Multitype branching processes with disasters I: The number of particles in the system

A multitype Markov branching process under the influence of disasters (catastrophes) arriving as a renewal process leading to possible mutation of particles to other types is considered. The integral equations for the probability generating function of the number of particles of each type are derived incorporating time dependent mutation rates. The asymptotic behaviours of the mean number of particles have been discussed for the general case. Explicit expressions for the mean number of particles of each type alive at time t are obtained for the Markovian case. Exponentially distributed disaster and constant mutation functions has been dealt with as a special case. The mean number of particles and the covariance matrix are obtained in explicit form for the reversible birth and death process with two types of particles. The results corresponding to Markov branching process involving only three types of particles are deduced. These models involving mutation have applications in the theory of carcinogenesis.