Asymptotic Speed Research Articles

Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations

This paper is concerned with the study of the large-time behavior of the solutions u of a class of one-dimensional reaction–diffusion equations with monostable reaction terms f, including in particular the classical Fisher-KPP nonlinearities. The nonnegative initial data u 0(x) are chiefly assumed to be exponentially bounded as x tends to + ∞ and separated away from the unstable steady state 0 as x tends to − ∞. On the one hand, we give some conditions on u 0 which guarantee that, for some λ > 0, the quantity c λ = λ +f′(0)/λ is the asymptotic spreading speed, in the sense that lim t→+∞ u(t, ct) = 1 (the stable steady state) if c < c λ and lim t→+∞ u(t, ct) = 0 if c > c λ. These conditions are fulfilled in particular when u 0(x) e λx is asymptotically periodic as x → + ∞. On the other hand, we also construct examples where the spreading speed is not uniquely determined. Namely, we show the existence of classes of initial conditions u 0 for which the ω-limit set of u(t, ct + x) as t tends to + ∞ is equal to the whole interval [0, 1] for all x ∈ ℝ and for all speeds c belonging to a given interval (γ1, γ2) with large enough γ1 < γ2 ≤ + ∞.

Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity

This paper is concerned with the spreading speeds and traveling wave solutions of a nonlocal dispersal equation with degenerate monostable nonlinearity. We first prove that the traveling wave solution ϕ(ξ) with critical minimal speed c=c⁎ decays exponentially as ξ→−∞, while other traveling wave solutions ϕ(ξ) with c>c⁎ do not decay exponentially as ξ→−∞. Then the monotonicity and uniqueness (up to translation) of traveling wave solution with critical minimal speed is established. Finally, we prove that the critical minimal wave speed c⁎ coincides with the asymptotic speed of spread.

Impact of Directed Movement on Invasive Spread in Periodic Patchy Environments

To address the effect of taxis of invasive animals on their spreading speed in heterogeneous environments, we deal with an advection-diffusion-reaction equation (ADR) in a periodic patchy environment. Two-types of advection that spatially vary depending on environmental heterogeneity are taken into consideration: a stepwise taxis function and a saw-like taxis function. We first analyze the ADR with the stepwise taxis advection, and derive an invasion criterion. When the invasion criterion holds, an initially localized population evolves to a traveling periodic wave (TPW). The asymptotic speed of the TPW is found to be equal to the minimal speed of the TPW analytically derived. Thus, we examine how the minimal speed is influenced by the taxis. The major results are: (1)As the magnitude of the taxis toward favorable patches increases, invasion becomes more feasible. However, the spreading speed increases at first, and then decreases to show a one-humped curve against the magnitude of the taxis; (2)As the scale of fragmentation in the patchy environment is increased, the spreading speed increases when the magnitude of the taxis is small, while it decreases when the magnitude of the taxis becomes sufficiently large. These characteristic features qualitatively apply to the ADR model with the saw-like taxis function.

Adapting Predictive Feedback Chaos Control for Optimal Convergence Speed

Stabilizing unstable periodic orbits in a chaotic invariant set not only reveals information about its structure but also leads to various interesting applications. For the successful application of a chaos control scheme, convergence speed is of crucial importance. Here we present a predictive feedback chaos control method that adapts a control parameter online to yield optimal asymptotic convergence speed. We study the adaptive control map both analytically and numerically and prove that it converges at least linearly to a value determined by the spectral radius of the control map at the periodic orbit to be stabilized. The method is easy to implement algorithmically and may find applications for adaptive online control of biological and engineering systems.

Asymptotic spreading fastened by inter-specific coupled nonlinearities: A cooperative system

This paper is concerned with the asymptotic spreading of a Lotka–Volterra cooperative system. By using the theory of asymptotic spreading of nonautonomous equations, the asymptotic speeds of spreading of unknown functions formulated by a coupled system are estimated. Our results imply that the asymptotic spreading of one species can be significantly fastened by introducing a mutual species, which indicates the role of cooperation described by the coupled nonlinearities.

A WEAK LIMIT SHAPE THEOREM FOR PLANAR ISOTROPIC BROWNIAN FLOWS

It has been shown by various authors under different assumptions that the diameter of a bounded nontrivial set γ under the action of a stochastic flow grows linearly in time. We show that the asymptotic linear expansion speed if properly defined is deterministic, i.e. we show for a two-dimensional isotropic Brownian flow Φ with a positive Lyapunov exponent that there exists a non-random set [Formula: see text] such that we have for ϵ &gt; 0, arbitrary connected γ ⊂⊂ ℝ2 consisting of at least two different points and arbitrarily large times T that [Formula: see text]

Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations

We consider a branching-selection system in $\mathbb {R}$ with $N$ particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as $N\to\infty$, the empirical measure process associated to the system converges in distribution to a deterministic measure-valued process whose densities solve a free boundary integro-differential equation. We also show that this equation has a unique traveling wave solution traveling at speed $c$ or no such solution depending on whether $c\geq a$ or $c<a$, where $a$ is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of Wiener-Hopf equations.

A periodic reaction-diffusion model with a quiescent stage

In this paper, we investigate the asymptotic behaviour for a periodic reaction-diffusion model with a quiescent stage. By appealing to the theory of asymptotic speeds of spread and traveling waves for monotone periodic semiflow, we establish the existence of the spreading speed and show that it coincides with the minimal wave speed for monotone periodic traveling waves. Finally, we consider the case where the spatial domain is bounded. A threshold result on the global attractivity of either zero or a positive periodic solution are established.

ORBIT-BASED DYNAMICAL MODELS OF THE SOMBRERO GALAXY (NGC 4594)

We present axisymmetric, orbit-based models to study the central black hole, stellar mass-to-light ratio, and dark matter halo of NGC 4594 (M104, the Sombrero Galaxy). For stellar kinematics, we use published high-resolution kinematics of the central region taken with the Hubble Space Telescope, newly obtained Gemini long-slit spectra of the major axis, and integral field kinematics from the SAURON instrument. At large radii, we use globular cluster kinematics to trace the mass profile and apply extra leverage to recovering the dark matter halo parameters. We find a black hole of mass M_{\bullet}=(6.6 +/- 0.4) x 10^8 M_{\odot}, and determine the stellar M/L_I=3.4 +/- 0.05 (uncertainties are the 68% confidence band marginalized over the other parameters). Our best fit dark matter halo is a cored logarithmic model with asymptotic circular speed V_c=376 +/- 12 km/s and core radius r_c= 4.7 +/- 0.6 kpc. The fraction of dark to total mass contained within the half-light radius is 0.52. Taking the bulge and disk components into account in our calculation of \sigma_e puts NGC 4594 squarely on the M-\sigma relation. We also determine that NGC 4594 lies directly on the M-L relation.

Crossing velocities for an annealed random walk in a random potential

We consider a random walk in an i.i.d. non-negative potential on the d -dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location y on the lattice. We prove that the expected time under the annealed path measure needed by the random walk to reach y grows only linearly in the distance from y to the origin. In dimension 1 we show the existence of the asymptotic positive speed.

Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction–diffusion equations

We consider front propagation in a family of scalar reaction–diffusion equations in the asymptotic limit where the polynomial degree of the potential function tends to infinity. We investigate the Gevrey properties of the corresponding critical propagation speed, proving that the formal series expansion for that speed is Gevrey-1 with respect to the inverse of the degree. Moreover, we discuss the question of optimal truncation. Finally, we present a reliable numerical algorithm for evaluating the coefficients in the expansion with arbitrary precision and to any desired order, and we illustrate that algorithm by calculating explicitly the first ten coefficients. Our analysis builds on results obtained previously in [F. Dumortier, N. Popović, T.J. Kaper, The asymptotic critical wave speed in a family of scalar reaction–diffusion equations, J. Math. Anal. Appl. 326 (2) (2007) 1007–1023], and makes use of the blow-up technique in combination with geometric singular perturbation theory and complex analysis, while the numerical evaluation of the coefficients in the expansion for the critical speed is based on rigorous interval arithmetic.

Asymptotic speed of propagation and traveling wavefronts for a lattice vector disease model

In this paper, we study a lattice vector disease model with infinite delay and global spatial interaction. We obtain the existence of solutions of the initial value problem, the asymptotic speed of propagation and the strictly monotonic traveling waves for the system. We also confirm that the asymptotic speed of propagation coincides with the minimal wave speed for traveling wavefronts. Furthermore, some asymptotic behaviors for the traveling waves as ξ = n + c t → − ∞ are described.

The TASEP speed process

In the multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of ℤ is occupied by a particle labeled with some number, and two neighboring particles are interchanged at rate one if their labels are in increasing order. Consider the process with the initial configuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on [−1, 1]. We study the joint distribution of these speeds: the TASEP speed process. We prove that the TASEP speed process is stationary with respect to the multi-type TASEP dynamics. Consequently, every ergodic stationary measure is given as a projection of the speed process measure. This generalizes previous descriptions restricted to finitely many classes. By combining this result with known stationary measures for TASEPs with finitely many types, we compute several marginals of the speed process, including the joint density of two and three consecutive speeds. One striking property of the distribution is that two speeds are equal with positive probability and for any given particle there are infinitely many others with the same speed. We also study the partially asymmetric simple exclusion process (ASEP). We prove that the states of the ASEP with the above initial configuration, seen as permutations of ℤ, are symmetric in distribution. This allows us to extend some of our results, including the stationarity and description of all ergodic stationary measures, also to the ASEP.

Non-linear determinacy of minimum wave speed for a Lotka–Volterra competition model

For a reaction–diffusion system that serves as a 2-species Lotka–Volterra diffusive competition model, suppose that the corresponding reaction system has one stable boundary equilibrium and one unstable boundary equilibrium. Then it is well known that there exists a positive number c ⁎ , called the minimum wave speed, such that, for each c larger than or equal to c ⁎ , the reaction–diffusion system has a positive traveling wave solution of wave speed c connecting these two equilibria if and only if c ⩾ c ⁎ . It has been shown that the minimum wave speed for this system is identical to another important quantity – the asymptotical speed of population spread towards the stable equilibrium. Hence to find the minimum wave speed c ⁎ not only is of the interest in mathematics but is of the importance in application. It has been conjectured that the minimum wave speed can be determined by studying the eigenvalues of the unstable equilibrium, called the linear determinacy. In this paper we will show that the conjecture on the linear determinacy is not true in general.

Solar winds along curved magnetic field lines

Both remote-sensing measurements using the interplanetary scintillation (IPS) technique and in situ measurements by the Ulysses spacecraft show a bimodal structure for the solar wind at solar minimum conditions. At present what makes the fast wind fast and the slow wind slow still remains to be answered. While a robust empirical correlation exists between the coronal expansion rate $f_c$ of the flow tubes and the speeds $v$ measured in situ, further data analysis suggests that $v$ depends on more than just $f_c$. We examine whether the non-radial shape of field lines, which naturally accompanies any non-radial expansion, could be an additional geometrical factor. We solved the transport equations incorporating the heating due to turbulent Alfv\'en waves for an electron-proton solar wind along curved field lines given by an analytical magnetic field model, representative of a solar minimum corona. The field line shape is found to influence substantially the solar wind parameters, reducing the asymptotic speed by up to $\sim 130$ km s$^{-1}$, or by $\sim 28%$ in relative terms, compared with the case neglecting the field line curvature. This effect was interpreted in the general framework of energy addition in the solar wind: Relative to the straight case, the field line curvature enhances the effective energy deposition to the subsonic flow, resulting in a higher proton flux and a lower terminal proton speed. Our computations suggest that the field line curvature could be a geometrical factor which, in addition to the tube expansion, substantially influences the solar wind speed. Furthermore, at solar minima although the field line curvature unlikely affects the polar fast solar wind, it does help make the wind at low latitudes slow, thereby helping better reproduce the Ulysses measurements.

Approximation by finitely supported measures

Given a compactly supported probability measure on a Riemannian manifold, we study the asymptotic speed at which it can be approximated (in Wasserstein distance of any exponent p) by finitely supported measure. This question has been studied under the names of ``quantization of distributions'' and, when p=1, ``location problem''. When p=2, it is linked with Centroidal Voronoi Tessellations.

General relativistic magnetohydrodynamic Bondi-Hoyle accretion

In this paper we present a fully relativistic study of axisymmetric magnetohydrodynamic Bondi--Hoyle accretion onto a moving Kerr black hole. The equations of general relativistic magnetohydrodynamics are solved using high resolution shock capturing methods. In this treatment we consider the ideal MHD limit. The parameters of interest in this study are the adiabatic constant $\Gamma$, the asymptotic speed of sound $c_{s}^{\infty}$, and the plasma beta parameter $\beta_{P}$. We focus the investigation on the parameter regime in which the flow is supersonic, or when $v_\infty \ge c_{s}^{\infty}$. In some cases, subsonic asymptotic flows are considered for comparison purposes. We study the accretion rates of the total energy and momenta, as well as the hydrodynamic energy and momentum accretion rates. The models presented in this study exhibit a matter density depletion in the downstream region of the black hole which tends to vacuum $(\rho_0=0)$ in convergence tests. This feature is due to the presence of the magnetic field, more specifically the magnetic pressure, and is not seen in previous purely hydrodynamic studies.

Spreading–vanishing dichotomy in a diffusive logistic model with a free boundary, II

We study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For simplicity, we assume that the environment and the solution are radially symmetric. In the special case of one space dimension and homogeneous environment, this free boundary problem was investigated in Du and Lin (2010) [10]. We prove that the spreading–vanishing dichotomy established in Du and Lin (2010) [10] still holds in the more general and ecologically realistic setting considered here. Moreover, when spreading occurs, we obtain best possible upper and lower bounds for the spreading speed of the expanding front. When the environment is asymptotically homogeneous at infinity, these two bounds coincide. Our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension.

Semiempirical Models of the Slow and Fast Solar Wind

Coronal holes can produce several types of solar wind with a variety of compositional properties, depending on the location and strength of the heating along their open magnetic field lines. High-speed wind is associated with (relatively) slowly diverging flux tubes rooted in the interiors of large holes with weak, uniform footpoint fields; heating is spread over a large radial distance, so that most of the energy is conducted outward and goes into accelerating the wind rather than increasing the mass flux. In the rapidly diverging open fields present at coronal hole boundaries and around active regions, the heating is concentrated at low heights and the temperature maximum is located near the coronal base, resulting in high oxygen freezing-in temperatures and low asymptotic wind speeds. Polar plumes have a strong additional source of heating at their bases, which generates a large downward conductive flux, raising the densities and enhancing the radiative losses. The relative constancy of the solar wind mass flux at Earth reflects the tendency for the heating rate in coronal holes to increase monotonically with the footpoint field strength, with very high mass fluxes at the Sun offsetting the enormous flux-tube expansion in active region holes. Although coronal holes are its main source, slow wind is also released continually from helmet streamer loops by reconnection processes, giving rise to plasma blobs (small flux ropes) and the heliospheric plasma sheet.

A viscosity solution method for the spreading speed formula in slowly varying media

In this paper, we consider reaction-diffusion-advection equations in slowly periodi-cally oscillating media. We prove the existence of and we give explicit expressions of the asymptotic spreading speeds of invasion of the unstable state 0 in any direction, when the period of the invaded medium becomes infinitely large. The limiting sprea-ding speeds involve families of 1-periodic Hamilton-Jacobi equations. In the case of one-dimensional reaction-diffusion equations, we analyze the relative effects of small perturbations of the diffusion and the reaction coefficients, and we compare the spreading speeds in slowly oscillating media to the homogenized spreading speeds in rapidly oscillating media.