Pure birth process

N(t) can be interpreted as the size of the population at time t. The most simple kind of birth-death process is that for which the rates of birth and death are time independent. A birth-death process N(t) is called a pure birth process (respectively pure death process) if Dm = 0 (Bm = 0) for any m. 

In the following, we derive the Kolmogorov differential equation on the basis of a simple model and report its various versions. In principle, this equation gives the rate at which a certain state is occupied by the system at a certain time. This equation is of a fundamental importance to obtain models discrete in space and continuous in time. The models, later discussed, are the Poisson Process, the Pure Birth Process, the Polya Process, the Simple Death Process and the Birth-and-Death Process. In section 2.1-3 this equation, i.e. Eq.2-30, has been derived for Markov chains discrete in space and time. 

The Pure Birth Process. The simplest generalization of the Poisson process is obtained by permitting the transition probabilities to depend on the actual state of the system. Thus, if at time t the system occupies state Sj = x (x = 0, 1,2,... 

The major conclusion drawn from Table 2-3 is that the increase of the population in the Pure Birth Process is significantly greater because the members don t become infertile after their first birth giving. For example, in the Pure Birth Process the population size becomes 10 after 2.83 time units where in the Poisson Process it takes 9 time units. 

We desire a mode/ that will reproduce and/or predict the experimental observations. Such models have been constructed by statisticians and mathematicians. The model selected here is the pure birth process, discussed in detail in many places [cf. Bharucha-Reid (B7) or Bailey (Bl)]. 

Let X(t) be the number of entities which have been replicated in a cell of age T. Of course, X t) is a discrete random variable, which assumes values 0, 1,2, N. If we put PJj) equal to P X x) = a), then it can be shown by a method essentially the same as used for the pure birth process (Section III, A), that P (t) satisfies the difference-differential equation... 

Note that the Poisson process plays a very important role in random walk theory. It can be defined in two ways (1) as a continuous-time Markov chain with constant intensity, i.e., as a pure birth process with constant birth rate k (2) as a renewal process. In the latter case, it can be represented as (3.25) with T = Here... 

This is a very simple model. It assumes that the reaction is a pure birth process. It corresponds to the case where the reaction term is F(p) = f p)p. This model was considered in [188, 189, 121]. The balance equations are... 

In the study of growth in a broad sense, birth may be liberally interpreted as an event whose probability is dependent on the number of parent events already in existenee. A speeifie value of random variable X(t) is denoted by k. For the pure birth process, the following are eonsidered, given that X(t) = k ... 

The pure death process is exactly analogous to the pure birth process, except that the pure death process X(t) is decreased rather than increased by the occurrence of an event. 

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]. 

Reduce this equation to a single-variable one-step process of pure birth type. [Compare the observation in connection with (VI.9.10).]... 

Table 2-3. Comparison between Pure Birth and Poisson Processes...

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... 

In this section pure birth-death processes will be discussed. Although such models have been dealt with extensively in the literature (see, for instance [4.1-8]), there still exist some seldom considered problems referring to the relation between the exact stationary or quasi-stationary solution of the master equation and the deterministic approach. This relation will be treated in Sect. 4.3.1 including an appUcation of the results obtained to observations on animal populations [4.15,17]. Further, the generalization of the stochastic standard model by including multistep birth or death processes will be investigated in Sect. 4.3.2. 

Invermectin is usually extracted from the soil of actinomycete Streptomyces avermitilis, the natural avermectins are 16-membered macrocyclic lactones and is found to be a mixture of 22, 23-dihydro structural analogues of avermectins and B, prepared by catalytic hydrogenation (reduction). In reality, avermectins are members of a family of rather structurally complex antibiotics obtained by fermentative process with the pure isolated strain of S. avermitilis. An intensive screening of cultures for the anthelmintic drugs exclusively from the natural products ultimately gave birth to this wonderful drug. 

Finally, it should not be left unnoted that in the UV-meter mentioned in the first paragraph, there was no pure oxalic acid at all. It contained a chemically bonded form of oxalic acid, which— from a toxicity viewpoint—is quite different from oxalic acid itself. This is by no means surprising, because common salt, which is added to most food, forms from a chemical reaction between a metal (sodium), which reacts vigorously with water, and a gas (chlorine), which was used as a toxic chemical weapon in World War I. The substance produced in the reaction, NaCl, has properties that are unrelated to the properties of the predecessors. This process of toxic parents giving birth to non-toxic children is a cornerstone of modem chemical science. 

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