Two logistic process models

In at least two places in this book, we discuss models that exhibit logistic growth patterns. As preparation, we describe two models that generate logistic curves as the expected values of the random number n - the number of agents of type 1 or choice 1, say.

4.4.1 Model 1: The aggregate dynamics and associated fluctuations

This model is described in Karlin and Taylor (1981, p. 144). We discuss a particular case of it with N₁ = 1 in their notation. See also Kendall (1949). Define the transition rates for the right move,

where n is the upper bound on k. (N₂ in Karlin and Taylor and in Kendall.) We may interpret the parameter λ as the constant of proportionality between the birth rate per individual and the deviation from the upper bound, n - k, and the parameter μ as that between the death rate per individual and the deviation from the floor, k — 1, when k is the number of individuals.

So far, this is a deterministic model. It may be converted into a stochastic model governed by the master equation

This master equation has the equilibrium distribution determined by

with 1 ≤ k ≤ n. The solution is

To obtain approximate solutions to the master equation as n goes to infinity, we change variables to

We have already seen several instances in which this type of change of variables was carried out. In this example we need to change the time scale as well, by

Define

and note that

where

In solving the master equation we again use the change of variables intro­duced in connection with Model 1.

One is to change the state variable from n to

where φ denotes the mean of the fraction of type 1 agents, and ξ denotes the fluctuation about the mean. We know from Aoki (1996a, p. 136) that the aggregate or average dynamics for φ is givvn by

where τ — nt, with α₀(φ) :— -ρ₀(φ) + γ₀(φ), and where

and

Exercise. Derive the ordinary differential equation for the mean and variance from the cumulant generating function. (First, obtain the partial differential equation for the probability generating function, G(z) say, from the master equation. Then,θ

4.5