Birth-death models

Let us now apply the above simultaneous birth-death model to a well-known process, i.e., a z-stage irreversible-consecutive first order chemical reaction, with a single initial substance. The various states are Sj = Aj (i = 0, 1,. .., Z) where Ai... 

Fig.2-57. The Birth-Death model for reversible-consecutive reactions...

Beyond the Eirst Layer Birth-Death Models.358... 

Beyond the First Layer Birth-Death Models... 

In order to model film growth accurately beyond the first monolayer, one can use the so-called birth-death models that track the relative coverages of each layer as well as interlayer transport [29]. Eignre 5.1.13 highlights the major components of these models. In these models, the molecnles that land on top of the nth layer (where the 0th layer is the substrate) may diffuse and incorporate at the step edge of the n + 1 layer or transfer down to the top of the n - 1 layer to be incorporated into the nth layer (see Eigure 5.1.13). Thermal desorption of molecules that have... 

Cohen, P.I., Petrich, G.S., Pukite, P.R., Whaley, G.J., and Arrott, A.S., Birth-death models of epitaxy I. Diffraction oscillations from low index surfaces. Surf. ScL, 216, 222, 1989. 

This kind of dephasing on limit cycle orbits was first discussed by Tomita and Tomita by means of a specific birth-death model of chemical reactions (Tomita and Tomita, 1974). 

In Litwiniszyn s pure birth model, the entire bed is considered as on state, and the number of blocked pores in a unit volume of the bed is considered as a random variable. In his birth-death model, the number of trapped particles over the entire bed is a random variable. Fan and his co-workers have extended the pure birth and birth-death process models by incorporating... 

Models of population growth are analogous to chemical reaction rate equations. In the model developed by Malthus in 1798, the rate of change of the population N of Earth is dN/dt = births — deaths. The numbers of births and deaths are proportional to the population, with proportionality constants b and d. Derive the integrated rate law for population change. How well does it fit the approximate data for the population of Earth over time given below ... 

Here va and va are the stoichiometric coefficients for the reaction. The formulation is easily extended to treat a set of coupled chemical reactions. Reactive MPC dynamics again consists of free streaming and collisions, which take place at discrete times x. We partition the system into cells in order to carry out the reactive multiparticle collisions. The partition of the multicomponent system into collision cells is shown schematically in Fig. 7. In each cell, independently of the other cells, reactive and nonreactive collisions occur at times x. The nonreactive collisions can be carried out as described earlier for multi-component systems. The reactive collisions occur by birth-death stochastic rules. Such rules can be constructed to conserve mass, momentum, and energy. This is especially useful for coupling reactions to fluid flow. The reactive collision model can also be applied to far-from-equilibrium situations, where certain species are held fixed by constraints. In this case conservation laws... 

The reaction transition probabilities in a cell are determined by birth-death probabilistic rules that model the changes in the species particle numbers in the reaction mechanism. Letting N = (iV(1, iV(2),..., iV(s)) be the set of all instantaneous cell species numbers, the reaction transition matrix can be written... 

Since these characteristics are time-dependent, let us assume particle birth-death and migration to be the Markov stochastic processes. Note that making use of the stochastic models, we discuss below in detail, does not contradict the deterministic equations employed for these processes. Say, the equations for nv t), Xu(r,t), Y(r,t) given in Section 2.3.1 are deterministic since both the concentrations and joint correlation functions are defined by equations (2.3.2), (2.3.4) just as ensemble average quantities. Note that the... 

To treat the stochastic Lotka and Lotka-Volterra models, we have now to extend the formalism presented in Section 2.2.2, where collective variables-numbers of particles iVA and Vg were used to describe reactions. The point is that this approach neglects local density fluctuations in small element volumes. To incorporate both these fluctuations and their correlations due to diffusive conjunction, we are in position now to reformulate these models in terms of the diffusion-controlled processes - in contrast to the rather primitive birth-death formalism used in Section 2.2.2. It permits also to demonstrate in the non-trivial way a role of diffusion in the autowave processes. The main results of this Chapter are published in [21, 25]. 

Reversible Markov Chains and Birth-and-Death Models 2156 6.1. Jackson Networks 2164... 

The next system is a simple model of population dynamics. Let p be the density of a population. The population density will change due to births, deaths, and migration. The simplest model has no migration and the birth and death terms are simply proportional to the density ... 

Under the prudent criterion (PC) trams that enter a degraded state drop-off passengers and reach the workshop (this strategy allows to minimize the risk of stop on line). In this case, the resulting model becomes a birth-death process on the c axis of Figure 6. 

In detail, in (Koutras VR Platis A. N., 2006), a birth-death process was proposed to model wehsite visitors arrival and service. An optimization problem was solved to determine the optimal trade off between resource availahihty for high priority visitors and free resource access to lower priority visitors. The major contribution of this paper consisted in deriving formulas for the probabihty that a website visitor has no further access to resources and in determining the optimal reserved resources assuring the above trade off. 

Therefore, the inventory process X( ) is a birth-death Markov chain. In the remainder of the paper, the term job completion will be used to refer to the production of a unit of finished good inventory. These model dynamics are based on George and Harrison (1999). 

As a simple illustration of how reactions may be treated stochastically on mesoscopic scales, we show how one may construct a mesoscopic Markov process model for the Schlogl reaction (Eq. [3]). As before, we assume that species A and B in the Schlogl mechanism are pool species and their concentrations do not change in time Their effect is incorporated into the rate constants of the reaction. A Markov model of the reactions is a birth-death process, where chemical species are born in some reactions (X is born in the reactions (1) A -> X and (3) 2X - - B 3X) and dies in others (X dies in the reactions (2) X A and (4) 3X 2X - - B). Suppose there are ttx molecules of species X in a cell. Then the probability that a reaction X — A occurs and changes the number of molecules in the cell from x to — 1 is just... 

Deep-bed filtration involves the flow of particles through randomly distributed passages thus, it tends to be stochastic in nature. The filtration process has been modeled as a pure birth process [12,13], a birth-death process [14-17], a random-walk process [18], and a stochastic diffusion process [19]. 

Most biochemical reactions are catalyzed by enzymes. Therefore, although quite instructive, the model for the birth-death process studied in the previous chapter is not good enough an approximation in many instances. Typically, an enzymatic process consists of a series of chemical reactions that occur at different rates, and in some occasions it is possible to identify two well-separated time scales. When this occurs, the time-scale separation can be exploited to simplify the analysis of the whole system. Below we introduce a methodology to perform such simplification. 

In this chapter a model of migration and birth-death processes for interacting populations will be developed which may be considered as a typical application of the general concepts of Chap. 3. 

There exists an extensive literature on birth-death and migration processes of biological species, of which [4.1-8] is only a small selection. This literature includes stochastic models as well as mean value models. Although there is, of course, some overlap with these models - the approach introduced here contains several important new features ... 

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