# A branching process in an alternating random environment

- Abstract
- Introduction
- The model description
- The population mean
- The particular case
- A numerical illustration

A continuous-time branching process model of a stochastic population is considered in which the life-time and the reproductive capacity of the individuals are dependent on a fluctuating environment. Using an imbedding technique, an integral equation is formulated for the probability generating function of the population. An explicit expression for the time-dependent population mean is obtained and particular cases are recovered. A numerical illustration is provided to highlight the influence of the random environment. The influence of the environment on the life-cycle of the individuals of the population from the point of view of their reproductive nature and the consequent fluctuations in the size of the population has been studied by taking up several experimental case studies with some species (see Andrewartha and Birch (1954), Andrewartha (1961), Silvertown (1987), Krebs (2002) and Lande et al. (2003)). Seasonality is quite often the dominant feature of environmental variability experienced by several biological populations due to which the population size and other demographic parameters fluctuate randomly over time. Keywords: Continuous-time branching process, alternating random environment, Probability generating function, population mean

[...] ( 2.6 ) We assume that the o?spring generating function of an individual while splitting when the environment is in level i is hi i = To describe the branching process, we de?ne the following generating functions Gi = E{?X(t) = = i = Using the regeneration point technique, we obtain t G0 = ? 1 t 0 {f00 + f01 du t t 0 + 0 f00 (u)h0 (G0 t u))du + t 0 f01 (u)h1 (G1 t t G1 = ? 1 + 0 {f10 + f11 du 0 f10 (u)h0 (G0 t u))du + f11 (u)h1 (G1 t u))du. [...]

[...] We assume that the environment has just entered into level 0 at time t = 0 so that 1 = lim = 0 = = lim The sojourn-times of the environment in the levels 0 and 1 form an alternating renewal process and we assume that the probability-density-function (p.d.f.) of the stay-in-times 2 of the environment in level i is ?i e??i t , i = Let ?i e??i t be the p.d.f. of the life-time of an individual of the population when the population in level i = Let fij (t)dt be the conditional probability that a particle which was born at time t = 0 while the environment was in level i has lived up to time t and branches in t + dt) and the environment is in level j at the time of branching. [...]

[...] In this case, we obtain + ?0 + ?1 ) + 2 {(?0 ) (?1 + ?1 = ? = + ?0 + ?1 ) b {(?0 ) (?1 + ?1 = Then, we have a = ?0 ?1 ? b a + ?0 + ?1 + ?1 = + ?0 + ?1 + ?1 = + ?1 + ? b a + ?0 + ?1 + ?0 = + ? + ?0 + ?1 + ?0 = ? b Consequently, from the equations ( 3.7 ) and ( 3.8 we deduce the classical results M0 = e??0 t , (?0 ?1 +?1 ?1 e??0 t M1 = . [...]