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Ipulatov

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- TL;DR Summary
- I'm doing a research project where I use birth-death processes to model population. But, I don't know what to do for when the population is so small, because I can't simulate an individual undergoing both a birth and death in the same time step. Because once it's dead, it should be dead. Does anyone know of a solution to this?

In birth-death processes for populations, what are solutions for preventing the same individual from both giving birth and dying simultaneously (meaning the same time-step) in your model?

I'll provide some context:

Let's say the timestep is a year. You might say - A certain percentage of individuals of a certain age will give birth and a certain percentage will die, and so I don’t need to care if any particular individual does one, both or neither; it’s the percentages that matter.

The problem with looking out for what happens after a year is that it potentially under-estimates the rate of extinction. This happens because a population size of zero is a special case that we need to treat differently. For example, let's look at a population of rabbits of size 1. Let's say they have averaged rates of 10 births/year and 5 deaths/year. In a population of size 1, clearly, the extinction rate is > 1/3 since there's is a 33.3% chance a death occurs first and there is a non-zero chance extinction occurs even if growth occurs first. However if we calculate Pois(10)-Pois(5) as a realization of the population size change, that value is less than zero 7.4% of the time. This gives a completely wrong extinction percent (>33.% vs 7.4%) since it is not taking into account the fact that anything that reaches a zero population size is extinct regardless of whatever is simulated to happen afterwards.

Another way to put this issue is that the distribution of the change in population size becomes skewed at small sizes since you can't treat size zero normally. The question we want answered is how we simulate this distribution.

I'll provide some context:

Let's say the timestep is a year. You might say - A certain percentage of individuals of a certain age will give birth and a certain percentage will die, and so I don’t need to care if any particular individual does one, both or neither; it’s the percentages that matter.

The problem with looking out for what happens after a year is that it potentially under-estimates the rate of extinction. This happens because a population size of zero is a special case that we need to treat differently. For example, let's look at a population of rabbits of size 1. Let's say they have averaged rates of 10 births/year and 5 deaths/year. In a population of size 1, clearly, the extinction rate is > 1/3 since there's is a 33.3% chance a death occurs first and there is a non-zero chance extinction occurs even if growth occurs first. However if we calculate Pois(10)-Pois(5) as a realization of the population size change, that value is less than zero 7.4% of the time. This gives a completely wrong extinction percent (>33.% vs 7.4%) since it is not taking into account the fact that anything that reaches a zero population size is extinct regardless of whatever is simulated to happen afterwards.

Another way to put this issue is that the distribution of the change in population size becomes skewed at small sizes since you can't treat size zero normally. The question we want answered is how we simulate this distribution.