| 1. |
- 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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| 2. |
- 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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| 3. |
- Jagers, Peter, 1941, et al.
(author)
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Amendment to: populations in environments with a soft carrying capacity are eventually extinct
- 2021
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In: Journal of Mathematical Biology. - : Springer Science and Business Media LLC. - 0303-6812 .- 1432-1416. ; 83:1
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Journal article (peer-reviewed)abstract
- This sharpens the result in the paperJagers and Zuyev (J Math Biol 81:845-851, 2020): consider a population changing at discrete (but arbitrary and possibly random) time points, the conditional expected change, given the complete past population history being negative, whenever population size exceeds a carrying capacity. Further assume that there is an epsilon > 0 such that the conditional probability of a population decrease at the next step, given the past, always exceeds epsilon if the population is not extinct but smaller than the carrying capacity. Then the population must die out.
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| 4. |
- Jagers, Peter, 1941, et al.
(author)
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Populations in environments with a soft carrying capacity are eventually extinct
- 2020
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In: Journal of Mathematical Biology. - : Springer Science and Business Media LLC. - 0303-6812 .- 1432-1416. ; 81:3, s. 845 -851
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Journal article (peer-reviewed)abstract
- Consider a population whose size changes stepwise by its members reproducing or dying (disappearing), but is otherwise quite general. Denote the initial (non-random) size by Z(0) and the size of the nth change by C-n, n = 1, 2, .... Population sizes hence develop successively as Z(1) = Z(0) + C-1, Z(2) = Z(1)+ C-2 and so on, indefinitely or until there are no further size changes, due to extinction. Extinction is thus assumed final, so that Z(n) = 0 implies that Z(n+1) = 0, without there being any other finite absorbing class of population sizes. We make no assumptions about the time durations between the successive changes. In the real world, or more specific models, those may be of varying length, depending upon individual life span distributions and their interdependencies, the age-distribution at hand and intervening circumstances. We could consider toy models of Galton-Watson type generation counting or of the birth-and-death type, with one individual acting per change, until extinction, or the most general multitype CMJ branching processes with, say, population size dependence of reproduction. Changes may have quite varying distributions. The basic assumption is that there is a carrying capacity, i.e. a non-negative number K such that the conditional expectation of the change, given the complete past history, is non-positive whenever the population exceeds the carrying capacity. Further, to avoid unnecessary technicalities, we assume that the change C-n equals -1 (one individual dying) with a conditional (given the past) probability uniformly bounded away from 0. It is a simple and not very restrictive way to avoid parity phenomena, it is related to irreducibility in Markov settings. The straightforward, but in contents and implications far-reaching, consequence is that all such populations must die out. Mathematically, it follows by a supermartingale convergence property and positive probability of reaching the absorbing extinction state.
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| 5. |
- Yen Fan, Jie, 1988, et al.
(author)
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LIMIT THEOREMS FOR MULTI-TYPE GENERAL BRANCHING PROCESSES WITH POPULATION DEPENDENCE
- 2020
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In: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 52:4, s. 1127-1163
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Journal article (peer-reviewed)abstract
- A general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population as a measure-valued process and obtain its asymptotics as the population grows with the environmental carrying capacity. Thus, a deterministic approximation is given, in the form of a law of large numbers, as well as a central limit theorem. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.
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