216
Fr´ ed´ eric Gosselin and Jean-Dominique Lebreton
(1994), Bart (1995), and McKelvey and co-workers (unpublished data) (Gosselin
1998b).
Second, if we want to investigate quasi-extinction (Ginzburg et al. 1982) of a
quasi-stationary BP, we can use a modified version of quasi-stationarity relative to
quasi-extinction by making the states below the quasi-extinction threshold absorbing. Although the quasi-extinction QSD and asymptotic growth rate are generally not linked in a simple way to the QSD and asymptotic growth rate relative
to extinction, they account for the same qualitative phenomena, such as the
asymptotically geometric probability distribution of the time to quasi-extinction.
Finally, note that further results about BPs are available (see Gosselin 1997,
1998a,b,d).
. . . That Are Underused in PVA
BPs have a long history among applied probabilists (e.g., Galton 1873; Yule
1924), and some of their features have been used implicitly in many models (e.g.,
North et al. 1988; Burkey 1989; Gabriel and B¨ urger 1992; Beissinger 1995).
However, BPs have seldom been used explicitly to model extinction (Lebreton
1978, 1982, 1990; Mode and Jacobson 1987a, b; Gosselin 1996). This current
state of affairs results partly from the complexity of the mathematics required,
partly from the difficulty in the transfer of knowledge on such multidisciplinary
matters, but also from the lack of results until recently for the most realistic BPs,
those with density-dependent regulation. The two main advantages of an explicit
use of BPs as extinction models, as opposed to ad hoc models, lie first in their
clearly defined structure, which simplifies the building of models, and second in
the availability of theoretical results.
The branching property, one of the specificities of BPs, makes it easy to build
the overall random transition from one time step to the next as the sum of
independent random individual contributions. These individual contributions can
be easily formulated, based on a life cycle graph of the species on which we
superimpose probability distributions (as in the White Stork example). BPs can
therefore be easily used to model specific population-environment systems.
Besides, the clearly defined structure of BPs and the independence assumption
in the branching property simplify computer programming of Monte Carlo simulations based on BPs.
Another advantage of BPs is the availability of theoretical results that can be
used to interpret PVA simulation results in the same way as, for instance, the
stable age theory enlightens the use of Leslie matrix models (Caswell 1989).
Consider as an illustration the White Stork BP model. Any other PVA of the
White Stork would have likely obtained a similar declining trend. Even a simple
deterministic Leslie matrix model (such as that in Table 13.1) would have found
exactly the same asymptotic growth rate λ. However, the Leslie matrix theory
does not tell anything about the meaning of λ relative to extinction. To the
contrary, the theoretical results presented in this chapter allow us to interpret this
declining trend as the interplay between extinction (at a rate that asymptotically
Fr´ ed´ eric Gosselin and Jean-Dominique Lebreton
(1994), Bart (1995), and McKelvey and co-workers (unpublished data) (Gosselin
1998b).
Second, if we want to investigate quasi-extinction (Ginzburg et al. 1982) of a
quasi-stationary BP, we can use a modified version of quasi-stationarity relative to
quasi-extinction by making the states below the quasi-extinction threshold absorbing. Although the quasi-extinction QSD and asymptotic growth rate are generally not linked in a simple way to the QSD and asymptotic growth rate relative
to extinction, they account for the same qualitative phenomena, such as the
asymptotically geometric probability distribution of the time to quasi-extinction.
Finally, note that further results about BPs are available (see Gosselin 1997,
1998a,b,d).
. . . That Are Underused in PVA
BPs have a long history among applied probabilists (e.g., Galton 1873; Yule
1924), and some of their features have been used implicitly in many models (e.g.,
North et al. 1988; Burkey 1989; Gabriel and B¨ urger 1992; Beissinger 1995).
However, BPs have seldom been used explicitly to model extinction (Lebreton
1978, 1982, 1990; Mode and Jacobson 1987a, b; Gosselin 1996). This current
state of affairs results partly from the complexity of the mathematics required,
partly from the difficulty in the transfer of knowledge on such multidisciplinary
matters, but also from the lack of results until recently for the most realistic BPs,
those with density-dependent regulation. The two main advantages of an explicit
use of BPs as extinction models, as opposed to ad hoc models, lie first in their
clearly defined structure, which simplifies the building of models, and second in
the availability of theoretical results.
The branching property, one of the specificities of BPs, makes it easy to build
the overall random transition from one time step to the next as the sum of
independent random individual contributions. These individual contributions can
be easily formulated, based on a life cycle graph of the species on which we
superimpose probability distributions (as in the White Stork example). BPs can
therefore be easily used to model specific population-environment systems.
Besides, the clearly defined structure of BPs and the independence assumption
in the branching property simplify computer programming of Monte Carlo simulations based on BPs.
Another advantage of BPs is the availability of theoretical results that can be
used to interpret PVA simulation results in the same way as, for instance, the
stable age theory enlightens the use of Leslie matrix models (Caswell 1989).
Consider as an illustration the White Stork BP model. Any other PVA of the
White Stork would have likely obtained a similar declining trend. Even a simple
deterministic Leslie matrix model (such as that in Table 13.1) would have found
exactly the same asymptotic growth rate λ. However, the Leslie matrix theory
does not tell anything about the meaning of λ relative to extinction. To the
contrary, the theoretical results presented in this chapter allow us to interpret this
declining trend as the interplay between extinction (at a rate that asymptotically