13. Potential of Branching Processes as a Modeling Tool
215
density dependence. The two main such results are the certainty of ultimate
extinction and quasi-stationarity.
Certainty of ultimate extinction is a general feature of BPs, as soon as population size is prevented from divergence to infinity (Equation [13.2]). This feature
seems realistic to us, although it has been open to discussion. Indeed for Chesson
(1978), “If population sizes are not allowed to become arbitrarily large it seems
that, in the absence of evolution, eventual extinction is a fact of life,” whereas the
mathematicians Athreya and Ney (1972) state that, because of certain ultimate
extinction, “an unmodified branching process is thus not a satisfactory model for
most biological situations.”
In our opinion, the biological realism of certain extinction comes simply from
the fact that it may be very likely or be unlikely over, for example, hundreds of
time steps. Actually, quasi-stationarity, defined as the existence of a QSD and of
an asymptotic growth rate, tells us how, asymptotically, the population goes
extinct. Because eventual extinction is certain, the asymptotic growth rate λ is less
than 1. Then, the asymptotic slow or rapid occurrence of extinction will depend on
whether λ is close to 1 or not.
Quasi-stationarity further implies the asymptotic geometric probability distribution of time to extinction, with a geometric rate of 1 − λ. It is satisfying that
this result, general in ad hoc extinction models studied by simulation (e.g., Goodman 1987; Woolfenden and Fitzpatrick 1991), holds for a reasonably wide class of
models. Indeed, certain extinction and quasi-stationarity are common features of
BPs that incorporate very different demographic features, as outlined in the sections above.
For instance, “realistic” DD BP share these two properties with subcritical
(density-independent) BGW BP. The key difference between decreasing densityindependent populations, modeled by subcritical BGW BPs, and densitydependent populations, modeled by DD BPs, may, however, lie in the asymptotic
risk of immediate extinction 1 − λ, which may become negligible when a “strong”
density dependence is introduced in a subcritical BGW BP. The sharp borderline
between stable populations and decreasing ones doomed to extinction, which
characterizes deterministic models, therefore vanishes in BPs. It is replaced by a
continuum of situations spreading (e.g., from a value of 1 − λ equal to 0.15 for the
decreasing density-independent multitype White Stork population through 1 − λ ≈
1.5 10
−3 for a weakly DD BP) (Gosselin 1996, model with no dispersal in scenario
I), to 1 − λ ≈ 10
−43 in the strongly DD BP treated by Lebreton (1981).
Two additional remarks widen the applicability of our results. First, if the
branching property were to be left, for example, because of dependencies between
mated individuals (e.g., McCarthy et al. 1994), certainty of extinction and quasistationarity persist in the more general framework of absorbing Markov chains
(Gosselin, 1998a, b,d). This renders the independence condition in the branching
property less stringent and makes our results apply to models of territorial species,
such as those of Woolfenden and Fitzpatrick (1991), 1 Schneider and Yodzis
1 Provided their reproduction number probability distribution fulfils certain conditions.
215
density dependence. The two main such results are the certainty of ultimate
extinction and quasi-stationarity.
Certainty of ultimate extinction is a general feature of BPs, as soon as population size is prevented from divergence to infinity (Equation [13.2]). This feature
seems realistic to us, although it has been open to discussion. Indeed for Chesson
(1978), “If population sizes are not allowed to become arbitrarily large it seems
that, in the absence of evolution, eventual extinction is a fact of life,” whereas the
mathematicians Athreya and Ney (1972) state that, because of certain ultimate
extinction, “an unmodified branching process is thus not a satisfactory model for
most biological situations.”
In our opinion, the biological realism of certain extinction comes simply from
the fact that it may be very likely or be unlikely over, for example, hundreds of
time steps. Actually, quasi-stationarity, defined as the existence of a QSD and of
an asymptotic growth rate, tells us how, asymptotically, the population goes
extinct. Because eventual extinction is certain, the asymptotic growth rate λ is less
than 1. Then, the asymptotic slow or rapid occurrence of extinction will depend on
whether λ is close to 1 or not.
Quasi-stationarity further implies the asymptotic geometric probability distribution of time to extinction, with a geometric rate of 1 − λ. It is satisfying that
this result, general in ad hoc extinction models studied by simulation (e.g., Goodman 1987; Woolfenden and Fitzpatrick 1991), holds for a reasonably wide class of
models. Indeed, certain extinction and quasi-stationarity are common features of
BPs that incorporate very different demographic features, as outlined in the sections above.
For instance, “realistic” DD BP share these two properties with subcritical
(density-independent) BGW BP. The key difference between decreasing densityindependent populations, modeled by subcritical BGW BPs, and densitydependent populations, modeled by DD BPs, may, however, lie in the asymptotic
risk of immediate extinction 1 − λ, which may become negligible when a “strong”
density dependence is introduced in a subcritical BGW BP. The sharp borderline
between stable populations and decreasing ones doomed to extinction, which
characterizes deterministic models, therefore vanishes in BPs. It is replaced by a
continuum of situations spreading (e.g., from a value of 1 − λ equal to 0.15 for the
decreasing density-independent multitype White Stork population through 1 − λ ≈
1.5 10
−3 for a weakly DD BP) (Gosselin 1996, model with no dispersal in scenario
I), to 1 − λ ≈ 10
−43 in the strongly DD BP treated by Lebreton (1981).
Two additional remarks widen the applicability of our results. First, if the
branching property were to be left, for example, because of dependencies between
mated individuals (e.g., McCarthy et al. 1994), certainty of extinction and quasistationarity persist in the more general framework of absorbing Markov chains
(Gosselin, 1998a, b,d). This renders the independence condition in the branching
property less stringent and makes our results apply to models of territorial species,
such as those of Woolfenden and Fitzpatrick (1991), 1 Schneider and Yodzis
1 Provided their reproduction number probability distribution fulfils certain conditions.