Is there anything known about the properties of birth-death processes where member organisms inherit birth rates with some noise?
If the birth rate per unit of time per organism is $\lambda$ and the death rate is $\mu$, then the expected number of offspring is $e^{\lambda-\mu}$, which explodes when $\lambda>\mu$. If new organisms instead inherit the birth rate from their ancestors with some noise, $\lambda' \leftarrow \lambda + N(0,\sigma^2)$ there will be a tendency to evolve towards higher $\lambda$. But if $\mu$ is large enough and $\sigma^2$ too small this evolution will rarely go far enough to get into the explosive $\lambda>\mu$ region if one starts with $\lambda<\mu$. Is there analysis of this or a similar case, especially giving the conditions on when runaway growth becomes likely?
Looking at the average $\lambda(t)$ suggest that it behaves like a biased random walk, but I suspect there is more structure to it due to correlations between related individuals.
Addendum: Looking at it in continuous time and space one can model it as $$\frac{dN(\lambda,t)}{dt} = -\mu N(\lambda,t) + \int_0^\infty u N(u,t) f(\lambda - u) du$$ where $f(x)$ is the normal PDF. This eventually blows up no matter how close to $\lambda=0$ one starts the population. Yet the time until exponential blow-up can become arbitrarily long and the population dip before arbitrarily close to 0, so this result only applies in the continuous setting. For discrete populations the dynamics becomes different since populations can actually crash.
