Nontrivial Traveling Wave Solutions Research Articles

Minimal wave speed and spreading speed in predator-prey systems with stage structure

This article studies the minimal wave speed of traveling wave solutions as well as spreading speed in predator-prey systems. The model divides the prey into the immaturity and maturity, and the predator feeds on the immature individuals. For the traveling wave solutions connecting the predator-free equilibrium with the coexistence state, we consider two different diffusion modes. For both modes, the minimal wave speed is shown by presenting the existence or nonexistence of nontrivial traveling wave solutions for all wave parameters. When the diffusion is local, the minimal wave speed is the spreading speed of the corresponding initial value problem, in which the initial condition implies that the predator is the invader while the prey is the aborigine in the whole habitat. Under the persistence assumption in the corresponding kinetic systems, our conclusion coincides with the fact that the diffusion of the prey may have less effect on the invasion ability of the predator.

Propagation dynamics of a nonlocal dispersal Zika transmission model with general incidence

In this paper, we are interested in propagation dynamics of a nonlocal dispersal Zika transmission model with general incidence. When the threshold is greater than one, we prove that there is a wave speed such that the model has a traveling wave solution with speed , and there is no traveling wave solution with speed less than . When the threshold is less than or equal to one, we show that there is no nontrivial traveling wave solution. The approaches we use here are Schauder's fixed point theorem with an explicit construction of a pair of upper and lower solutions, the contradictory approach, and the two‐sided Laplace transform.

Time periodic traveling wave solutions of a time-periodic reaction–diffusion SEIR epidemic model with periodic recruitment

This paper focuses on the existence and nonexistence of a time periodic traveling wave solution of a time-periodic reaction–diffusion SEIR epidemic model. The main feature of the model is the possible deficiency of the classical comparison principle such that many known results do not directly work. If the basic reproduction number of the model, denoted by R0, is larger than one, there exists a minimal wave speed c∗>0 satisfying for each c>c∗, the system admits a nontrivial time periodic traveling wave solution with wave speed c and for c<c∗, there exists no nontrivial time periodic traveling waves such that the system; if R0<1, the system admits no nontrivial time periodic traveling waves.

Bifurcation of finger-like structures in traveling waves of epithelial tissues spreading

We consider a continuous active polar fluid model for the spreading of epithelial monolayers introduced by R. Alert, C. Blanch-Mercader, and J. Casademunt, 2019. The corresponding free boundary problem possesses flat front traveling wave solutions. Linear stability of these solutions under periodic perturbations is considered. It is shown that the solutions are stable for short-wave perturbations while exhibiting long-wave instability under certain conditions on the model parameters (if the traction force is sufficiently strong). Then, considering the prescribed period as the bifurcation parameter, we establish the emergence of nontrivial traveling wave solutions with a finger-like periodic structure (pattern). We also construct asymptotic expansions of the solutions in the vicinity of the bifurcation point and study their stability. We show that, depending on the value of the contractility coefficient, the bifurcation can be a subcritical or a supercritical pitchfork.

Bifurcation Theory, Lie Group-Invariant Solutions of Subalgebras and Conservation Laws of a Generalized (2+1)-Dimensional BK Equation Type II in Plasma Physics and Fluid Mechanics

The nonlinear phenomena in numbers are modelled in a wide range of fields such as chemical physics, ocean physics, optical fibres, plasma physics, fluid dynamics, solid-state physics, biological physics and marine engineering. This research article systematically investigates a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation. We achieve a five-dimensional Lie algebra of the equation through Lie group analysis. This, in turn, affords us the opportunity to compute an optimal system of fourteen-dimensional Lie subalgebras related to the underlying equation. As a consequence, the various subalgebras are engaged in performing symmetry reductions of the equation leading to many solvable nonlinear ordinary differential equations. Thus, we secure different types of solitary wave solutions including periodic (Weierstrass and elliptic integral), topological kink and anti-kink, complex, trigonometry and hyperbolic functions. Moreover, we utilize the bifurcation theory of dynamical systems to obtain diverse nontrivial travelling wave solutions consisting of both bounded as well as unbounded solution-types to the equation under consideration. Consequently, we generate solutions that are algebraic, periodic, constant and trigonometric in nature. The various results gained in the study are further analyzed through numerical simulation. Finally, we achieve conservation laws of the equation under study by engaging the standard multiplier method with the inclusion of the homotopy integral formula related to the obtained multipliers. In addition, more conserved currents of the equation are secured through Noether’s theorem.

Traveling wave solutions for an integrodifference equation of higher order

&lt;abstract&gt;&lt;p&gt;This article is concerned with the minimal wave speed of traveling wave solutions for an integrodifference equation of higher order. Besides the operator may be nonmonotone, the kernel functions may be not Lebesgue measurable and integrable such that the equation has lower regularity. By constructing a proper set of potential wave profiles, we obtain the existence of smooth traveling wave solutions when the wave speed is larger than a threshold. Here, the profile set is obtained by giving a pair of upper and lower solutions. When the wave speed is the threshold, the existence of nontrivial traveling wave solutions is proved by passing to a limit function. Moreover, we obtain the nonexistence of nontrivial traveling wave solutions when the wave speed is smaller than the threshold.&lt;/p&gt;&lt;/abstract&gt;

Traveling waves of a differential-difference diffusive Kermack-McKendrick epidemic model with age-structured protection phase

We consider a general class of diffusive Kermack-McKendrick SIR epidemic models with an age-structured protection phase with limited duration, for example due to vaccination or drugs with temporary immunity. A saturated incidence rate is also considered which is more realistic than the bilinear rate. The characteristics method reduces the model to a coupled system of a reaction-diffusion equation and a continuous difference equation with a time-delay and a nonlocal spatial term caused by individuals moving during their protection phase. We study the existence and non-existence of non-trivial traveling wave solutions. We get almost complete information on the threshold and the minimal wave speed that describes the transition between the existence and non-existence of non-trivial traveling waves that indicate whether the epidemic can spread or not. We discuss how model parameters, such as protection rates, affect the minimal wave speed. The difficulty of our model is to combine a reaction-diffusion system with a continuous difference equation. We deal with our problem mainly by using Schauder's fixed point theorem. More precisely, we reduce the problem of the existence of non-trivial traveling wave solutions to the existence of an admissible pair of upper and lower solutions.

Wave propagation in a diffusive SAIV epidemic model with time delays

Based on the fact that the incubation periods of epidemic disease in asymptomatically infected and infected individuals are inevitable and different, we propose a diffusive susceptible, asymptomatically infected, symptomatically infected and vaccinated (SAIV) epidemic model with delays in this paper. To see whether epidemic disease can propagate spatially with a constant speed, we focus on the travelling wave solution for this model. When the basic reproduction number of the corresponding spatial-homogenous delayed differential system is greater than one and the wave speed is greater than or equal to the critical speed, we prove that this model admits nontrivial positive travelling wave solutions. Our theoretical results are of benefit to the prevention and control of epidemic.

Traveling waves for a discrete diffusive SIR epidemic model with treatment

The main purpose of this paper is to study the existence of traveling waves for a discrete diffusive SIR epidemic model with treatment. Compared to the work in Zhang and Wang (2014), more accurate results about the existence and nonexistence of nontrivial traveling wave solutions are obtained. We prove that when the basic reproduction number R0>1, there exists a critical number c∗>0 such that for each c>c∗, the system admits a nontrivial traveling wave solution with speed c, and for 0<c<c∗, the system has no nontrivial traveling wave solution. When R0<1, we show that there exists no nontrivial traveling wave solution by an integration argument. In addition, based on Deng and Zhang (2020), we obtain the existence of traveling waves with the critical speed c=c∗ under some assumptions.

Existence of traveling waves for a nonlocal dispersal SIR epidemic model with treatment

This paper is concerned with the existence and nonexistence of traveling wave solutions for a nonlocal dispersal epidemic model with treatment. The existence of traveling wave solutions is established by Schauder's fixed point theorem, while the nonexistence of traveling wave solutions is proved by two-sided Laplace transform. From the results, we conclude the minimal wave speed, which is an important threshold to predict how fast the disease invades. Compared with the work in [35], we obtain more accurate results about the existence and nonexistence of nontrivial traveling wave solutions. We prove that when the basic reproduction number R0>1, there exists a critical number c1⁎>0 such that for each c>c1⁎, the system has a nontrivial traveling wave solution with speed c, while for 0<c<c1⁎ the system admits no nontrivial traveling wave solution. When R0<1, we show that there exists no nontrivial traveling wave solution. In addition, based on [24], we obtain the existence of traveling waves with the critical speed c=c1⁎ under certain conditions.

Traveling Waves for a Discrete Diffusion Epidemic Model with Delay

This paper is concerned with traveling wave solutions in a discrete diffusion epidemic model with delayed transmission. Employing the way of contradictory discussions and the bilateral Laplace transform, we obtain the nonexistence of nontrivial positive bounded traveling wave solutions. Utilizing the super-/sub-solutions method and the fixed point theory, we derive the existence of nontrivial positive traveling wave solutions with both super-critical and critical speeds. Our results indicate that the critical speed is the minimal speed.

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.

Existence of traveling wave solutions of a deterministic vector-host epidemic model with direct transmission

We consider an epidemic model with direct transmission given by a system of nonlinear partial differential equations and study the existence of traveling wave solutions. When the basic reproductive number of the considered model is less than one, we show that there is no nontrivial traveling wave solution. On the other hand, when the basic reproductive number is greater than one, we prove that there is a minimum wave speed c⁎ such that the system has a traveling wave solution with speed c connecting both equilibrium points for any c≥c⁎. Moreover, under suitable assumption on the diffusion rates, we show that there is no traveling wave solution with speed less than c⁎. We conclude with numerical simulations to illustrate our findings. The numerical experiments support the validity of our theoretical results.

Barotropic instability of shear flows

AbstractWe consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm‐Liouville theory and hypergeometric functions.

Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay

In this paper, we investigate the existence and nonexistence of traveling wave solutions in a nonlocal dispersal epidemic model with spatio-temporal delay. It is shown that this model admits a nontrivial positive traveling wave solution when the basic reproduction number $ R_0&gt;1 $ and the wave speed $ c\geq c^* $ ($ c^* $ is the critical speed) and this model has no traveling wave solutions when $ R_0\leq1 $ or $ c&lt;c^* $. This indicates that $ c^* $ is the minimal wave speed.

Propagation thresholds of competitive integrodifference systems

ABSTRACTThis paper deals with the propagation thresholds of competitive integrodifference systems, which may be non-monotone even if the interspecific competition vanishes. The purpose is to establish the propagation thresholds that two competitors invade a new habitat. We first obtain the minimal wave speed by presenting the existence and non-existence of non-trivial travelling wave solutions. Then we confirm the minimal wave speed is the spreading speed of one competitor if the initial condition admits non-empty compact support.

Traveling Wave Solutions of a Diffusive SEIR Epidemic Model with Nonlinear Incidence Rate

This paper is concerned with the existence and nonexistence of traveling wave solutions of a diffusive SEIR epidemic model with nonlinear incidence rate, which are determined by the basic reproduction number $R_0$ and the minimal wave speed $c^*$. Namely, the system admits a nontrivial traveling wave solution if $R_0 \gt 1$ and $c \geq c^*$ and then the non-existence of traveling wave solutions of the system is established if $R_0 \gt 1$ and $0 \lt c \lt c^*$. Especially, using numerical simulation, we give the basic framework of traveling wave solutions of the system.

Traveling waves for a two-group epidemic model with latent period in a patchy environment

In this paper, we derive a two-group SIR epidemic model with latent period in a patchy environment by applying discrete Fourier transform. It is assumed that the infectious disease spreads between two groups and it has a fixed latent period. When the basic reproduction number R0>1, we prove that the system admits a nontrivial traveling wave solution for each admissible speed c (namely, c>c⁎, where c⁎ is the minimal wave speed). We also show that there is no positive traveling wave solution (ϕ1,ϕ2,φ1,φ2) satisfying φi(±∞)=0,ϕi(−∞)=Si0andϕi(+∞)=S⁎i when R0≤1 and c>0, or R0>1 and c∈(0,c⁎), where i=1,2.

Traveling wave solutions in a two-group epidemic model with latent period

In this paper, we propose a susceptible-infective-recovered (SIR) epidemic model to describe the geographic spread of an infectious disease in two groups/sub-populations living in a spatially continuous habitat. It is assumed that the susceptibility of individuals for infection and the infectivity of individuals are distinct between these two groups/sub-populations. It is also assumed that the infectious disease has a fixed latent period and the latent individuals may diffuse. We investigate the traveling wave solutions and obtain complete information about the existence and nonexistence of nontrivial traveling wave solutions. We prove that when the basic reproduction number at the disease free equilibrium , there exists a critical number c* > 0 such that for each c > c*, the system admits a nontrivial traveling wave solution with wave speed c, and for c < c*, the system admits no nontrivial traveling wave solution. When , we show that there exists no nontrivial traveling wave solution. In addition, for the case and c > c*, we also find that the final sizes of susceptible individuals, denoted by , satisfies , which means that there is no outbreak of this the infectious disease anymore. At last, we analyze and simulate the continuous dependence of the minimal speed c* on the parameters.

Traveling waves in an SEIR epidemic model with the variable total population

In the present paper, we propose a simple diffusive SEIR epidemic model where the total population is variable. We first give the explicit formula of the basic reproduction number $\mathcal{R}_0$ for the model. And hence, we show that if $\mathcal{R}_0>1$, then there exists a constant $c^*>0$ such that for any $c>c^*$, the model admits a nontrivial traveling wave solution, and if $\mathcal{R}_0 0$ (or, $\mathcal{R}_0>1$ and $c\in(0,c^*)$), then the model has no nontrivial traveling wave solution. Consequently, we obtain the full information about the existence and non-existence of traveling wave solutions of the model by determined by the constants $\mathcal{R}_0$ and $c^*$. The proof of the main results is mainly based on Schauder fixed point theorem and Laplace transform.