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 oﬀspring generating function of an individual while splitting when the environment is in level i is hi i = To describe the branching process, we deﬁne the following generating functions Gi = E{θX(t) = = i = Using the regeneration point technique, we obtain t G0 = θ 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 = θ 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 αi e−αi t , i = Let λi e−λi 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 + λ0 + λ1 ) + 2 {(α0 + λ0 ) (λ1 = = + λ0 + λ1 ) b Then, we have {(α0 + λ0 ) (λ1 = λ a = λ0 λ1 + α b a + α0 + α1 + λ1 = α + α0 + α1 + λ1 = λ1 λ b a + α0 + α1 + λ0 = + λ0 + α + α0 + α1 + λ0 = 0. [...]
[...] In section a numerical illustration is provided The Model Description We consider a stochastic population which evolves in a random environment. We assume that the environment stays in level 0 and in level 1 alternately for random lengths of time. Let be the number of individuals in the population and let be the level of the environment at any time t. For simplicity, we let that there is just a single new born individual in the population at time t = 0 so that = 1. [...]
[...] η f10 (η)M0 + m1 f11 (η)}M1 = ( 3.2 ) 5 From ( 2.3 we have 1 f00 f01 η + α0 + α1 + λ1 = , η (η a)(η 1 f10 f11 η + α0 + α1 + λ0 = , η (η a)(η (η + α0 + λ0 + α1 + λ1 m1 α0 α1 1 m0 f00 = , (η a)(η {η + α0 + λ0 m0 + α1 + λ1 ) α0 α1 m1 f11 = (η a)(η The simultaneous equations in ( 3.2 ) can be solved explicitly for the Laplace transforms i = we obtain = where A0 = 1 [m1 f01 f10 f11 + m1 f11 f00 f01 , η 1 = [m1 λ1 α0 (η + α0 + α1 + λ0 ) 2 (η b)2 (η Ai , i = B(η) ( 3.3 ) + α0 + λ0 + α1 + λ1 m1 α0 α1 + α0 + α1 + λ1 η + α0 + α1 + λ1 m1 ) , = (η a)(η 1 A1 = [m0 f10 f00 f01 + m0 f00 f10 f11 , η 1 [m0 λ0 α1 (η + α0 + α1 + λ1 ) = 2 (η b)2 (η + α0 + λ0 m0 + α1 + λ1 ) α0 α1 + α0 + α1 + λ0 , η + α0 + α1 + λ0 m0 ) = , (η a)(η B(η) = m0 f00 m1 f11 m0 m1 f01 (η)f10 1 = + α0 + λ0 + α1 + λ1 m1 α0 α1 } (η a)2 (η b)2 + α0 + λ0 m0 + α1 + λ1 ) α0 α1 } m0 m1 α0 α1 λ0 λ1 ] = (η + α0 + λ0 m0 + α1 + λ1 m1 α0 α (η a)(η ( 3.4 ) Substituting ( 3.4 ) in ( 3.3 we obtain M0 = η + α0 + α1 + λ1 m1 ) , (η a)(η η + α0 + α1 + λ0 m0 ) , (η a)(η 6 ( 3.5 ) ( 3.6 ) M1 = where a and are the real distinct roots of the equation b {η + α0 + λ0 m0 + α1 + λ1 m1 α0 α1 = 0. [...]
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