Is this a non-homogeneous Poisson process?

Question

Consider the population growth model where $P'(t) = rP(t)$, where $P(t)$ is the population at time $t$ with the constant growth rate $r$. Modeling this as a homogeneous Poisson process, at time $t$, $P(t+1)$ can be obtained by simulating from a Poisson distribution with mean $\lambda = rP(t)$. So far so good.

Now, if $r$ is a function of $P(t)$ with constants $\alpha$ and $K$, such that $$ r(t) = \alpha \log(K/P(t)), $$ is it still a non-homogeneous Poisson process? If not what process this might follow?

If the process is not known, suppose we know $P(0)$, choosing initial $\alpha$ and $K$, can we simulate this as a homogeneous Poisson process starting with $r(0)$ and iterating $r$ and $P$ as the process evolves?

Thanks!

Answer 1

Inhomogeneous Poisson processes are typically understood to have the following features:

1. the number of points in each finite interval has a Poisson distribution,
2. the number of points in disjoint intervals are independent random variables,
3. the distributions depend on the time-varying (deterministic) rate $\lambda(t)$, instead of the stationary assumptions for the homogeneous Poisson process

Your specification seems to violate (2), as the number of points depend on the history of the process. You may want to have a look at cluster processes, where the offspring in your context would represent the points in a cluster. Hawkes processes are another class that is widely used which depend on the process history.

Regarding simulation, you should be able to simulate the process by using Ogata's modified thinning algorithm, as you can compute the intensity of the process at each point.

For more background on theory and the simulation algorithm, this is an excellent reference:

• Daley, D. J.; Vere-Jones, D., An introduction to the theory of point processes. Vol. I: Elementary theory and methods., Probability and Its Applications. New York, NY: Springer. xxi, 469 p. (2003). ZBL1026.60061.