| 1. |
- Athreya, K. B., et al.
(author)
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Foreword
- 2016
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In: Lecture notes in statistics. Workshop on Branching Processes and their Applications, WBPA 2015; Badajoz; Spain; 7 April 2015 through 10 April 2015. - 0930-0325. ; 219
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Journal article (other academic/artistic)
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| 2. |
- Baker, J., et al.
(author)
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On the establishment of a mutant
- 2020
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In: Journal of Mathematical Biology. - : Springer Science and Business Media LLC. - 0303-6812 .- 1432-1416. ; 80, s. 1733-1757
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Journal article (peer-reviewed)abstract
- How long does it take for an initially advantageous mutant to establish itself in a resident population, and what does the population composition look like then? We approach these questions in the framework of the so called Bare Bones evolution model (Klebaner et al. in J Biol Dyn 5(2):147-162, 2011. https://doi.org/ 10.1080/ 17513758.2010.506041) that provides a simplified approach to the adaptive population dynamics of binary splitting cells. As the mutant population grows, cell division becomes less probable, and it may in fact turn less likely than that of residents. Our analysis rests on the assumption of the process starting from resident populations, with sizes proportional to a large carrying capacity K. Actually, we assume carrying capacities to be a(1)K and a(2)K for the resident and the mutant populations, respectively, and study the dynamics for K -> infinity. We find conditions for the mutant to be successful in establishing itself alongside the resident. The time it takes turns out to be proportional to log K. We introduce the time of establishment through the asymptotic behaviour of the stochastic nonlinear dynamics describing the evolution, and show that it is indeed 1/rho log K, where rho is twice the probability of successful division of the mutant at its appearance. Looking at the composition of the population, at times 1/rho log K + n, n is an element of Z(+), we find that the densities (i.e. sizes relative to carrying capacities) of both populations follow closely the corresponding two dimensional nonlinear deterministic dynamics that starts at a random point. We characterise this random initial condition in terms of the scaling limit of the corresponding dynamics, and the limit of the properly scaled initial binary splitting process of the mutant. The deterministic approximation with random initial condition is in fact valid asymptotically at all times 1/rho log K + n with n is an element of Z.
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| 3. |
- Chigansky, Pavel, et al.
(author)
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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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In: Stochastic Models. - : Informa UK Limited. - 1532-6349 .- 1532-4214. ; 35:2, s. 108-118
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Journal article (peer-reviewed)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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| 4. |
- Chigansky, P., et al.
(author)
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What can be observed in real time PCR and when does it show?
- 2018
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In: Journal of Mathematical Biology. - : Springer Science and Business Media LLC. - 0303-6812 .- 1432-1416. ; 76:3, s. 679-695
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Journal article (peer-reviewed)abstract
- Real time, or quantitative, PCR typically starts from a very low concentration of initial DNA strands. During iterations the numbers increase, first essentially by doubling, later predominantly in a linear way. Observation of the number of DNA molecules in the experiment becomes possible only when it is substantially larger than initial numbers, and then possibly affected by the randomness in individual replication. Can the initial copy number still be determined? This is a classical problem and, indeed, a concrete special case of the general problem of determining the number of ancestors, mutants or invaders, of a population observed only later. We approach it through a generalised version of the branching process model introduced in Jagers and Klebaner (J Theor Biol 224(3):299-304, 2003. doi: 10.1016/S0022-5193(03) 001668), and based on Michaelis-Menten type enzyme kinetical considerations from Schnell and Mendoza (J Theor Biol 184(4):433-440, 1997). A crucial role is played by the Michaelis-Menten constant being large, as compared to initial copy numbers. In a strange way, determination of the initial number turns out to be completely possible if the initial rate v is one, i.e all DNA strands replicate, but only partly so when v < 1, and thus the initial rate or probability of succesful replication is lower than one. Then, the starting molecule number becomes hidden behind a "veil of uncertainty". This is a special case, of a hitherto unobserved general phenomenon in population growth processes, which will be adressed elsewhere.
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| 5. |
- Fan, J. Y., et al.
(author)
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Convergence of the age structure of general schemes of population processes
- 2020
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In: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265. ; 26:2, s. 893-926
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Journal article (peer-reviewed)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 (Proc. Steklov Inst. Math. 282 (2013) 90-105), the Law of Large Numbers as K -> infinity 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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| 6. |
- Grandin, Karl, et al.
(author)
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Nuclear Energy
- 2010
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In: AMBIO. - : Springer Science and Business Media LLC. - 0044-7447 .- 1654-7209. ; 39:Suppl. 1, s. 26-30
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Journal article (peer-reviewed)abstract
- Nuclear energy can play a role in carbon free production of electrical energy, thus making it interesting for tomorrow’s energy mix. However, several issues have to be addressed. In fission technology, the design of so-called fourth generation reactors show great promise, in particular in addressing materials efficiency and safety issues. If successfully developed, such reactors may have an important and sustainable part in future energy production. Working fusion reactors may be even more materials efficient and environmental friendly, but also need more development and research. The roadmap for development of fourth generation fission and fusion reactors, therefore, asks for attention and research in these fields must be strengthened.
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| 7. |
- Hamza, K., et al.
(author)
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On the establishment, persistence, and inevitable extinction of populations
- 2016
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In: Journal of Mathematical Biology. - : Springer Science and Business Media LLC. - 0303-6812 .- 1432-1416. ; 72:4, s. 797-820
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Journal article (peer-reviewed)abstract
- Comprehensive models of stochastic, clonally reproducing populations are defined in terms of general branching processes, allowing birth during maternal life, as for higher organisms, or by splitting, as in cell division. The populations are assumed to start small, by mutation or immigration, reproduce supercritically while smaller than the habitat carrying capacity but subcritically above it. Such populations establish themselves with a probability wellknown from branching process theory. Once established, they grow up to a band around the carrying capacity in a time that is logarithmic in the latter, assumed large. There they prevail during a time period whose duration is exponential in the carrying capacity. Even populations whose life style is sustainble in the sense that the habitat carrying capacity is not eroded but remains the same, ultimately enter an extinction phase, which again lasts for a time logarithmic in the carrying capacity. However, if the habitat can carry a population which is large, say millions of individuals, and it manages to avoid early extinction, time in generations to extinction will be exorbitantly long, and during it, population composition over ages, types, lineage etc. will have time to stabilise. This paper aims at an exhaustive description of the life cycle of such populations, from inception to extinction, extending and overviewing earlier results. We shall also say some words on persistence times of populations with smaller carrying capacities and short life cycles, where the population may indeed be in danger in spite of not eroding its environment.
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