I am reading a recently released book entitled Branching Processes: Variation, Growth, and Extinction of Populations by Haccou, Jagers, and Vatutin. One key concern of the book is the connection between population models and discrete dynamical systems.
They authors differentiate between individual level versus population level models. Individual level models try to model the behaviour of individual agents in the system using ideas from stochastic processes and branching processes. From these individual level models, stochastic process theory provides methods to estimate population level behaviors. Alternatively, discrete and even continuous dynamical systems model actually model directly at the population level using differential equations. The claim is that modeling at the population level directly makes some assumptions and can lead to divergences between the predictions of both types of models.
My question is, can anyone tell me if there is a stochastic process/branching model that is equivalent to the Logistic Growth equation? I have a quote from the book below, but the author make an explicit example of logistic growth, but they don't provide any citation or analysis. Hence I wanted to experiment with or derive this behavior myself.
The models we consider are individual-based. They start from descriptions of, or assumptions about, individual life and reproduction, and deduce the behavior of populations. Such models are sometimes called mechanistic, and the whole ap- proach reductionist, since properties of populations are brought back to the under- lying mechanisms of individual life. Population models can also be based directly on phenomena that appear at the population level; these are called phenomeno- logical. For instance, the effects of population density may be hard to describe at the individual level but established much more easily at the population level; an example is the well known phenomenon of “logistic growth.”(Haccou, et al., p.4)
So the logistic growth equation, in discrete form, is:
$$ x_{n+1} = rx_n(1 - \frac{Nx_n}{K}) $$
I am trying to figure out what the suitable equivalent branching model is to represent the logistic growth model? Would this be some sort of density dependent Galton-Watson process? Is that the right model or am I missing something. I am also not sure where the $r$ parameter would fit in the branching process. Would that be something as below?
$$ Z_{n+1} = \sum_{j=1}^{Z_n}X_{j,n}(\frac{Z_n}{K}) $$
Then would we just follow the same derivation for the Galton-Watson process using generating functions, or is there a better way to deal with having that $Z_n$ term in the sum.
There is no such thing as "stochastic equivalent" of the deterministic logistic equation. The reason is that the stochastic counterpart is usually formulated as a birth and death process, and the birth rates and death rates can be chosen in different (and nonequivalent ways), and yet the deterministic limit in the case of infinite population size is the same. For more details you can read Extinction and quasi-stationarity in the stochastic logistic SIS model.