Distinguishing two Population Processes with Identical Equilibrium Densities
Eric Jakeman · AIP conference proceedings · 2003
We analyze the relationship between the evolution of simple population processes and the rate of emigration of individuals. An external monitoring scheme is defined by counting the number leaving the population in fixed time intervals. This is the analogue of photon counting in quantum optics. It is a reasonable measurement in many situations of interest and also has the merit of being analytically tractable. The formalism we develop is used to investigate the statistical and correlation properties of two stochastic population models that give rise to identical first order probability densities. The first is the birth‐death‐ immigration process for which many well‐known results can be found in the literature. The second is based on a population sustained by multiple immigration. This model is a generalization of the pair process investigated previously [1]. It can be used to generate populations with a range of equilibrium densities including those with power law tails to be described in a companion paper. Here we show that, in the case of a geometric distribution of multiples, the equilibrium density is negative binomial and higher order joint statistical properties must be used to distinguish the model from the conventional birth‐death‐immigration process. Formulae characterizing the integrated counting statistics of the two models are derived and it is shown how they may be exploited to achieve this objective.