Asymptotic Speed Of Propagation Research Articles

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.

Influence of the mode of reproduction on dispersal evolution during species invasion

We consider a reaction-diffusion-reproduction equation, modeling a population which is spatially heterogeneous. The dispersion of each individuals is influenced by its phenotype. In the literature, the asymptotic propagation speed of an asexual population has already been rigorously determined. In this paper we focus on the difference between the asexual reproduction case, and the sexual reproduction case, involving a non-local term modeling the reproduction. This comparison leads to a different invasion speed according to the reproduction. After a formal analysis of both cases, leading to a heuristic of the asymptotic behaviour of the invasion fronts, we give some numerical evidence that the acceleration rate of the spatial spreading of a sexual population is slower than the acceleration rate of an asexual one. The main difficulty to get sharper results on a transient comes from the non-local sexual reproduction term.

Asymptotic Spreading for General Heterogeneous Fisher-KPP Type Equations

In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous reaction-diffusion equations: ∂ t u − ∑ i , j = 1 N a i , j ( t , x ) ∂ i j u − ∑ i = 1 N q i ( t , x ) ∂ i u = f ( t , x , u ) . \begin{equation*} \partial _{t} u - \sum _{i,j=1}^N a_{i,j}(t,x)\partial _{ij}u-\sum _{i=1}^N q_i(t,x)\partial _i u=f(t,x,u). \end{equation*} These are concerned with the dynamics of the solution starting from initial data with compact support. The nonlinearity f f is of Fisher-KPP type, and admits 0 0 as an unstable steady state and 1 1 as a globally attractive one (or, more generally, admits entire solutions p ± ( t , x ) p^\pm (t,x) , where p − p^- is unstable and p + p^+ is globally attractive). Here, the coefficients a i , j , q i , f a_{i,j}, q_i, f are only assumed to be uniformly elliptic, continuous and bounded in ( t , x ) (t,x) . To describe the spreading dynamics, we construct two non-empty star-shaped compact sets S _ ⊂ S ¯ ⊂ R N \underline {\mathcal {S}}\subset \overline {\mathcal {S}} \subset \mathbb {R}^N such that for all compact set K ⊂ i n t ( S _ ) K\subset \mathrm {int}(\underline {\mathcal {S}}) (resp. all closed set F ⊂ R N ∖ S ¯ F\subset \mathbb {R}^N\backslash \overline {\mathcal {S}} ), one has lim t → + ∞ sup x ∈ t K | u ( t , x ) − 1 | = 0 \lim _{t\to +\infty } \sup _{x\in tK} |u(t,x)-1| = 0 (resp. lim t → + ∞ sup x ∈ t F | u ( t , x ) | = 0 \lim _{t\to +\infty } \sup _{x\in tF} |u(t,x)| =0 ). The characterizations of these sets involve two new notions of generalized principal eigenvalues for linear parabolic operators in unbounded domains. In particular, it allows us to show that S ¯ = S _ \overline {\mathcal {S}}=\underline {\mathcal {S}} and to establish an exact asymptotic speed of propagation in various frameworks. These include: almost periodic, asymptotically almost periodic, uniquely ergodic, slowly varying, radially periodic and random stationary ergodic equations. In dimension N N , if the coefficients converge in radial segments, again we show that S ¯ = S _ \overline {\mathcal {S}}=\underline {\mathcal {S}} and this set is characterized using some geometric optics minimization problem. Lastly, we construct an explicit example of non-convex expansion sets.

Using effective boundary conditions to model fast diffusion on a road in a large field

We consider a logistic diffusion equation on the plane consisting of two components, a ‘road’ that is parallel to the x-axis, and a ‘field’, in each of which the diffusion rate differs significantly. Compared to the size of the field, the width δ of the road is assumed to be small. Thus in this diffusion equation multiple scales appear in two places: the spatial variable and the diffusion parameter. Such an equation is not easy to solve numerically, and it is not easy to see the effects of the road. Recently, Berestycki, Roquejoffre and Rossi provide a model which is meant to resolve these issues. In this paper we first use the idea of effective boundary conditions (EBCs) to propose, rigorously, a different model: we study the limit of the solution of the original logistic equation as , obtaining a limiting model, in which the road now is the x-axis with EBCs imposed on it. This effective problem has no multiple scales and hence should be easier to solve numerically. Moreover, to see the effects of the road, we further investigate the asymptotic propagation speed of the effective model, showing that the road indeed enhances the spreading speed along its direction, provided that the diffusion rate on the road is of order .

The Effect on Fisher-KPP Propagation in a Cylinder with Fast Diffusion on the Boundary

In this paper we consider a reaction-diffusion equation of Fisher-KPP type inside an infinite cylindrical domain in $\mathbb{R}^{N+1}$, coupled with a reaction-diffusion equation on the boundary of the domain, where potentially fast diffusion is allowed. We will study the existence of an asymptotic speed of propagation for solutions of the Cauchy problem associated with such a system, as well as the dependence of this speed on the diffusivity at the boundary and the amplitude of the cylinder. When $N=1$ the domain reduces to a strip between two straight lines. This models the effect of two roads with fast diffusion on a strip-shaped field bounded by them.

On the preservation of cooperation in two-strategy games with nonlocal interactions

Nonlocal interactions such as spatial interaction are ubiquitous in nature and may alter the equilibrium in evolutionary dynamics. Models including nonlocal spatial interactions can provide a further understanding on the preservation and emergence of cooperation in evolutionary dynamics. In this paper, we consider a variety of two-strategy evolutionary spatial games with nonlocal interactions based on an integro-differential replicator equation. By defining the invasion speed and minimal traveling wave speed for the derived model, we study the effects of the payoffs, the selection pressure and the spatial parameter on the preservation of cooperation. One of our most interesting findings is that, for the Prisoners Dilemma games in which the defection is the only evolutionary stable strategy for unstructured populations, analyses on its asymptotic speed of propagation suggest that, in contrast with spatially homogeneous games, the cooperators can invade the habitat under proper conditions. Other two-strategy evolutionary spatial games are also explored. Both our theoretical and numerical studies show that the nonlocal spatial interaction favors diversity in strategies in a population and is able to preserve cooperation in a competing environment. A real data application in a virus mutation study echoes our theoretical observations. In addition, we compare the results of our model to the partial differential equation approach to demonstrate the importance of including non-local interaction component in evolutionary game models.

On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ<sup>3</sup>

This paper is concerned with the following two-species Lotka-Volterra competition-diffusion system in the three-dimensional spatial space {partial derivative/partial derivative tu(1) (x, t) = Delta u(1) (x, t) + u(1) (x, t) [1 - u(1) (x, t) - k(1)u(2) (x, t)] , {partial derivative/partial derivative tu(2) (x, t) = d Delta u(2) (x, t) + ru(2) (x, t) [1 - u(2) (x, t) - k(2)u(1) (x, t)] , where x is an element of R-3 and t > 0. For the bistable case, namely k(1) , k(2) > 1, it is well known that the system admits a one-dimensional monotone traveling front Phi(x + ct) = (Phi(1) (x + ct), Phi(2) (x + ct)) connecting two stable equilibria E-u = (1, 0) and E-v = (0, 1), where c is an element of R is the unique wave speed. Recently, two-dimensional V-shaped fronts and high-dimensional pyramidal traveling fronts have been studied under the assumption c > 0. In this paper it is shown that for any s > c > 0, the system admits axisymmetric traveling fronts psi(x', x(3) + st) = Phi(1)(x ', x(3) + st), Phi(2)(x ', x(3) + st) in R-3 connecting E-u = (1, 0) and E-v = (0, 1), where x' is an element of R-2. Here an axisymmetric traveling front means a traveling front which is axially symmetric with respect to the x 3-axis. Moreover, some important qualitative properties of the axisymmetric traveling fronts are given. When s tends to c, it is proven that the axisymmetric traveling fronts converge locally uniformly to planar traveling wave fronts in R-3. The existence of axisymmetric traveling fronts is obtained by constructing a sequence of pyramidal traveling fronts and taking its limit. The qualitative properties are established by using the comparison principle and appealing to the asymptotic speed of propagation for the resulting system. Finally, the nonexistence of axisymmetric traveling fronts with concave/convex level set is discussed.

Asymptotic Speeds of Spread for a Nonlocal Diffusion Equation

The current paper is devoted to the study of spatial spreading dynamics of a class of nonlocal diffusion equation. It is known that there exists a critical speed \(c^{*}>0\) such that this nonlocal diffusion equation has a unique regular traveling wave solution if and only if \(c>c^{*}\). In this paper we show that this \(c^{*}\) is the asymptotic speed of propagation.

Asymptotic speed of propagation for a viscous semilinear parabolic equation

We characterize the asymptotic speed of propagation of almost planar solutions to a semilinear viscous parabolic equation, with periodic nonlinearity.

Propagation in a non local reaction diffusion equation with spatial and genetic trait structure

We study existence and uniqueness of traveling fronts, and asymptotic speed of propagation for a non local reaction diffusion equation with spatial and genetic trait structure.

The Shape of Expansion Induced by a Line with Fast Diffusion in Fisher-KPP Equations

We establish a new property of Fisher-KPP type propagation in a plane, in the presence of a line with fast diffusion. We prove that the line enhances the asymptotic speed of propagation in a cone of directions. Past the critical angle given by this cone, the asymptotic speed of propagation coincides with the classical Fisher-KPP invasion speed. Several qualitative properties are further derived, such as the limiting behaviour when the diffusion on the line goes to infinity.

Propagation speed in a strip bounded by a line with different diffusion

In this paper we consider a model for the diffusion of a population in a strip-shaped field, where the growth of the species is governed by a Fisher–KPP equation and which is bounded on one side by a road where the species can have a different diffusion coefficient. Dirichlet homogeneous boundary conditions are imposed on the other side of the strip. We prove the existence of an asymptotic speed of propagation which is greater than the one of the case without road and study its behavior for small and large diffusions on the road. Finally we prove that, when the width of the strip goes to infinity, the asymptotic speed of propagation approaches the one of a half-plane bounded by a road, case that has been recently studied in [2,3].

Nonlinear Diffusion Problems with Free Boundaries: Convergence, Transition Speed, and Zero Number Arguments

This paper continues the investigation of [Du and Lou, J. Eur. Math. Soc. (JEMS), arXiv:1301.5373, 2013], where the long-time behavior of positive solutions to a nonlinear diffusion equation of the form $u_t=u_{xx}+f(u)$ for $x$ over a varying interval $(g(t), h(t))$ was examined. Here $x=g(t)$ and $x=h(t)$ are free boundaries evolving according to $g'(t)=-\mu u_x(t, g(t))$, $h'(t)=-\mu u_x(t,h(t))$, and $u(t, g(t))=u(t,h(t))=0$. We answer several intriguing questions left open in that investigation. First we prove the conjectured convergence result for the general case that $f$ is $C^1$ and $f(0)=0$. Second, for bistable and combustion types of $f$, we determine the asymptotic propagation speed of $h(t)$ and $g(t)$ in the transition case. More presicely, we show that when the transition case happens, for bistable type of $f$ there exists a uniquely determined $c_1>0$ such that $\lim_{t\to\infty} h(t)/\ln t=\lim_{t\to\infty} -g(t)/\ln t=c_1$, and for combustion type of $f$, there exists a uniquely determi...

Asymptotic pattern for a periodic lattice model with age structure and global interaction

In this paper, we derive a time-periodic lattice model for a single species in a patchy environment, which has age structure and an infinite number of patches connected locally by diffusion. By appealing to the theory of asymptotic speed of propagation and monotonic periodic semiflows for travelling waves, we establish the existence of periodic travelling wave and spreading speed of the model.

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.

Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model

In this article, the well-posedness of the initial value problem, the existence of traveling wavefronts and the asymptotic speed of propagation for a SIR epidemic model with stage structure and nonlocal response are studied. We further show that the minimum wave speed in fact coincides with the asymptotic speed of propagation.

Asymptotic speed of propagation and traveling wave solutions for a lattice integral equation

We investigate the asymptotic speed of propagation and monotone traveling wave solutions for a lattice integral equation which is an epidemic model while the population is distributed on one-dimensional lattice Z . It is proved that the asymptotic speed of propagation c ∗ coincides with the minimal wave speed.

Asymptotic results for a multi-type contact birth–death process and related SIS epidemic

Exact results concerning the asymptotic speed of propagation of infection have recently been obtained for the multi-type SIS epidemic in continuous space when the contact distributions are assumed to be symmetric with the Laplace transforms finite for all entries. There is a link between the equations for this epidemic and the equations for a multi-type contact birth–death process. This enables methods developed for the epidemic to be used to obtain the asymptotic speed of translation for the contact birth–death process. Symmetry of the contact distributions is required but no existence constraint is placed on their Laplace transforms. The method for removing this constraint may also be used for the SIS epidemic. Results are given for both processes when the basic reproduction ratio is at most one.

Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation

In this paper, we study a very general non-local lattice differential equation with delay. We obtain the existence of the asymptotic speed of propagation, the existence and uniqueness of the traveling wavefront and the minimal speed of the traveling wavefront for the system. We also confirm that the asymptotic speed of propagation and the minimal speed of the traveling wavefront coincide.

Homogenization and flame propagation in periodic excitable media: The asymptotic speed of propagation

AbstractWe study the effect of homogenization on flame propagation in periodic excitable media when the width of the flame is much smaller than the characteristic size of the heterogeneities. © 2005 Wiley Periodicals, Inc.