CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 2, Number 1, Winter 1994

TRAVELLING FRONTS FOR CORRELATED RANDOM WALKS

K.P.HADELER

ABSTRACT. Scalar reaction diffusion equations describe Brownian motion and multiplication of particles. These equa- tions are well understood, in particular, asymptotic behavior of solutions and existence of travelling fronts. If Brownian motion is replaced by a correlated random walk, then one obtains nonlinear hyperbolic systems. The form of these sys- t e m depends eeeentially on the underlying assumptions. In some cases these systems can be reduced to single hyperbolic equations of which reaction diffusion equations appear aa limit cases. In other cases such reductions do not seem possible. In the special case of one hyperbolic equation and in the general case of a hyperbolic system the existence problem for travel- ling front solutions is studied in detail.

1. Derivation of the model. We consider particles which multiply and, at the same time, move in a spatial continuum. There are several ways to look at such a system. The heat equation (or diffusion equation) ut = u,, describes Brownian motion of a particle. The solution ~ ( t , x) is the probability density of the location of the particle at time t, i.e., the quantity J',u(t,x) dz is the probability that at time t the particle is located in the interval [a,b]. This solution has the properties u(t, x) 2 0, JvWw u(t, x) dx = 1. One can also consider the function v(t,x) = ST, u(t, y)dy. Then v(t,x) is the probability that at time t the position < of the particle satisfies the inequality < < x.

In another view the solution u(t, x) of the equation ut = u,, describes the quantity (or frequency) of some substance at time t at the location x, i.e., ~ ' ,u(t ,x) dx is the amount of matter in [a,b] at time t. In this case u(t, x) 2 0, and u need not be integrable.

Both concepts have been carried over to nonlinear problems. In Fisher's [7]considerations on the advance of advantageous traits in a population with a certain genetical structure the solution u(t, x) of the equation ut = uzz+u(l-u) describes the frequency of the advantageous

Received by the editor on August 27, 1992, and in revised form on July 9, 1993.

Copyright 01994 Rocky Mountain Mathematic. Consortium

28 K.P. HADELER

trait at the position x at time t. In the subsequent work of Kolmogorov, Petrovskij, Piskunov [20] (in the following quoted as KPP) u(t,z) is the density of matter which di&lses and grows at the same time, in the sense that J: u(t, x) dx is the amount of matter in the interval [a, b] at time t.

Many authors, e.g., Davis (31, have considered processes where parti- cles multiply according to a branching process and, at the same time, undergo Brownian motion. Then, at time t, there are n(t) particles with positions &(t), . . . , <n(t) (t) (notice that n and the & are ran- dom variables). McKean [21] has shown that the solution v(t, x) of vt = v,, + v(1- v) is the probability that, at time t, the inequality

< x holds for i = 1,. . . ,n(t).

In these models Brownian motion can be replaced by other concepts of motion, e.g., by a correlated random walk. Consider a particle which moves left or right on the real line, with constant absolute speed 7. The change of direction of the motion is described by a Poisson process. The probability that the particle changes direction in a time interval At is pAt + o(At). Thus the process depends on two parameters 7 ,p > 0. Apparently Taylor [25] and Goldstein [8] were the first to consider the probability density u(t, x) for the position of a moving particle. Goldstein investigated the motion on a grid and in the continuous limit case. He derived a telegraph equation utt + 2put = 72u,, for the probability density. We refer to Zauderer [27] for a detailed exposition of the linear case, and to Kac (171, Kaplan [19], Pinsky [23], Griego and Hersh [9], Janssen and Siebert [16] and Janssen [15] for more abstract treatments.

All views, the view of Fisher, and KPP, and of McKean, have been carried over to correlated random walks. Dunbar and Othmer [5] have considered a process where finitely many particles perform a correlated random walk and, at the same time, undergo a branching process. For the function v(t,x) which describes the probability that &(t) < x for i = 1,. . . , n(t) they derive a nonlinear telegraph equation of the form

(1.1) vtt + m(v)vt = a2v,, + f (v).

Dunbar [4] has given a rigorous derivation and analysis similar to McKean's [21] for the correlated random walk model. Othmer, Dunbar and Alt [22] have studied more general processes in several space dimensions.