| 1. |
- Bartoszek, Krzysztof
(författare)
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A Central Limit Theorem for punctuated equilibrium
- 2020
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Ingår i: Stochastic Models. - : TAYLOR & FRANCIS INC. - 1532-6349 .- 1532-4214. ; 36:3, s. 473-517
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Tidskriftsartikel (refereegranskat)abstract
- Current evolutionary biology models usually assume that a phenotype undergoes gradual change. This is in stark contrast to biological intuition, which indicates that change can also be punctuated-the phenotype can jump. Such a jump could especially occur at speciation, i.e., dramatic change occurs that drives the species apart. Here we derive a Central Limit Theorem for punctuated equilibrium. We show that, if adaptation is fast, for weak convergence to normality to hold, the variability in the occurrence of change has to disappear with time.
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| 2. |
- 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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| 3. |
- El Maroufy, H., et al.
(författare)
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TRANSITION PROBABILITIES FOR GENERALIZED SIR EPIDEMIC MODEL
- 2012
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Ingår i: Stochastic Models. - : Informa UK Limited. - 1532-6349 .- 1532-4214. ; 28:1, s. 15-28
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Tidskriftsartikel (refereegranskat)abstract
- Gani and Purdue outlined a matrix-geometric method for determining the total size distribution of an epidemic in a recursive manner. In this article, we explore how this method can be used to study an SIR epidemic model with a generalized mechanism of infection. We are able to obtain an explicit formula for the Laplace transform of the transition probabilities. Using this we derive various other quantities explicitly. Examples of such quantities are the transition probabilities and the expectation of the duration of the epidemic.
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| 4. |
- Lindgren, Georg, et al.
(författare)
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Stochastic asymmetry properties of 3D Gauss-Lagrange ocean waves with directional spreading
- 2011
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Ingår i: Stochastic Models. - : Informa UK Limited. - 1532-6349 .- 1532-4214. ; 27:3, s. 490-520
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Tidskriftsartikel (refereegranskat)abstract
- In the stochastic Lagrange model for ocean waves the vertical and horizontal location of surface water particles are modeled as correlated Gaussian processes. In this article we investigate the statistical properties of wave characteristics related to wave asymmetry in the 3D Lagrange model. We present a modification of the original Lagrange model that can produce front-back asymmetry both of the space waves, i.e. observation of the sea surface at a fixed time, and of the time waves, observed at a fixed measuring station. The results, which are based on a multivariate form of Rice’s formula for the expected number of level crossings, are given in the form of the cumulative distribution functions for the slopes observed either by asynchronous sampling in space, or at synchronous sampling at upcrossings and down-crossings, respectively, of a specified fixed level. The theory is illustrated in a numerical section, showing how the degree of wave asymmetry depends on the directional spectral spreading and on the mean wave direction. It is seen that the asymmetry is more accentuated for high waves, a fact that may be of importance in safety analysis of capsizing risk.
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| 5. |
- Muratov, Anton, 1988, et al.
(författare)
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LISA: LOCALLY INTERACTING SEQUENTIAL ADSORPTION
- 2013
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Ingår i: Stochastic Models. - : Informa UK Limited. - 1532-6349 .- 1532-4214. ; 29:4, s. 475-496
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Tidskriftsartikel (refereegranskat)abstract
- We study a class of dynamically constructed point processes in which at every step a new point (particle) is added to the current configuration with a distribution depending on the local structure around a uniformly chosen particle. This class covers, in particular, generalized Polya urn scheme, Dubins-Freedman random measures, and cooperative sequential adsorption models studied previously. Specifically, we address models where the distribution of a newly added particle is determined by the distance to the closest particle from the chosen one. We address boundedness of the processes and convergence properties of the corresponding sample measure. We show that, in general, the limiting measure is random when it exists and that this is the case for a wide class of almost surely bounded processes.
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| 6. |
- Nyström, Kaj, et al.
(författare)
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Reflected BSDE of Wiener-Poisson type in time-dependent domains
- 2016
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Ingår i: Stochastic Models. - : Informa UK Limited. - 1532-6349 .- 1532-4214. ; 32:2, s. 275-300
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Tidskriftsartikel (refereegranskat)abstract
- In the paper we study multi-dimensional reflected backward stochasticdifferential equations driven by Wiener-Poisson type processes. We prove existence and uniqueness of solutions, with reflection in the inward spatial normal direction, in the setting of certain time-dependent domains.
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| 7. |
- Pihlsgård, Mats
(författare)
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Loss rate asymptotics in a GI/G/1 queue with finite buffer
- 2005
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Ingår i: Stochastic Models. - : Informa UK Limited. - 1532-6349 .- 1532-4214. ; 21:4, s. 913-931
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Tidskriftsartikel (refereegranskat)abstract
- We consider the stationary loss rate l(K) of a GI/G/1 queue with finite buffer of size K. Let X-n = U-n - T-n, n >= 1 where U-n is the service time, T-n is the interarrival time and let rho be the traffic intensity. We derive sharp asymptotics for the loss rate as K -> infinity, in the cases (i): rho > 1, and (ii): rho < 1 and X-n non-lattice with light tails. We also look at another reflection, related to Moran's dam model. As an example, we look at the PH/PH/1 case, where we show how to compute the asymptotic loss rate as well as the exact one and illustrate our results numerically.
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| 8. |
- Sagitov, Serik, 1956, et al.
(författare)
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Defective Galton-Watson processes
- 2017
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Ingår i: Stochastic Models. - : Informa UK Limited. - 1532-6349 .- 1532-4214. ; 33:3, s. 451-472
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Tidskriftsartikel (refereegranskat)abstract
- The Galton-Watson process is a Markov chain modeling the population size of independently reproducing particles giving birth to k offspring with probability p(k), k 0. In this paper, we consider defective Galton-Watson processes having defective reproduction laws, so that Sigma(k 0)p(k) = 1 - E for some E (0, 1). In this setting, each particle may send the process to a graveyard state with probability E. Such a Markov chain, having an enhanced state space {0, 1, ...}{}, gets eventually absorbed either at 0 or at . Assuming that the process has avoided absorption until the observation time t, we are interested in its trajectories as t and E 0.
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| 9. |
- Stefansson, Sigurdur O., et al.
(författare)
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Almost sure convergence of vertex degree densities in the vertex splitting model
- 2016
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Ingår i: Stochastic Models. - : Informa UK Limited. - 1532-6349 .- 1532-4214. ; 32:4, s. 575-592
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Tidskriftsartikel (refereegranskat)abstract
- We study the limiting degree distribution of the vertex splitting model introduced in Ref.([3]). This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this, we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature.
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| 10. |
- Vallier, Thomas
(författare)
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Robustness of preferential attachment under deletion of edges
- 2007
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Ingår i: Stochastic Models. - : Informa UK Limited. - 1532-6349 .- 1532-4214. ; 23:2, s. 265-276
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Tidskriftsartikel (refereegranskat)abstract
- The effect of deletion of old edges in the preferential attachment model introduced by Barabasi and Alberti([1]) is studied. We consider a model where every edge is deleted after a time Delta. The resulting graph has only Delta edges and with a high probability (1 + c)Delta nodes for some positive c. However its structure don't resemble the structure of the former model even for large Delta. In particular, we prove that the expected degrees of the resulting graph are uniformly bounded by a constant that does not depend on Delta. We discuss applications of our model for the evolution of networks where competition occurs.
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