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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?

N(t) as function of t for some realizations.

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.

Mean lambda as function of t. Red circles indicate explosive growth.

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.

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    $\begingroup$ What happens exactly if $\lambda$ falls to zero or below? Or are you drawing from a one-sided normal? $\endgroup$
    – Ian
    Jan 9, 2020 at 2:30
  • $\begingroup$ Yes, the integral form does cut the normal at 0. After all, negative rates make little sense. $\endgroup$ Jan 9, 2020 at 10:46
  • $\begingroup$ But the rate can fall as low as zero, and it can actually get there in the discrete time setting right? Also for conservation of probability you need to normalize the truncated normal. (A normal may not really be the right model here anyway.) $\endgroup$
    – Ian
    Jan 9, 2020 at 12:30
  • $\begingroup$ @Ian - Yes, a bit more careful normalisation is needed to make the formalization well-posed. It still looks like a discrete population has a fundamentally different and probably more interesting dynamics. $\endgroup$ Jan 9, 2020 at 23:18
  • $\begingroup$ I'm not sure if the structure is sort of "intrinsic", but if you label each newly born organism in the discrete time simulation with the $\lambda$ of its original ancestor then it could be interesting to track the survival times of particular "species". $\endgroup$
    – Ian
    Jan 10, 2020 at 0:47

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