Asymptotic Behaviour of Extinction Probability of Interacting Branching Collision Processes

Although the exact expressions for the extinction probabilities of the Interacting Branching Collision Processes (IBCP) are very recently given by Chen et al [4], some of these expressions are very complicated and hence useful information regarding asymptotic behaviour, for example, is hardly to be obtained. Also, these exact expressions take very different forms for different cases and thus seem lacking of 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 {an} follows an harmonious and simple asymptotic law of an ∼ knnc as n→ ∞ where k and are two constants and 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 takes a very simple and uniform form which could be interpreted as the “spectrum”, ranging from minus infinity to positive infinity