A multivariate immigration with multiple death process and applications to lunar craters

Abstract

SUMMARY We consider a stochastic model of a population composed of several types of individuals in which the Poissonwise arrival of an individual of 'large' type can cause the immediate death or loss of a certain number of individuals of smaller type. The survival of various small-type individuals is assumed independent, though possibly a function of the size of victim and killer. The probability generating function version of the forward Kolmogorov equation is derived, and from this explicit formulae for the first two factorial moments derived in the general case. The joint probability generating function of the numbers of individuals of various sizes is obtained explicitly in the case of two types of individuals by means of a direct probabilistic argument. The limiting distribution of small-type individuals is studied numerically and shown to be well approximated by the negative binomial distribution. This stochastic process is then used to approximate the formation and survival of lunar craters, a stochastic covering process. We consider here a stochastic model of a population composed of several types of individuals. The individuals arrive independently of each other according to Poisson processes with different rates. If an individual of 'large' type enters the population, he may at the instant of his arrival kill off a certain number of individuals of 'smaller' types already existing in the population. It is assumed that small-type individuals are killed independently of each other, with probabilities depending on the size of both the victim and the killer. In ?2 we derive a partial differential equation for the joint probability generating function (p.g.f.) for several types of individuals. The arguments of the p.g.f. in some terms of this differential equation are shifted, making an analytic solution very difficult. However, it is possible to obtain explicit formulae for the moments. In ?3 the joint p.g.f. is obtained by a direct probabilistic argument in the case of two types of individuals. The limiting marginal distribution of the number of small-type individuals is studied numerically in ?4, where it is shown that the negative binomial distribution provides a fairly good approximation to this distribution. We conclude with a discussion of the formation and survival of lunar craters, a stochastic covering process which can be approximated by a multivariate immigration with multiple death process. The limiting form of the expected number density of craters, as predicted by the meteoroidal impact hypothesis of their origin, is shown to be inversely proportional to the third power of crater diameter, in agreement with observations. The non-randomness