Fig. 3.

Monte–Carlo simulations of densities in age- and number-dependent birth-death processes. Column a shows results for a death-only process with a linear death rate function $\mathit{\mu }(a)=a$. We initiated all simulations from 10 individuals with initial age drawn from distribution $P(a)=128a^3{e}^{-4a}/3$. In column b, we consider a budding-only birth process with a carrying capacity $K=5$ (in Eq. 10). Here, simulations were initiated with a single parent individual with an initial age also drawn from the distribution P(a). In (i), we plot the total number density ${\mathit{\rho }}_n^{(0)}(t)=\int \mathrm{d}\mathbf{a}{\mathit{\rho }}_n(\mathbf{a};t)$ for both processes. We also plot the single-particle density function ${\mathit{\rho }}_{n=1,5,9}(a;t=2)$ for the pure death process in a(ii–iv) and ${\mathit{\rho }}_{n=1,3,5}(a;t=5)$ for the limited budding process in b(ii–iv). Finally, the population-summed two-point correlations functions ${\sum }_n{\mathit{\rho }}_n^{(2)}(a_1,a_2;t)$ for pure death and pure budding are shown in panels a(v) and b(v)