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Critical birth-death processes with $n$ distinct types are investigated. Eachtype $i$ cell divides independently $(i)\to(i)+(i)$ or mutates $(i)\to(i+1)$ atthe same rate. The total number of cells grows exponentially as a Yule processuntil the maximal type $n$ cells appear, which cannot mutate but die at rateone. The last type makes the process critical and hence after the exponentiallygrowing phase eventually all cells die with probability one. The process mimicsthe accumulation of mutations in a growing population where too many mutationsare lethal. This has applications for understanding the mutational burden andso-called error catastrophe in cancer, bacteria, or virus. We presentlarge-time asymptotic results for the general $n$-type critical birth-deathprocess. We find that the mass function of the number of cells of type $k$ hasalgebraic and stationary tail $(\text{size})^{-1-\chi_k}$, with$\chi_k=2^{1-k}$, for $k=2,\dots,n$, in sharp contrast to the exponential tailof the first type. The same exponents describe the tail of the asymptoticsurvival probability $(\text{time})^{-\chi_n}$. We discuss the consequences andapplications of the results for studying extinction due to mutational burden inbiological populations.

Multi-type critical process: a birth-death model of error catastrophe