| 1. |
- Chigansky, Pavel, et al.
(författare)
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Populations with interaction and environmental dependence: From few, (almost) independent, members into deterministic evolution of high densities
- 2019
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Ingår i: Stochastic Models. - : Informa UK Limited. - 1532-6349 .- 1532-4214. ; 35:2, s. 108-118
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Tidskriftsartikel (refereegranskat)abstract
- Many populations, e.g. not only of cells, bacteria, viruses, or replicating DNA molecules, but also of species invading a habitat, or physical systems of elements generating new elements, start small, from a few lndividuals, and grow large into a noticeable fraction of the environmental carrying capacity K or some corresponding regulating or system scale unit. Typically, the elements of the initiating, sparse set will not be hampering each other and their number will grow from Z0 = z0 in a branching process or Malthusian like, roughly exponential fashion, Zt ~atW, where Z t is the size at discrete time t → ∞, a > 1 is the offspring mean per individual (at the low starting density of elements, and large K), and W a sum of z0 i.i.d. random variables. It will, thus, become detectable (i.e. of the same order as K) only after around K generations, when its density Xt := Z t /K will tend to be strictly positive. Typically, this entity will be random, even if the very beginning was not at all stochastic, as indicated by lower case z0 , due to variations during the early development. However, from that time onwards, law of large numbers effects will render the process deterministic, though inititiated by the random density at time log K, expressed through the variable W. Thus, W acts both as a random veil concealing the start and a stochastic initial value for later, deterministic population density development. We make such arguments precise, studying general density and also system-size dependent, processes, as K → ∞. As an intrinsic size parameter, K may also be chosen to be the time unit. The fundamental ideas are to couple the initial system to a branching process and to show that late densities develop very much like iterates of a conditional expectation operator. The “random veil”, hiding the start, was first observed in the very concrete special case of finding the initial copy number in quantitative PCR under Michaelis-Menten enzyme kinetics, where the initial individual replication variance is nil if and only if the efficiency is one, i.e. all molecules replicate.
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| 2. |
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| 3. |
- Jagers, Peter, 1941, et al.
(författare)
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Branching processes in near-critical random environments
- 2004
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 41A, s. 17-23
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Tidskriftsartikel (refereegranskat)abstract
- Branching processes are studied in random environments that are influenced by the population size and approach criticality as the population gets large. Results are applied to the polymerase chain reaction (PCR), which is empirically known to exhibit first exponential and then linear growth of molecule numbers.
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| 4. |
- Jagers, Peter, 1941, et al.
(författare)
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Convergence of the age structure of general schemes of population processes
- 2020
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Ingår i: Bernoulli. - 1350-7265. ; 26:2, s. 893-926
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Tidskriftsartikel (refereegranskat)abstract
- We consider a family of general branching processes with reproduction parameters depending on the age of the individual as well as the population age structure and a parameter K, which may represent the carrying capacity. These processes are Markovian in the age structure. In a previous paper [8] the Law of Large Numbers as K Ñ 8 was derived. Here we prove the Central Limit Theorem, namely the weak convergence of the fluctuation processes in an appropriate Skorokhod space. We also show that the limit is driven by a stochastic partial differential equation.
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| 5. |
- Jagers, Peter, 1941, et al.
(författare)
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Dependence and interaction in branching processes
- 2013
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Ingår i: Springer Proceedings in Mathematics and Statistics. - Berlin, Heidelberg : Springer Berlin Heidelberg. - 2194-1009 .- 2194-1017. - 9783642335488 ; 33, s. 325-333
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Konferensbidrag (refereegranskat)abstract
- Independence of reproducing individuals can be viewed as the very defining property of branching processes. It is crucial for the most famous results of the theory, the determination of the extinction probability and the dichotomy between extinction and exponential increase. In general processes, stabilisation of the age-distribution under growth follows, and indeed of the over-all population composition, and so do the many fine results of the area, like conditional stabilisation of the size of non-extinct subcritical processes. The last two decades have witnessed repeated attempts at treating branching processes with various kinds of dependence between individuals, ranging from local dependence between close relatives only to population size dependence. Of particular interest are very recent findings on processes that change from being supercritical to subcriticality at some threshold size, the carrying capacity of the habitat. We overview the development with an emphasis on these recent results.
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| 6. |
- Jagers, Peter, 1941, et al.
(författare)
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From Size to Age and Type Structure Dependent Branching: A First Step to Sexual Reproduction in General Population Processes
- 2016
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Ingår i: Lecture Notes in Statistics: Branching Processes and Their Applications, Workshop on Branching Processes and their Applications, WBPA 2015, Badajoz, Spain, 7-10 April 2015. - Cham : Springer International Publishing. - 0930-0325 .- 2197-7186. - 9783319316390 ; 219, s. 137-148
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- Classical branching processes, even the most general, share the property that individuals are supposed to multiply independently of one another, at least given some environment that in its turn is supposed to be unaffected by the population. Only more recently have birth-and-death and branching processes been considered which allow individual reproduction to be influenced by population size. The first results, due to Klebaner, deal with Galton-Watson processes. Work on general, age-structured processes and habitats with a threshold, a so called carrying capacity, came only decades later, inspired by deterministic population dynamics. Multi-type such processes have only been analysed recently. It turns out that a two-type population (males, females) where only the latter can give birth, their fecundity however influenced by the number of males present, provides an approach to sexual reproduction without mating assumptions, like random mating, which are artificial in genereral branching processes.
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| 7. |
- Jagers, Peter, 1941, et al.
(författare)
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Markovian Paths to Extinction
- 2007
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Ingår i: Adv. Appl. Prob.. ; 39, s. 569-587
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Tidskriftsartikel (refereegranskat)
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| 8. |
- Jagers, Peter, 1941, et al.
(författare)
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On the Path to Extinction
- 2007
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Ingår i: PNAS. ; 104
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Tidskriftsartikel (refereegranskat)
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| 9. |
- Jagers, Peter, 1941, et al.
(författare)
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Population-size-dependent, age-structured branching processes linger around their carrying capacity
- 2011
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Ingår i: Journal of Applied Probability. - 0021-9002. ; 48A, s. 249-260
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Tidskriftsartikel (refereegranskat)abstract
- Dependence of individual reproduction upon the size of the whole population is studied in a general branching process context. The particular feature under scrutiny is that of reproduction changing from supercritical in small populations to subcritical in large ones. The transition occurs when population size passes a critical threshold, known in ecology as the carrying capacity. We show that populations either die out directly, never coming close to the carrying capacity, or else they grow quickly towards the latter, subsequently lingering around it for a time that is expected to be exponentially long in terms of a carrying capacity tending to infinity.
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| 10. |
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