200
Fr´ ed´ eric Gosselin and Jean-Dominique Lebreton
1. Extinction models must consider several distinct types of variability in population processes (e.g., Chesson 1978; Shaffer 1981; Gilpin 1987), namely,
demographic stochasticity (or within-individual variation), environmental stochasticity over time, density dependence, dispersal over spatial units, and
environmental variability over space and time.
2. Extinction models should emphasize, at least in a first step, demographic
aspects prior to genetical ones (Lande 1988).
3. To be useful in practice, extinction models should be written in terms of
demographic parameters (Boyce 1992). In this context, discrete-time models
are more realistic to account for the seasonality and for the age structure of
most species, in particular for vertebrates, the subject of many PVA (Eberhardt
1985; Lebreton and Clobert 1991).
The aim of this chapter is to introduce a class of extinction models, called
discrete-time branching processes (BP) and to present mathematical results about
them that are useful in the context of population extinction. In particular, we
emphasize a paradoxical form of stability when ultimate extinction is certain,
called quasi-stationarity, which provides a clear conceptual background to the
interplay of persistence and extinction. Quasi-stationarity is often implicit in
many PVAs, especially in relation to a geometric probability distribution of time
to extinction (Goodman 1987; Woolfenden and Fitzpatrick 1991; Gabriel and
B¨ urger 1992). Although quasi-stationarity has already been explicitly used in
some stochastic finite-state population models (e.g., Verboom et al. 1991; Day and
Possingham 1995), BPs are among the simplest individual-based infinite-state
models in which quasi-stationarity can be studied formally. We hope, in turn, to
convince the reader that BPs are suitable for playing a theoretical and practical
role in the study of population extinction similar to that of matrix models (e.g.,
Caswell 1989; Heppell et al., this volume) in the study of population growth.
Our chapter is organized as follows: after having first recalled the general
features of BPs, we consider the simplest case of density-independent growth and
introduce the key notion of quasi-stationarity. Then we investigate BPs that account for an age structure and apply such a BP to a population of White Storks. We
then introduce density dependence and random environment to a BP, first separately, then simultaneously, together with an age structure. Finally, we discuss the
relevance of BPs as extinction models.
The notation and abbreviations used are given in Appendix 13.1. In particular,
we denote by Pr(A) the probability of the event A and by E(X) the expectation of
the random variable X. Furthermore, we denote by N = {0, 1, 2, . . . } and N* = {1,
2, . . . } the sets of non-negative and positive integers, respectively. In this chapter, time t takes discrete values, in N.
General Features of BPs
Characteristics Relevant to PVA
BPs are stochastic processes built to model simultaneously the multiplicative
nature of population growth and random differences between individual demo-
Fr´ ed´ eric Gosselin and Jean-Dominique Lebreton
1. Extinction models must consider several distinct types of variability in population processes (e.g., Chesson 1978; Shaffer 1981; Gilpin 1987), namely,
demographic stochasticity (or within-individual variation), environmental stochasticity over time, density dependence, dispersal over spatial units, and
environmental variability over space and time.
2. Extinction models should emphasize, at least in a first step, demographic
aspects prior to genetical ones (Lande 1988).
3. To be useful in practice, extinction models should be written in terms of
demographic parameters (Boyce 1992). In this context, discrete-time models
are more realistic to account for the seasonality and for the age structure of
most species, in particular for vertebrates, the subject of many PVA (Eberhardt
1985; Lebreton and Clobert 1991).
The aim of this chapter is to introduce a class of extinction models, called
discrete-time branching processes (BP) and to present mathematical results about
them that are useful in the context of population extinction. In particular, we
emphasize a paradoxical form of stability when ultimate extinction is certain,
called quasi-stationarity, which provides a clear conceptual background to the
interplay of persistence and extinction. Quasi-stationarity is often implicit in
many PVAs, especially in relation to a geometric probability distribution of time
to extinction (Goodman 1987; Woolfenden and Fitzpatrick 1991; Gabriel and
B¨ urger 1992). Although quasi-stationarity has already been explicitly used in
some stochastic finite-state population models (e.g., Verboom et al. 1991; Day and
Possingham 1995), BPs are among the simplest individual-based infinite-state
models in which quasi-stationarity can be studied formally. We hope, in turn, to
convince the reader that BPs are suitable for playing a theoretical and practical
role in the study of population extinction similar to that of matrix models (e.g.,
Caswell 1989; Heppell et al., this volume) in the study of population growth.
Our chapter is organized as follows: after having first recalled the general
features of BPs, we consider the simplest case of density-independent growth and
introduce the key notion of quasi-stationarity. Then we investigate BPs that account for an age structure and apply such a BP to a population of White Storks. We
then introduce density dependence and random environment to a BP, first separately, then simultaneously, together with an age structure. Finally, we discuss the
relevance of BPs as extinction models.
The notation and abbreviations used are given in Appendix 13.1. In particular,
we denote by Pr(A) the probability of the event A and by E(X) the expectation of
the random variable X. Furthermore, we denote by N = {0, 1, 2, . . . } and N* = {1,
2, . . . } the sets of non-negative and positive integers, respectively. In this chapter, time t takes discrete values, in N.
General Features of BPs
Characteristics Relevant to PVA
BPs are stochastic processes built to model simultaneously the multiplicative
nature of population growth and random differences between individual demo-