13. Potential of Branching Processes as a Modeling Tool
203
present or the case in which it is replaced by a new one. More sophisticated rules,
accounting for the age of individuals, will be developed with multitype BPs.
The stochastic replacement of individuals can be summarized in a single equation giving Z t , conditional on Z t−1 :
Z t =
Z t−1
͚
j=1
X t−1,j
(13.3)
The branching property states moreover that the individuals’ contributions [i.e.,
the (X t−1, j ) (j = 1, . . . , Z t − 1 )] are independent of each other. An example of
realization of this transition from time t − 1 to time t is given in Figure 13.1.
Density-Independent BP: The Bienaym´ e-Galton-Watson BP
Definition
The Bienaym´ e-Galton-Watson BP (BGW BP) is the simplest BP, because the
individuals’ performances are identical and constant in time (i.e., the X t−1, j are
identically distributed over time t − 1 = 0, 1, . . . and over individuals j = 1, . . .,
Z t−1 ). In particular, there is no age structure in this model, and the expected
individual performance, denoted by m = E(X t−1, j ), is constant. This definition as
well as the properties that we will state about BGW BPs are well known and can
be found in, for example, the work of Athreya and Ney (1972), Jagers (1975), or
Asmussen and Hering (1983).
Demographic Background: BGW BP as a Single Age
Class Model
The BGW BP can be expressed easily in terms of demographic parameters, as we
now show by a simple example. Let us consider the females of a sexually reproducing population such that a female survives with probability s until next year
and, independently, gives birth to 0, 1, 2, . . . 1-year-old females with respective
probabilities π 0 , π 1 , π 2 , . . . . For a female to be replaced by k individuals, where k
is in N, she has to give birth to k − 1 individuals and survive (with probability
s π k−1 ), or, exclusively, give birth to k individuals and die (with probability (1 − s)
π k . Hence, the common probability distribution of the random variables (X t−1, j ) is
given by
PR(X = k) = s π k−1 + (1 − s) π k
(13.4)
One can then show that m = s + h, where h =
∞
͚
k=1
k π k is the average net
fecundity, expressed in females aged 1 per female. This model has a single age
class, because the survivors of newborn individuals are considered as adults at age
1. It could be easily modified to split the net fecundity, with expected value h, into
203
present or the case in which it is replaced by a new one. More sophisticated rules,
accounting for the age of individuals, will be developed with multitype BPs.
The stochastic replacement of individuals can be summarized in a single equation giving Z t , conditional on Z t−1 :
Z t =
Z t−1
͚
j=1
X t−1,j
(13.3)
The branching property states moreover that the individuals’ contributions [i.e.,
the (X t−1, j ) (j = 1, . . . , Z t − 1 )] are independent of each other. An example of
realization of this transition from time t − 1 to time t is given in Figure 13.1.
Density-Independent BP: The Bienaym´ e-Galton-Watson BP
Definition
The Bienaym´ e-Galton-Watson BP (BGW BP) is the simplest BP, because the
individuals’ performances are identical and constant in time (i.e., the X t−1, j are
identically distributed over time t − 1 = 0, 1, . . . and over individuals j = 1, . . .,
Z t−1 ). In particular, there is no age structure in this model, and the expected
individual performance, denoted by m = E(X t−1, j ), is constant. This definition as
well as the properties that we will state about BGW BPs are well known and can
be found in, for example, the work of Athreya and Ney (1972), Jagers (1975), or
Asmussen and Hering (1983).
Demographic Background: BGW BP as a Single Age
Class Model
The BGW BP can be expressed easily in terms of demographic parameters, as we
now show by a simple example. Let us consider the females of a sexually reproducing population such that a female survives with probability s until next year
and, independently, gives birth to 0, 1, 2, . . . 1-year-old females with respective
probabilities π 0 , π 1 , π 2 , . . . . For a female to be replaced by k individuals, where k
is in N, she has to give birth to k − 1 individuals and survive (with probability
s π k−1 ), or, exclusively, give birth to k individuals and die (with probability (1 − s)
π k . Hence, the common probability distribution of the random variables (X t−1, j ) is
given by
PR(X = k) = s π k−1 + (1 − s) π k
(13.4)
One can then show that m = s + h, where h =
∞
͚
k=1
k π k is the average net
fecundity, expressed in females aged 1 per female. This model has a single age
class, because the survivors of newborn individuals are considered as adults at age
1. It could be easily modified to split the net fecundity, with expected value h, into