Branching process

A stochastic process in which individuals independently produce offspring according to specified distributions. It models growing populations and gives an approximation to sparse percolation clusters when collisions between branches can be neglected. On a lattice, loops and overlapping branches prevent this approximation from being an exact identity.

A branching process with mean offspring number one. In a continuous-time birth–death example with equal per-cell birth and death rates $\beta$, one initial individual has survival probability $1/(1+\beta t)$ and mean size $1+\beta t$ conditional on survival.

For equal individual birth and death rates $\lambda$ and one initial individual, the probability generating function is $F(z,t)=1-(1-z)/[1+\lambda t(1-z)]$. It solves $F_t=\lambda (z-1{)}^2{F}_z$ with $F(z,0)=z$, and its geometric expansion gives the displayed probabilities. The mean remains one and the variance is $2\lambda t$, although extinction occurs with probability tending to one. The surviving fraction $1/(1+\lambda t)$ has conditional mean population $1+\lambda t$.

A particle of type $i$ has independently a Poisson distribution of type-$j$ offspring with mean ${\lambda }_{ij}$. Different particles reproduce independently. If $p_i^{(t)}$ is the probability generation $t$ is nonempty from one type-$i$ ancestor, then $p^{(0)}=1$ and $p^{(t+1)}=T(p^{(t)})$, where $T_i(x)=1-\exp [-(\Lambda x{)}_i]$. This iteration decreases to the survival probability of a branching process, the greatest nonnegative fixed point of $T$. The expected type-count row vector in generation $t$ is $e_i^T{\Lambda }^t$.

This criterion holds even for reducible nonnegative matrices. If a fixed point $p\ne 0$ exists, $1-e^{-z}<z$ for $z>0$ gives $(\Lambda p{)}_i>p_i$ on its support. Some $\alpha >1$ therefore satisfies $\Lambda p\ge \alpha p$, forcing the spectral radius above one by the Perron–Frobenius theorem. Conversely take $v\ge 0$, $v\ne 0$, with $\Lambda v=\rho v$ and $\rho >1$. For sufficiently small $\eta >0$, $1-e^{-\eta \rho v_i}\ge \eta v_i$. Thus $\eta v$ is a subsolution; monotone iteration under $T$ gives a positive fixed point below $1$. Strictness of $1-e^{-z}<z$ excludes survival at the critical value for this Poisson offspring law.