THE MARKOV BRANCHING RANDOM-WALK AND SYSTEMS OF REACTION-DIFFUSION (KOLMOGOROV-PETROVSKII-PISKUNOV) EQUATIONS

A general model of a branching random walk in R(1) is considered, with several types of particles, where the branching occurs with probabilities determined by the type of a parent particle. Each new particle starts moving from the place where it was born, independently of other particles. The distribution of the displacement of a particle, before it splits, depends on its type. A necessary and sufficient condition is given for the random variable X(0) = sup max X(n,k) n greater than or equal to 0 1 less than or equal to k less than or equal to N-n to be finite. Here, X(n,k) is the position of the k(th) particle in the n(th) generation, N-n is the number of particles in the n(th) generation (regardless of their type). It turns out that the distribution of X(0) gives a minimal solution to a natural system of stochastic equations which has a linearly ordered continuum of other solutions. The last fact is used for proving the existence of a monotone travelling-wave solution to systems of coupled non-linear parabolic PDE's.