Asymptotic Speed Of Propagation Research Articles

A Delay Reaction-Diffusion Model of the Spread of Bacteriophage Infection

This paper is a continuation of recent attempts to understand, via mathematical modeling, the dynamics of marine bacteriophage infections. Previous authors have proposed systems of ordinary differential delay equations with delay dependent coefficients. In this paper we continue these studies in two respects. First, we show that the dynamics is sensitive to the phage mortality function, and in particular to the parameter we use to measure the density dependent phage mortality rate. Second, we incorporate spatial effects by deriving, in one spatial dimension, a delay reaction- diffusion model in which the delay term is rigorously derived by solving a von Foerster equation. Using this model, we formally compute the speed at which the viral infection spreads through the domain and investigate how this speed depends on the system parameters. Numerical simulations suggest that the minimum speed according to linear theory is the asymptotic speed of propagation. − KS(t)P (t), dI dt = −µiI(t )+ KS(t)P (t) − e −µiT KS(t − T )P (t − T ), dP dt = β − µpP (t) − KS(t)P (t )+ be −µiT KS(t − T )P (t − T ).

Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction

In this paper, we derive a lattice model for a single species in a one-dimensional patchy environment with infinite number of patches connected locally by diffusion. Under the assumption that the death and diffusion rates of the mature population are age independent, we show that the dynamics of the mature population is governed by a lattice delay differential equation with global interactions. We study the well-posedness of the initial-value problem and obtain the existence of monotone travelling waves for wave speeds c > c * . We show that the minimal wave speed c * is also the asymptotic speed of propagation, which depends on the maturation period and the diffusion rate of mature population monotonically.

Wavefronts and global stability in a time-delayed population model with stage structure

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

A Rigorous Upper Bound on the Propagation Speed for the Swift–Hohenberg and Related Equations

We prove that if the initial condition of the Swift–Hohenberg equation $$\partial _t u(x,t) = (\varepsilon ^2 - (1 + \partial _x^2 )^2 ){\text{ }}u(x,t) - u^3 (x,t)$$ is bounded in modulus by Ce −βx as x→+∞, the solution cannot propagate to the right with a speed greater than $$\mathop {\sup }\limits_{0 < {\gamma } \leqslant \beta } {\gamma }^{ - 1} (\varepsilon ^2 + 4{\gamma }^2 + 8{\gamma }^4 ).$$ This settles a long-standing conjecture about the possible asymptotic propagation speed of the Swift–Hohenberg equation. The proof does not use the maximum principle and is simple enough to generalize easily to other equations. We illustrate this with an example of a modified Ginzburg–Landau equation, where the critical speed is not determined by the linearization alone.

Counterexample to a conjecture of Goriely for the speed of fronts of the reaction-diffusion equation

In a recent paper Goriely considers the one--dimensional scalar reaction--diffusion equation $u_t = u_{xx} + f(u)$ with a polynomial reaction term $f(u)$ and conjectures the existence of a relation between a global resonance of the hamiltonian system $ u_{xx} + f(u) = 0$ and the asymptotic speed of propagation of fronts of the reaction diffusion equation. Based on this conjecture an explicit expression for the speed of the front is given. We give a counterexample to this conjecture and conclude that additional restrictions should be placed on the reaction terms for which it may hold.

On a System of Integrodifference Equations Modelling the Propagation of Genes

The author considers the gene spread of a certain diploid plant, one of whose gene loci has two alleles. A mathematical model is formulated as a system of integrodifference equations $[N^{(n + 1)} ,u^{(n + 1)} ] = Q[N^{(n)} ,u^{(n)} ]$, where Q is a system of nonlinear convolution integral operators. The dynamics of Q in the heterozygote intermediate case is considered in a one-dimensional habitat. When the initial disadvantageous gene is present only in a finite region of the habitat, the dynamics of Q tends to a constant equilibrium dominated by the advantageous gene. Under certain conditions, the system has traveling wave solutions with speed not less than a minimal wave speed. The method of the Laplace transform is used to determine the minimal wave speed explicitly. The Contraction Mapping Theorem is applied to prove the existence and the uniqueness of the traveling waves. Moreover, it is proved that the minimal wave speed is the asymptotic speed of propagation of the advantageous gene, and the dynam...

The asymptotic speed of propagation of the deterministic non-reducible n-type epidemic.

A model has been formulated in to describe the spatial spread of an epidemic involving n types of individuals, when triggered by the introduction of infectives from outside. Wave solutions for such a model have been investigated in and have been shown only to exist at certain speeds. This paper establishes that the asymptotic speed of propagation, as defined in Aronson and Weinberger, of such an epidemic is in fact c0, the minimum speed at which wave solutions exist. This extends the known result for the one-type and host-vector epidemics.

The rate of spread of infection in models for the deterministic host-vector epidemic

Wave solutions for models of the deterministic host-vector epidemic have been considered previously. Under certain conditions the existence of a critical speed c 0 was established. Wave solutions were shown to exist for all positive speeds c⩾ c 0, the solution at each speed being unique up to translation. If c 0>0, then no solution exists at any speed c for 0< c< c 0. This paper considers models for an epidemic which is triggered by the introduction of infectives from outside into the populations of hosts and vectors. It establishes that c 0 is the asymptotic speed of propagation of infection for these models.

Travelling wave solutions of diffusive Lotka-Volterra equations

We establish the existence of travelling wave solutions for two reaction diffusion systems based on the Lotka-Volterra model for predator and prey interactions. For simplicity, we consider only 1 space dimension. The waves are of transition front type, analogous to the travelling wave solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction diffusion equation. The waves discussed here are not necessarily monotone. For any speed c there is a travelling wave solution of transition front type. For one of the systems discussed here, there is a distinguished speed c* dividing the waves into two types, waves of speed c < c* being one type, waves of speed c ⩾ c* being of the other type. We present numerical evidence that for this system the wave of speed c* is stable, and that c* is an asymptotic speed of propagation in some sense. For the other system, waves of all speeds are in some sense stable. The proof of existence uses a shooting argument and a Lyapunov function. We also discuss some possible biological implications of the existence of these waves.

Run for your life. A note on the asymptotic speed of propagation of an epidemic

We study the large-time behaviour of the solution of a nonlinear integral equation of mixed Volterra-Fredholm type describing the spatio-temporal development of an epidemic. For this model it is known that there exists a minimal wave speed c 0 (i.e., travelling wave solutions with speed c exist if ¦c¦ > c 0 and do not exist if ¦c¦ < c 0 ). In this paper we show that c 0 is the asymptotic speed of propagation (i.e., for any c 1, c 2with 0 < c 1 < c 0 < c 2 the solution tends to zero uniformly in the region ¦x¦ ⩾ c 2t , whereas it is bounded away from zero uniformly in the region ¦x¦ ⩽ c 1t for t sufficiently large).

Perturbation Effects on the Decay of Discontinuous Solutions of Nonlinear First Order Wave Equations

The decay of discontinuous solutions of quasi-linear hyperbolic equations of the form \[ u_t + g(u)u_x + \lambda h(u) = 0,\quad \lambda > 0,\quad g_u (u) > 0,\quad h_u (u) > 0\quad {\text{for}}\,u > 0\], where subscripts denote differentiation and $g(u), h(u)$ are nonnegative, is considered from a pedagogical point of view. For a given finite initial disturbance, zero outside a finite range in x, it is shown that if $h(u) = O(u^\alpha ),\alpha > 0$, for $0 < u \ll 1$, the initial disturbance decays (i) within a finite time and finite distance for $0 < \alpha < 1$ and is unique under certain conditions, (ii) within an infinite time, like $O(\exp - \lambda t)$, and in a finite distance for $\alpha = 1$, and (iii) within an infinite time and distance like $O(t^{ - 1/(\alpha - 1)} )$ for $1 < \alpha \leqq 3$, and $O(t^{ - 1/2} )$ for $\alpha \geqq 3$. The asymptotic speed of propagation of the discontinuity is given in each case together with its role in the decay process. A boundary value problem for stress wave propagation in a mildly nonlinear Maxwell rod with a finite nonlinear viscous damping is considered. The first term in an asymptotic solution, in the magnitude of the wave speed nonlinearity, is shown to be equivalent to that of the above equation with all its implications on decay now applying to the stress wave in the rod. The effect of a higher order perturbation on decay is studied by considering, by way of example, the initial value problem for \[ u_t + g(u)u_x = \varepsilon u_{xx} ,\quad \varepsilon \ll 1,\quad g_u (u) > 0, \] with $g(u)$ nonnegative. For initial data comparable to the above the effect on the asymptotic speed of propagation for t large is found to be $O(\varepsilon t^{ - 1/ 2} \log t)$ while it is $O(\varepsilon t^{ - 1} \log t)$ for the decay of the solution.