Branching Processes: Variation, Growth, and Extinction of by Patsy Haccou

By Patsy Haccou

Biology takes a different position one of the different typical sciences simply because organic devices, be they items of DNA, cells or organisms, reproduce kind of faithfully. As for the other organic techniques, replica has a wide random part. the speculation of branching strategies used to be constructed in particular as a mathematical counterpart to this so much basic of organic strategies. This energetic and wealthy learn sector permits us to make predictions approximately either extinction hazards and the advance of inhabitants composition, and likewise uncovers points of a population's historical past from its present genetic composition. Branching procedures play an more and more vital function in versions of genetics, molecular biology, microbiology, ecology and evolutionary thought. This ebook provides this physique of mathematical principles for a organic viewers, yet also needs to be stress-free to mathematicians.

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Such descriptions are called Markovian, and in much literature the use of the word “state” implies Markovianness. Components of (or partial information about) a state are called state variables. ) For instance, an individual’s type may be one of the state variables. , past reproduction). , how many children they have had) is often used in this way. However, especially with long-lived species, such models can become quite complicated and difficult to analyze. , the complete life history of an individual is always a state, but not a very parsimonious one).

The independence between population members was not called for, but to deduce further properties this is, however, important; independence is required in the calculation of variances. As we shall see in Chapter 5, it plays a crucial role in the determination of extinction risks. 12). Then, of course, independence (or at least correlation zero) is needed for the additivity of the variance, Var[Z n |Z n−1 ] = Var[ξ1 + · · · + ξ Z n−1 |Z n−1 ] = Var[ξ1 |Z n−1 ] + · · · + Var[ξ Z n−1 |Z n−1 ] = σ 2 Z n−1 .

Using a more complicated version of this model, Yakovlev et al. (1998) examined whether cell differentiation depended solely on cell age (the intrinsic biological-clock model) or was also affected by an external hormonal signal. By comparison of model predictions to data they showed that thyroid hormone reduced the mean duration of the mitotic cycle of progenitor cells and increased the probability of cell differentiation, which provided evidence in favor of the latter hypothesis. 6 Dependence on the Population Itself Up to this point, we have assumed that individuals reproduce independently from one another.

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