boundary model explicitly describes the motion of the sharp front between the cancer and surrounding tissues without needing to introduce degenerate nonlinear diffusion. Numerical simulations suggest the model gives rise to very interesting travelling wave solutions that move with speed c, and the model supports both malignant invasion and malignant retreat, where the travelling wave can move in either the positive or negative x-directions. Unlike the well-studied Fisher–Kolmogorov and Porous-Fisher models where travelling waves move with a minimum wave speed c≥c∗>0, the moving boundary model leads to travelling wave solutions with |c|<c∗∗. We interpret these travelling wave solutions in the phase plane and show that they are associated with several features of the classical Fisher–Kolmogorov phase plane that are often disregarded as being nonphysical. We show, numerically, that the phase plane analysis compares well with long time solutions from the full partial differential equation model as well as providing accurate perturbation approximations for the shape of the travelling waves.

Bounds for Wave Speeds in the Riemann Problem: Direct Theoretical Estimates

In this paper we provide bound estimates for the two fastest wave speeds emerging from the solution of the Riemann problem for three well-known hyperbolic systems, namely the Euler equations of gas dynamics, the shallow water equations and the blood flow equations for arteries. Several approaches are presented, all being direct, that is non-iterative. The resulting bounds range from crude but simple estimates to accurate but sophisticated estimates that make limited use of information from the solution of the Riemann problem. Through a carefully chosen suite of test problems we asses our wave speed estimates against exact solutions and against previously proposed wave speed estimates. The results confirm that the derived theoretical bounds are actually so, from below and above, for minimal and maximal wave speeds respectively. The results also show that popular previously proposed estimates do not bound the true wave speeds in general. Applications in mind, but not pursued here, include (i) reliable implementation of the Courant condition to determine a stable time step in all explicit methods for hyperbolic equations; (ii) use in local time stepping algorithms and (iii) construction of HLL-type numerical fluxes for hyperbolic equations.

Exact minimum speed of traveling waves in a Keller–Segel model

In this letter we present a Keller–Segel model with logistic growth dynamics arising in the study of chemotactic pattern formation. We prove the existence of a minimum wave speed for which the model exhibits nonnegative traveling wave solutions at all speeds above this value and none below. The exact value of the minimum wave speed is given for all biologically relevant parameter values. These results strengthen recent results where non-sharp upper and lower bounds on the minimum wave speed were derived in a restricted parameter regime.

Spreading speed and traveling waves for an epidemic model in a periodic patchy environment

In this paper, we study an epidemic model in a periodic patchy environment. It is assumed that all the parameters share the same period N∈N. When the basic reproduction number R0>1, we first establish the spreading speed of traveling wave, which is denoted by c*. Then we show that for each c ≥ c*, the system admits a nontrivial traveling wave solution with the speed c if R0>1. We also show that there is no nontrivial traveling wave solution when R0≤1 and c > 0, or R0>1 and c ∈ (0, c*). This indicates that the spreading speed coincides with the minimum wave speed of traveling waves. Finally, we discuss the dependence of R0 and c* on the heterogeneity and amplitude of the parameters of our model.

Minimal Wave Speed in an Integrodifference System of Predator-Prey Type

This article studies the minimal wave speed of traveling wave solutions in an integrodifference predator-prey system that does not have the comparison principle. By constructing generalized upper and lower solutions and utilizing the theory of asymptotic spreading, we show the minimal wave speed of traveling wave solutions modeling the invasion process of two species by presenting the existence and nonexistence of nonconstant traveling wave solutions with any wave speeds.

Minimal wave speed and spread speed in a system modelling the geographic spread of black-legged tick Ixodes scapularis

In a recent work [1], a mathematical model is derived to explore the role that the white-tail deer plays in the geographic spread of the black-legged tick Ixodes scapularis in the northeast of the United States of America. The threshold dynamics is rigorously investigated in terms of the basic reproduction number, for the cases of the 1-D whole space Ω=R and general bounded spatial domain Ω with homogeneous Neumann and Direchlet boundary conditions. However, the minimal wave speed and spread speed of the model, which are the motivation of this model and thus most important, are only explored numerically. In the present paper, we offer a rigorous theoretical confirmation of what are numerically observed in [1], concluding that if the basic reproduction number is larger than one, the model allows a spread speed which is also the minimal speed of traveling wave fronts, and this speed is linearly deterministic.

Crossover in spreading behavior due to memory in population dynamics

The reaction-diffusion equation is one of the possible ways for modeling animal movement, where the reactive part stands for the population growth and the diffusive part for random dispersal of the population. However, a reaction-diffusion model may not represent all aspects of the spatial dynamics, because of the existence of distinct mechanisms that can affect the movement, such as spatial memory, which results in a bias for one direction of dispersal. This bias is modeled through an advective term on an advection-reaction-diffusion equation. Thus, considering the effects of memory on the population spread, we propose a model composed of a coupled partial differential equation system with two equations: one for the population dynamics and the other for the memory density distribution. For the population growth, we use either the exponential or logistic growth function. The analytic approach shows that for the exponential and logistic growth, the minimum traveling wave speeds are the same with or without memory dynamics in which the variation of memory is infinitesimal. From the numerical analysis, we explore how our parameters, memory, growth rate, and carrying capacity, affect the population redistribution. The combinations of these parameters result in a redistribution pattern of the population associated with either diffusive or superdiffusive and imply the dispersal is faster than the diffusion. Further, in the parameter-space defined by memory and growth rate, we have shown that memory is a factor that switches the dynamics between two spreading behaviors, one faster than the other.

Analytical and numerical detection of traveling wave and wave-train solutions in a prey–predator model with weak Allee effect

The traveling wave and wave-train fronts correspond to the transition zones from a trivial steady state to a coexistence steady state and a limit cycle around the coexistence steady state, respectively. In this study, we establish the existence of both types of fronts for a reaction–diffusion system of prey–predator interactions with weak additive Allee effect in prey growth, Holling type II functional response and density-dependent death rate for predators. For analytical simplicity, we consider immobile prey population. Under this consideration, the existence of both the traveling fronts is mathematically equivalent to the existence of point-to-point and point-to-cycle heteroclinic connections in $$\mathbb {R}^{3}$$. The proof for the existence of traveling fronts relies on the construction of an unbounded wedged region, a shooting argument and an appropriate Lyapunov function. Also, by employing the Hopf bifurcation theorem we show the existence of traveling wave-train solution. In both the cases, we find successful invasion by the predator species. Further, we provide adequate numerical illustrations in order to corroborate our theoretical predictions. Our numerical simulations also suggest that the theoretically obtained minimum wave speed serves as an asymptotic speed of traveling wave propagation.

Traveling waves in a nonlocal dispersal predator-prey model

&lt;p style='text-indent:20px;'&gt;This paper is concerned with the traveling wave solutions for a class of predator-prey model with nonlocal dispersal. By adopting the truncation method, we use Schauder's fixed-point theorem to obtain the existence of traveling waves connecting the semi-trivial equilibrium to non-trivial leftover concentrations for &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ c&amp;gt;c_{*} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, in which &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ c_* $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is the minimal wave speed. Meanwhile, through the limiting approach and the delicate analysis, we establish the existence of traveling wave solutions with the critical speed. Finally, we show the nonexistence of traveling waves for &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ 0&amp;lt;c&amp;lt;c_{*} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; by the characteristic equation corresponding to the linearization of original system at the semi-trivial equilibrium. Throughout the whole paper, we need to overcome the difficulties brought by the nonlocal dispersal and the non-preserving of system itself.&lt;/p&gt;