The Markov branching random walk and systems of reaction-diffusion (Kolmogorov-Petrovskii-Piskunov) equations

A general model of a branching random walk inR 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 $$X^0 = \mathop {\sup max}\limits_{ n \geqq 0 1 \leqq k \leqq N_n } X_{n,k} $$ to be finite. Here,X _{n, k} is the position of thek ^th particle in then ^th generation,N _n is the number of particles in then ^th generation (regardless of their type). It turns out that the distribution ofX 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.