Monostable Nonlinearity Research Articles

Precise rates of propagation in reaction–diffusion equations with logarithmic Allee effect

This paper focuses on propagation phenomena in reaction–diffusion equations with a weakly degenerate monostable nonlinearity. The kind of reaction term we consider can be seen as an intermediate between the classical logistic one (or Fisher–KPP) and more usual power laws that usually model Allee effects. We investigate the effect of the decay rate of the initial data on the propagation rate. When the tail of the initial data is sub-exponential, both finite speed propagation and acceleration may happen. We derive the exact separation between the two situations. When the initial data is sub-exponentially unbounded, acceleration unconditionally occurs. Estimates for the locations of the level sets are expressed in terms of the decay of the initial data. In addition, sharp exponents of acceleration for initial data with sub-exponential and algebraic tails are given. Numerical simulations are presented to illustrate the above findings.

Curved fronts in time-periodic reaction-diffusion equations with monostable nonlinearity

Recently, it was found that there exists various curved fronts with nonplanar level sets, which has been attracting increasing attention. In this paper, we study periodic traveling curved fronts for reaction-diffusion equations with time-periodic monostable nonlinearity in $ \mathbb{R}^2 $. By using the comparison principle and constructing super-sub solutions, we first establish the existence of periodic traveling curved fronts. Furthermore, we show that when the given perturbation decays at infinity under some weighted sense, the periodic traveling curved front is the globally asymptotic stability by introducing mild subsolutions.

Reaction–Diffusion Problems on Time-Periodic Domains

AbstractReaction–diffusion equations are studied on bounded, time-periodic domains with zero Dirichlet boundary conditions. The long-time behaviour is shown to depend on the principal periodic eigenvalue of a transformed periodic-parabolic problem. We prove upper and lower bounds on this eigenvalue under a range of different assumptions on the domain, and apply them to examples. The principal eigenvalue is considered as a function of the frequency, and results are given regarding its behaviour in the small and large frequency limits. A monotonicity property with respect to frequency is also proven. A reaction–diffusion problem with a class of monostable nonlinearity is then studied on a periodic domain, and we prove convergence to either zero or a unique positive periodic solution.

Propagation dynamics of degenerate monostable equations in space–time periodic media

This paper is concerned with the multidimensional pulsating waves and spreading speed for a class of reaction–advection–diffusion equations with degenerate monostable nonlinearity in space–time periodic media. Firstly, we show that the pulsating wave with the critical speed is unique up to translation, monotone and decays to zero exponentially, while the pulsating waves with noncritical speeds decay to zero nonexponentially. Then, combining the super- and sub-solution method and the asymptotic behavior of the pulsating waves, it is proved that the spreading speed of the degenerate monostable equations with compactly supported initial data is characterized by the critical speed under appropriate assumptions.

Pushed fronts of monostable reaction-diffusion-advection equations

In this paper, we prove some qualitative properties of pushed fronts for the periodic reaction-diffusion-equation with general monostable nonlinearities. Especially, we prove the exponential behavior of pushed fronts when they are approaching their unstable state. The proof also allows us to get the exponential behavior of pulsating fronts with speed c larger than the minimal speed. Through the exponential behavior, we finally prove the stability of pushed fronts.

Global stability of critical-speed pulsating fronts for degenerate monostable reactions

This paper is devoted to the global exponential stability of pulsating fronts for a kind of reaction–advection–diffusion equations with degenerate monostable nonlinearity in a periodic medium. We focus on the pulsating front with critical-speed, which is unique up to translation and decays exponentially when it approaches the unstable steady state 0. Based on the sub-super solution and the squeezing methods, we show that the solution will converge to a shift of the front exponentially in time when the general initial data decays faster than a positive exponential function at one end and has a positive lower bound at the other end.

Qualitative properties of traveling wave solutions in delayed reaction-diffusion equations with degenerate monostable nonlinearity

This paper deals with the asymptotic behavior, uniqueness up to translation and Lyapunov stability of traveling wave solutions with non-critical speeds in a class of delayed reaction-diffusion equations with degenerate monostable nonlinearity. With the help of Harnack inequalities and the detailed asymptotic higher order expansion, the precise asymptotic behavior of traveling wave solutions is established. Our results show that these traveling wave solutions decay algebraically, and the corresponding decaying rates of traveling wave solutions are independent of the delay. To illustrate our results on the asymptotic behavior, several degenerate equations are investigated. For a special equation, we present that the delay could affect the asymptotic behavior of traveling wave solutions. Furthermore, based on the special asymptotic behavior, we obtain the uniqueness up to translation and Lyapunov stability of traveling wave solutions.

Propagation Dynamics for Time-Periodic and Partially Degenerate Reaction-Diffusion Systems

This paper is concerned with the propagation dynamics for partially degenerate reaction-diffusion systems with monostable and time-periodic nonlinearity. In the cooperative case, we prove the existence of periodic traveling fronts and the exponential stability of continuous periodic traveling fronts. In the noncooperative case, we establish the existence of the minimal wave speed of periodic traveling waves and its coincidence with the spreading speed. More specifically, when the system is nondegenerate, the existence of periodic traveling waves is proved by using Schauder's fixed point theorem and the regularity of analytic semigroup, while in the partially degenerate case, due to the lack of compactness and standard parabolic estimates, the existence result is obtained by appealing to the asymptotic fixed point theorem with the help of some properties of the Kuratowski measure of noncompactness. It may be the first work to study periodic traveling waves of partially degenerate reaction-diffusion systems with noncooperative and time-periodic nonlinearity.

Minimal-speed selection of traveling fronts to a three components lattice competition system

In this paper, we study the traveling wave problem for a three species lattice competition system with monostable nonlinearity. We are interested in the linear or nonlinear determinacy of the minimal wave speed. Some conditions on the linear determinacy has been obtained in [J.-S. Guo, Y. Wang, C.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math. 19 (2015) 1805–1829]. In this paper, we first provided some general conditions for the linear determinacy. Then, we derive some novel results on the nonlinear determinacy. Finally, some numerical simulations are carried out to illustrate our analytic results. To our knowledge, this may be the first investigation on the nonlinear determinacy of the three species competition system.

Control of reaction-diffusion models in biology and social sciences

<p style='text-indent:20px;'>These lecture notes address the controllability under state constraints of reaction-diffusion equations arising in socio-biological contexts. We restrict our study to scalar equations with monostable and bistable nonlinearities.</p><p style='text-indent:20px;'>The uncontrolled models describing, for instance, population dynamics, concentrations of chemicals, temperatures, etc., intrinsically preserve pointwise bounds of the states that represent a proportion, volume-fraction, or density. This is guaranteed, in the absence of control, by the maximum or comparison principle.</p><p style='text-indent:20px;'>We focus on the classical controllability problem, in which one aims to drive the system to a final target, for instance, a steady-state. In this context the state is required to preserve, in the presence of controls, the pointwise bounds of the uncontrolled dynamics.</p><p style='text-indent:20px;'>The presence of constraints introduces significant added complexity for the control process. They may force the needed control-time to be large enough or even make some natural targets to be unreachable, due to the presence of barriers that the controlled trajectories might not be able to overcome.</p><p style='text-indent:20px;'>We develop and present a general strategy to analyze these problems. We show how the combination of the various intrinsic qualitative properties of the systems' dynamics and, in particular, the use of traveling waves and steady-states' paths, can be employed to build controls driving the system to the desired target.</p><p style='text-indent:20px;'>We also show how, depending on the value of the Allee parameter and on the size of the domain in which the process evolves, some natural targets might become unreachable. This is consistent with empirical observations in the context of endangered minoritized languages and species at risk of extinction.</p><p style='text-indent:20px;'>Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to illustrate the main methods and results.</p>

Transition fronts of Fisher–KPP equations in locally spatially inhomogeneous patchy environments

This paper is devoted to the study of spatial propagation dynamics of species in locally spatially inhomogeneous patchy environments or media. For a lattice differential equation with monostable nonlinearity in a discrete homogeneous media, it is well-known that there exists a minimal wave speed such that a traveling front exists if and only if the wave speed is not slower than this minimal wave speed. We shall show that strongly localized spatial inhomogeneous patchy environments may prevent the existence of transition fronts (generalized traveling fronts). Transition fronts may exist in weakly localized spatial inhomogeneous patchy environments but only in a finite range of speeds, which implies that it is plausible to obtain a maximal wave speed of existence of transition fronts.

Qualitative properties of pulsating fronts for reaction–advection–diffusion equations in periodic excitable media

In this paper, we study the pulsating fronts of reaction–advection-diffusion equations with two types of nonlinear term in periodic excitable media. Firstly, for the case with combustion nonlinearity, the unique front is proved to decay exponentially when it approaches the unstable limiting state. Secondly, for the degenerate monostable type nonlinearity, it is shown that the front with critical speed is unique, monotone and decays exponentially at negative end, while the fronts of noncritical speeds decay to zero non-exponentially.

Asymptotic spreading of a time periodic diffusion equation with degenerate monostable nonlinearity

This paper deals with the asymptotic spreading for a class of diffusion equations with degenerate monostable nonlinearity, where the initial value is asymptotically front-like. By the method of squeezing technique and super- and sub-solutions, we prove that the speed of asymptotic spreading may be finite or infinite, which depends on the degeneracy of nonlinearity as well as the decaying rate of the initial value. Different from the non-degenerate case, the traveling wave solution with the critical speed is globally asymptotically stable if the initial value has light tails, while an acceleration propagation may occur with the initial value having heavy tails in the degenerate case. Furthermore, if the initial value has heavy tails but lighter than algebraic, then the speed of the asymptotic spreading is finite, and the balance between finite or infinite speed is established if the initial value has algebraic tails. Our results reflect that the degeneracy may be harmful to the acceleration propagation.

When the Allee threshold is an evolutionary trait: Persistence vs. extinction

We consider a nonlocal parabolic equation describing the dynamics of a population structured by a spatial position and a phenotypic trait, submitted to dispersion, mutations and growth. The growth term may be of the Fisher-KPP type but may also be subject to an Allee effect which can be weak (non-KPP monostable nonlinearity, possibly degenerate) or strong (bistable nonlinearity). The type of growth depends on the value of a variable θ: the Allee threshold, which is considered here as an evolutionary trait. After proving the well-posedness of the Cauchy problem, we study the long time behavior of the solutions. Due to the richness of the model and the interplay between the various phenomena and the nonlocality of the growth term, the outcomes (extinction vs. persistence) are various and in sharp contrast with earlier results of the existing literature on local reaction-diffusion equations.

Enhanced frequency synchronization for concurrent aeroelastic and base vibratory energy harvesting using a softening nonlinear galloping energy harvester

This paper proposes a softening nonlinear aeroelastic galloping energy harvester for enhanced energy harvesting from concurrent wind flow and base vibration. Traditional linear aeroelastic energy harvesters have poor performance with quasi-periodic oscillations when the base vibration frequency deviates from the aeroelastic frequency. The softening nonlinearity in the proposed harvester alters the self-excited galloping frequency and simultaneously extends the large-amplitude base-excited oscillation to a wider frequency range, achieving frequency synchronization over a remarkably broadened bandwidth with periodic oscillations for efficient energy conversion from dual sources. A fully coupled aero-electro-mechanical model is built and validated with measurements on a devised prototype. At a wind speed of 5.5 m/s and base acceleration of 0.1 g, the proposed harvester improves the performance by widening the effective bandwidth by 300% compared to the linear counterpart without sacrificing the voltage level. The influences of nonlinearity configuration, excitation magnitude, and electromechanical coupling strength on the mechanical and electrical behavior are examined. The results of this paper form a baseline for future efficiency enhancement of energy harvesting from concurrent wind and base vibration utilizing monostable stiffness nonlinearities.

Propagation acceleration in reaction diffusion equations with anomalous diffusions

In this paper we consider the propagation speed in a reaction diffusion system with an anomalous Lévy process diffusion, modeled by a nonlocal equation with a fractional Laplacian and a generalized monostable or ignition nonlinearity. Given a typical Heaviside initial datum, we show that the speed of interface propagation displays an algebraic rate behavior in time, in contrast to the known linear rate in the classical model of Brownian motion and the exponential rate in the KPP model with the anomalous diffusion, and depends on the sensitive balance between the anomaly of the diffusion process and the strength of monostable reaction. In particular, for the combustion model with a fractional Laplacian (−Δ) s , we show that the speed of propagation transits continuously from being linear in time, when a traveling wave solution exists for s ∈ (1/2, 1), to being algebraic in time with a power reciprocal to 2s, when no traveling wave solution exists for s ∈ (0, 1/2).

Exact asymptotic behavior of pulsating traveling waves for a periodic monostable lattice dynamical system

This paper is concerned with the pulsating traveling waves of a periodic lattice dynamical system with monostable nonlinearity. We first establish the exponential upper and lower bounds of the pulsating wave profiles at minus infinity. Then we prove the uniqueness result and derive the asymptotic behavior of all non-critical monostable pulsating traveling waves. This might be the first time to obtain the exact asymptotic behavior of the pulsating traveling waves for periodic discrete systems towards the unstable steady state.

The Minimal Wave Speed of a Lotka-Volterra Competition Model

The minimal wave speed selection for the traveling waves of a 3-species competition system with monostable nonlinearity was considered. First, a comparison method was used to establish minimal-speed selection mechanisms through construction of subtle upper and lower solutions. Then, the speed selection about the threshold values of competition coefficients and the estimations were obtained. Finally, some numerical simulations were performed to show that the obtained results extend those in the previous literatures.

Spreading speed of the periodic Lotka-Volterra competition model

In this paper, we study the minimal speed (spreading speed) selection mechanism of traveling waves to a periodic diffusive Lotka-Volterra model with monostable nonlinearity. The method of upper and lower solutions pair is applied to establish the existence of traveling waves. We prove that the nature of nonlinear selection is to find a lower solution pair with the first species decaying in a faster rate. By novel constructions of various upper and lower solutions, we obtain a number of new results on the minimal speed determinacy. Numerical simulations are carried out to illustrate all of our discoveries.

Magnet-induced monostable nonlinearity for improving the VIV-galloping-coupled wind energy harvesting using combined cross-sectioned bluff body

This study proposes a novel wind energy harvester that exploits both vortex-induced vibration (VIV) and galloping phenomena with a hybrid cross-sectioned bluff body. Magnetic-induced monostable nonlinearity is then introduced in the harvester to improve its energy harvesting performance. The bluff-body cross section uses the combination of 3/4 round and 1/4 square to introduce the coupled VIV and galloping, and the magnetic repulsive force of the two magnets is employed to realize the monostable nonlinearity. Governing equations bringing together the VIV, galloping and magnet-induced monostable nonlinearity are established. Wind tunnel experiment is first conducted to validate the aerodynamic model of the governing equations without the monostable nonlinearity, which reveals the advantage of the coupled VIV and galloping for performance improvement. Consequently, the numerical simulations are performed to reveal the benefit of the magnet-induced monostable nonlinearity for wind energy harvesting. Results show that compared to the coupled VIV-galloping harvester without the monostable nonlinearity, the proposed monostable harvester can reduce the cut-in speed of galloping, which boosts the voltage output in a wider wind speed range and further promotes the advantages of the combination of VIV and galloping.