Asymptotic behaviour of extinction probability of interacting branching collision processes

Abstract

Although the exact expressions for the extinction probabilities of the Interacting Branching Collision Processes (IBCP) were very recently given by Chen et al. [4], some of these expressions are very complicated; hence, useful information regarding asymptotic behaviour, for example, is harder to obtain. Also, these exact expressions take very different forms for different cases and thus seem lacking in homogeneity. In this paper, we show that the asymptotic behaviour of these extremely complicated and tangled expressions for extinction probabilities of IBCP follows an elegant and homogenous power law which takes a very simple form. In fact, we are able to show that if the extinction is not certain then the extinction probabilities {a_n} follow an harmonious and simple asymptotic law of a_n ∼ kn^-αρ_cⁿ as n → ∞, where k and α are two constants, ρ_c is the unique positive zero of the C(s), and C(s) is the generating function of the infinitesimal collision rates. Moreover, the interesting and important quantity α takes a very simple and uniform form which could be interpreted as the `spectrum', ranging from -∞ to +∞, of the interaction between the two components of branching and collision of the IBCP.