P. Jagers, Branching Processes with Biological Applications, Wiley, London, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Thus the Perron Frobenius theorem is used to obtain bounds on R, and R, v) which are functions of (r,z) and thus can be esti mated consistently. Similar methods were used for a multitype epidemic model without household structure by Britton [20] By the Perron Frobenius theorem (e.g. Jagers [27], page 92) it follows that there is a unique (up to normalisation)vector (x,x2, x j) satisfying R, xj : E Ximij i=1 (j = 1,2, J) Substituting the expression (2.1) for mij yields J R,xj = E xi i(n)tlun,i, A)A = E 1 i,n,k k=l kZk Define the final sum by r = x, n)pm, k(A) ....

P. Jagers, Branching Processes with Biological Applications, Wiley, London, 1975.

...., i n 1 ) number of children of #i 1 , i n 1 #. Thus, not all #i 1 , i n # with i 1 , i n # 1 occur as vertices of T , but only those which correspond to individuals which are actually born or realized . For more details, see Harris (1963) Ch. VI.2 or Jagers (1975), Ch. 1.2 or Grimmett and Kesten (1983) T 0 : #0# is called the 0th generation of T , and for n # 1 the collection of vertices #i 1 , i n # of T is called the nth generation of T , and denoted by T n . T n , the cardinality of T n , is just Z n . The subtree of T ....

Jagers, P., Branching Processes with Biological Applications, John Wiley & Sons, 1975.

....in section 5.1 as a model for phylogenetic trees. b) Branching processes. On can view the Yule process as the simplest continuous time branching process. There is a huge mathematical theory of random branching processes, represented by texts such as Harris (1963) Athreya and Ney (1972) Jagers (1975) and Asmussen and Hering (1983) Abstractly, a branching process is equivalent to a random tree, though the bulk of the branching process literature has a different focus from the random tree literature. Recent research in stochastic modeling of phylogenetic trees (see section 5.2) implicitly uses ....

JAGERS, P. (1975). Branching Processes with Biological Applications. Wiley. KARLIN, S. and TAYLOR, H.M. (1975). A First Course in Stochastic Processes. Academic Press, second edition.

....and the extinction probability was first obtained by Watson [14] by considering the probability generating function of the number of children in the nth generation. This mathematical model was known as Galton Watson branching process, and had been studied thoroughly in literature, for example, [12, 2, 5, 6, 9]. For the interesting details on the early history of branching processes, please consult [8] Another model for the branching process was based on the interpretation of the random walk S n n and the branching process in terms of queuing theory which is due to Kendall [7] Here S n = X 1 X ....

P. Jagers. Branching Processes with Biological Applications. Wiley, 1975.

....= 1 and (2.1) hold, then for any x 2 R, lim n 1 P Yn Gamma b(n=M(n) a(n=M(n) x j Zn Gamma1 0 = 1 Gamma (1 h Gammaff (x) Gamma1=ff ; 3.6) where M(n) is a s.v.f. defined by (3.4) Maximal number of offspring in branching processes 5 Proof. A) It is well known (see e.g. [9], Thm 2.4.2) that for x 0, lim n 1 P Zn Gamma1 n x j Zn Gamma1 0 = P (Z x) where E expf GammauZg = 1= 1 oe 2 u=2) Now, 3.5) follows by Lemma 2.1. B) Since (2.1) and (3.2) hold, we have by Lemma 2.1 lim n 1 P Yn Gamma b(Q Gamma1 n ) a(Q Gamma1 n ) x = ....

....(3.14) implies that G(x) belongs to one of three types corresponding to the three max stable laws. The explicit expression for g(x) in (3. 14) as well as necessary and sufficient conditions for F 2 SD(G) were established in [8] In the subcritical case when 0 m 1, it is known (see e.g. [9], Thm 2.6.2) that lim n 1 P (Zn = j j Zn 0) p j ; j = 0; 1; 3.16) where fp j g is a probability distribution whose generating function fl(s) P 1 j=0 p j s j , j s j 1, is the unique generation function solution of the functional equation fl(f(s) mfl(s) 1 Gamma m; fl(0) ....

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Jagers, P. (1975) Branching Processes with Biological Applications. John Wiley, London.

....a population model. In order to imagine how large this class of models is the following two cases are considered. Assume that the offspring variables (r) i are independent. Then this model is a typical Bienaym e Galton Watson branching process and many scientist work in this field ( 2] 10] [13]) Another interesting scenario appears when the considered population is in some biological equilibrium. This can be modeled by the assumption that the population size is fixed to some constant N in each generation. In this case for fixed r the offspring variables (r) i tend to be negatively ....

.... variable population sizes ( 20] 21] Unfortunately for the general case of stochastic variable population size the literature is rather spare and not a lot is known about the connection to the branching process models, where the offspring variables are assumed to be independent [2] 10] [13]. 4 Generalizations for more complex population models In recent years the number of publications about the coalescent increased enormously. Theoretical and mathematical aspects are covered as well as applied statistical and biological topics and even numerical questions in computer science. The ....

Jagers, P.: Branching Processes with Biological Applications. Wiley (1975).

....in Russia and Harris and Bellman in the United States. Harris authoritative book [38] appeared in 1960 and stimulated much re search on the subject. The book by Mode [51] on multitype branching processes came out in 1969. Then in 1972 the book by Athreya and Ney [23] was published. Jagers [43] wrote a book on branching processes with biological applications in mind. On a more ab stract level, the book by Asmussen and Herring [4] came out in 1982. Senata and Heyde [56] wrote a scholarly book in the 70 s on the early history of branching processes. The subject of branching processes has ....

Jagers,P.(1975): Branching processes with biological applications, Wiley Interscience, New York.

....be defined by (t) X x2U x (t Gamma oe x ) This gives the total weight (as measured by ) of the population at time t. Individuals make no contribution before they are born; that is is zero for negative arguments. This process has been extensively studied; see for example [14] [23], 34] In particular the well developed theory of its exponential growth plays a role in deriving the main results here. In the next section a weak result on the growth of this general branching process will be described. It remains to incorporate the final ingredient, the movement process. ....

Jagers, P. (1975), Branching Processes with Biological Applications. Wiley, New York.

....of branching processes in evolution is a more recent development, BRANCHING AND INFERENCE 3 and seems very promising; Jagers, Nerman and Taib [7, 9, 18, 19] have done a considerable amount of work on this topic. For general background on biological applications of branching processes, see [8]. The relationship between branching and traditional models for evolution is well known [17, 14, 3] Qualitatively, both models display similar behaviour unless there is some kind of spatial structure, in which case the behaviour can be radically different. We will discuss these issues in x5. The ....

P. Jagers. Branching Processes with Biological Applications. Wiley, Chichester, 1975. BRANCHING AND INFERENCE 12

...., where C 1 is the order of the component of G(n; n; q) containing the vertex 1 (cf. Grimmett (1989) Thm 4. 2) We note the implication that E(Z Gamma1 ) 1 Gamma 1 2 , where Z is the total size of a branching process with Poisson distributed family sizes having parameter ( 1) cf. Jagers (1975), Thm 2.11.2. Returning to (8.2) we find that ae( OE( qe ) Gamma OE( q) is convex, implying that ( ff; q) 0 if and only if ff Gamma ae 0 (0 ) 0. However, ae 0 (0 ) lim #0 i 0 ( log q) q) and the proof is complete. Analogous results may be obtained for ....

Jagers, P. (1975), Branching Processes with Biological Applications, John Wiley, Chichester.

....Israel. 1 INTRODUCTION Consider a growing tree of which each vertex generates additional vertices according to some probabilistic reproduction law. Growing trees arise naturally in many applications, such as searching and sorting [8] multiaccess communication [2] and growth of populations [3] [4]. Often, the tree that arises is growing in presence of a stochastic process, the random environment, which determines the reproduction law of each vertex. In addition, the tree may consist of vertices of different types, and the reproduction law of each vertex may depend on the type of the ....

....of randomly growing trees do not assume the existence of a random environment, and are based on the assumption that the vertices reproduce independently of each other. Growing trees in a random environment were considered so far only in the context of branching processes in a random environment [4], with the restriction that the state of the environment can change only at every generation, so that vertices that belong to the same generation (and are of the same type) have always the same reproduction law [1] The binary trees considered here are growing in a random environment where the ....

P. Jagers, Branching Processes with Biological Applications, London: Wiley & Sons, 1975.

....Z 0 . For each g 0, it is assumed that given the evolution of the population up to the gth generation, the Z g individuals in the gth generation have independent random numbers of children distributed according to (p i ) These children are the Z g 1 members of the (g 1)th generation. See [30, 33, 5] for background. Let S n : X 1 Delta Delta Delta X n (1) be the sum of n independent random variables X j with common distribution (p i ) From the description of (Z 0 ; Z 1 ; this sequence is a Markov chain with state space f0; 1; 2; g and time homogeneous transition ....

P. Jagers. Branching Processes with Biological Applications. Wiley, 1975.

.... Bellman Harris processes and Markov branching processes appear in Asmussen and Hering (1976) General branching processes with immigration have not attracted much interest; a result for single type processes under second moment conditions for both the reproduction and immigration may be found in Jagers (1975). This paper uses ideas and methods from Asmussen and Hering (1976) to prove immigration results for general multi type branching processes. In order to make it reasonably self contained, we give a short description of such processes in the next section, following Jagers (1992) 2. General ....

JAGERS, P. (1975) Branching Processes with Biological Applications. Wiley, Chichester.

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P. Jagers, Branching Processes with Biological Applications, Wiley, London, 1975.

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P. Jagers. Branching Processes with Biological Applications. Wiley, 1975.

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P. Jagers, Branching Processes with Biological Applications, Wiley, London, 1975.

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Jagers, P. (1975). Branching Processes with Biological Applications, John Wiley & Sons, Ltd., London.

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Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, New York.

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P. Jagers, Branching Processes with Biological Applications, Wiley, London, 1975.

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Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, New York.

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