An environment-dependent branching process

Abstract

A mathematical model of a biological population, taking into account the effect of environmental in uences both on the life-time distribution and on the reproductive capacity of the individuals of the population, is considered and analyzed. It is assumed that the environment stays in level 0 and in level 1 alternately for random lengths of time. The sojourn-times of the environment in the levels 0 and 1 form an alternating renewal process and the probability density function (p.d.f.) of the stay-in times of the environment in level i is i = 0, 1. Further, assuming that the p.d.f. of the life-time of an individual of the population when the population is in level i,i = 0,1, is an explicit expression for the time-dependent mean size of the population is obtained. The particular case corresponding to the environment independent population is deduced and two other particular cases, corresponding to partial interaction of the environment, are analysed. The coef cient of variation of the population size is also obtained and a numerical illustration is provided to highlight the impact of environment on the population size.