Birth-death processes in small populations

[Ipulatov]

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.

 

[fresh_42]

You cannot treat such small population sizes the same as big ones. Everything under a certain size, and certainly $0,1$ or $2$ do not behave "by percentages". When it comes down from population to a number of individuals, then properties like risk aversion, age, health, strength etc. do not level out anymore, they become the dominant factor. I don't know which model you are using, so the minimum size might depend on the model, but individual rates of birth and death are definitely different from an overall estimation. You can use Lotka -Volterra with small population sizes, however, I don't think it is very accurate. And your question doesn't exist for bigger populations.

For small populations you could consider Markov processes, as every individual directly affects the others, e.g. mating with one individual is not possible, same for sick individuals. Or death after giving birth leads to the death of the offspring. Such thoughts are irrelevant for bigger populations as they are included in the death rates.

 

[Ipulatov]

Aren't birth-death processes a specific type of markov process? Can you elaborate on what you mean by "For small populations you could consider Markov processes"?

 

[fresh_42]

Aren't birth-death processes a specific type of markov process? Can you elaborate on what you mean by "For small populations you could consider Markov processes"?

You haven't said how you model it. You can use predator-prey models, or Markov processes for individuals. If an individual dies, then your transition matrix should reflect this, i.e. the individual Markov chain ends here. You can model it by a multiplication with $0$. A life insurance company will probably not use Markov processes.

 

[StoneTemplePython]

I actually inferred / assumed that by birth-death you were doing a markov chain/process. Except your 2nd to last paragraph in OP reads like a non-sequitor.

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.

The most baffling part is why you compute Pois(10)-Pois(5) (or even what that means since neither $\lambda$ nor $k$ are specified).

It sounds like you're confusing a single communicating class (/possibly recurrent) chain formulation of birthdeath processes with what you want to be an absorbing chain which has two distinct classes -- the single absorbing node, and all the rest. E.g. the classic telecoms queuing formulation of birth death processes can go to state 0 calls in the queue and back up to one, two... etc. In your problem once something reaches state zero it is absorbed, never to be heard from again. It's a very different chain.

But for such a basic issue: I can't figure out why this thread is tagged A.

Recognizing birthdeath processes being the same as simple random walks (with a one sided absorbing barrier --or if you prefer gamblers ruin with infinitely rich adversary) then using whatever method you like for random walks seems prudent. I like martingale methods; generating function methods are popular too.

All that being said, the way your problem is actually described feels more like a branching process to me; it's closely related to random walks in any case.

 

[FactChecker]

Your simulation process is not clear to me. I would assume that in each step you calculate the deaths from the remainder after the prior step before you calculate and add in new births in this step.