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.
