Critical branching process with two types of particles evolving in asynchronous random environments (Q2845213)

The model of branching processes in a random environment which is considered in the paper is a two-type pure decomposable version of the process introduced by \textit{W. L. Smith} and \textit{W. E. Wilkinson} [Ann. Math. Stat. 40, 814--827 (1969; Zbl 0184.21103)]. Each particle may produce offspring of its own type only and \(\exp(X_k(i))\) is the mean number of children produced by a particle of type \(i\) of generation \(k\). It is assumed that \(\{X_k(i), \,k=1,2,\dots\}\) are i.i.d. random variables and \(X_k(1)=-X_k(2)\), \(k=1,2,\dots\), with probability 1. It means that the processes are dependent through the environment only and a favorable environment for the first process (\(X_k(1)>0\)) is unfavorable for the second (\(X_k(2)<0\)) and vice versa.NEWLINENEWLINEUnder the assumption that the random walk \(X_1(1)+\dots+X_k(1)\) is oscillating, the joint conditional distribution of the number of particles in the population is studied at time points \(nt\), \(0<t<1\), given that both processes survive up to the time point \(n\) (\(n \to \infty\)).NEWLINENEWLINEUnder the same assumptions, the problem is considered at the time points when the environment is very unfavorable for the particles of the first type. At these points of time the number of particles of the first type is small while the number of particles of the second type is very large. It looks like bottlenecks and periods of growth in the model of predator-prey coexistence.