13. Potential of Branching Processes as a Modeling Tool
207
call the limit λ the asymptotic growth rate of the population. Technically, under
mild conditions, the asymptotic growth rate is the dominant eigenvalue of the
substochastic matrix Q (Appendix 13.2).
We define quasi-stationarity of a BP as the simultaneous existence of a QSD
and of an asymptotic growth rate λ. We then say that the BP is quasi-stationary.
Because quasi-stationarity implies that the probability of immediate extinction
conditioned on nonextinction at present converges to 1 − λ, quasi-stationarity
further entails that the probability distribution of the time to extinction will be
asymptotically geometric, with reason 1 − λ.
Multitype BGW BP and a Case Study: The Alsace
White Stork
Despite the theoretical interest of the previous results, more general models are
needed for reaching some realism and applicability. We propose in this section a
first such generalization, which consists of considering different kinds of individuals. This will allow us to apply these theoretical results to an age-structured
declining White Stork (Ciconia ciconia) population.
Multitype BGW BP
We now distinguish individuals according to characteristics such as their age, so
that individuals are classified into d types (d ∈ N*). The possible states of the
stochastic random variable Z t representing the population at time t are no longer
single integers, but d-uplets of non-negative integer numbers, representing the
number of individuals in the d different types at time t: Z t is now a random vector
with values in N
d . Each type then has its own offspring probability distribution
(i.e., the probability distribution of the X variables introduced above now depends
on the type of the individual considered). Moreover, the X variables are multivariate with d integer-valued components, because an individual may produce offspring individuals of several types (e.g., an adult aged i may survive as an adult
aged i + 1 and give birth to young, that is, to individuals aged 0).
A BP that relies on the same assumptions as a BGW BP except that it allows for
different types of individuals is called a multitype BGW BP (MT BP). The scalar
m is then replaced by a non-negative d × d matrix M = (m α,β ) 1≤α,β≤d so that
E(Z t | Z t−1 ) = MZ t−1 and E(Z t ) = ME(Z t−1 )
(13.7)
The entry m α,β of the matrix M is the average number of individuals of type α
produced by an individual of type β at the next time step. In particular, with age
intervals whose length equals the time step, M is a Leslie matrix. As a consequence, “the exact stochastic analogue of Leslie’s theory . . . can be regarded as a
special case of the general theory of the multi-type Galton-Watson process”
(Pollard 1966).
207
call the limit λ the asymptotic growth rate of the population. Technically, under
mild conditions, the asymptotic growth rate is the dominant eigenvalue of the
substochastic matrix Q (Appendix 13.2).
We define quasi-stationarity of a BP as the simultaneous existence of a QSD
and of an asymptotic growth rate λ. We then say that the BP is quasi-stationary.
Because quasi-stationarity implies that the probability of immediate extinction
conditioned on nonextinction at present converges to 1 − λ, quasi-stationarity
further entails that the probability distribution of the time to extinction will be
asymptotically geometric, with reason 1 − λ.
Multitype BGW BP and a Case Study: The Alsace
White Stork
Despite the theoretical interest of the previous results, more general models are
needed for reaching some realism and applicability. We propose in this section a
first such generalization, which consists of considering different kinds of individuals. This will allow us to apply these theoretical results to an age-structured
declining White Stork (Ciconia ciconia) population.
Multitype BGW BP
We now distinguish individuals according to characteristics such as their age, so
that individuals are classified into d types (d ∈ N*). The possible states of the
stochastic random variable Z t representing the population at time t are no longer
single integers, but d-uplets of non-negative integer numbers, representing the
number of individuals in the d different types at time t: Z t is now a random vector
with values in N
d . Each type then has its own offspring probability distribution
(i.e., the probability distribution of the X variables introduced above now depends
on the type of the individual considered). Moreover, the X variables are multivariate with d integer-valued components, because an individual may produce offspring individuals of several types (e.g., an adult aged i may survive as an adult
aged i + 1 and give birth to young, that is, to individuals aged 0).
A BP that relies on the same assumptions as a BGW BP except that it allows for
different types of individuals is called a multitype BGW BP (MT BP). The scalar
m is then replaced by a non-negative d × d matrix M = (m α,β ) 1≤α,β≤d so that
E(Z t | Z t−1 ) = MZ t−1 and E(Z t ) = ME(Z t−1 )
(13.7)
The entry m α,β of the matrix M is the average number of individuals of type α
produced by an individual of type β at the next time step. In particular, with age
intervals whose length equals the time step, M is a Leslie matrix. As a consequence, “the exact stochastic analogue of Leslie’s theory . . . can be regarded as a
special case of the general theory of the multi-type Galton-Watson process”
(Pollard 1966).