13. Potential of Branching Processes as a Modeling Tool
213
BGW BPs in Random Environments
Another major limitation of BGW BPs is the absence of environmental stochasticity (i.e., of random variation of demographic rates over time), despite the key
role it plays in extinction processes (e.g., Shaffer 1987; Lande 1993).
Such environmental variation can, however, be introduced to the BP via a
sequence of random variables, called the environment process, that accounts for
the random variability of the environment. Then the offspring number probability
distribution varies randomly over time t, depending on the environment variable
at time t. In particular, the mean of the offspring probability distribution at time t,
m t , is now a random variable. BPs that are like BGW BPs except that they include
such random environmental variability are called BGW BPs in random environments (BP RE).
The asymptotic results presented above still hold when the environment process is independent and identically distributed over time. The new criteria for
supercriticality, criticality, and subcriticality are, respectively, E(ln m t ) > 0, E(ln
m t ) = 0, and E(ln m t ) < 0, under additional assumptions (Smith and Wilkinson
1969; Athreya and Karlin 1971). Thus, provided E(ln m t ) < 0, the BP RE certainly
ultimately goes extinct and has a QSD and an asymptotic growth rate. Results
about more general environmental processes were obtained by Athreya and Karlin
(1971) but under somewhat more stringent conditions that make these extensions
of little help in biology. For instance, the time reversibility (or exchangeability in
the terms of Athreya and Karlin 1971) required is not met when the environment
process is an autoregressive moving average (ARMA; e.g., Box and Jenkins
1970) process.
Closer to Realism
Mathematical results are now available for BPs that combine density dependence,
environmental variability, and several kinds of individuals (Gosselin 1998b). For
instance, sufficient conditions for certainty of ultimate extinction and quasistationarity to hold are that (1) the density dependence and the random environment considered are such that, in some way, the mean number of offspring per
individual tends to zero when population size tends to infinity; and (2) the random
environment does not have any temporal autocorrelation (for more details and
further results, see Gosselin 1998b). However, spatial autocorrelation of the
random environment (as found in, e.g., Hanski and Woiwod 1993) can be
incorporated.
Discussion
We discuss the relevance of BPs as extinction models, first in terms of structure
and second in terms of theoretical results available. We then discuss their use as
PVA extinction models and the further work needed in this direction.
213
BGW BPs in Random Environments
Another major limitation of BGW BPs is the absence of environmental stochasticity (i.e., of random variation of demographic rates over time), despite the key
role it plays in extinction processes (e.g., Shaffer 1987; Lande 1993).
Such environmental variation can, however, be introduced to the BP via a
sequence of random variables, called the environment process, that accounts for
the random variability of the environment. Then the offspring number probability
distribution varies randomly over time t, depending on the environment variable
at time t. In particular, the mean of the offspring probability distribution at time t,
m t , is now a random variable. BPs that are like BGW BPs except that they include
such random environmental variability are called BGW BPs in random environments (BP RE).
The asymptotic results presented above still hold when the environment process is independent and identically distributed over time. The new criteria for
supercriticality, criticality, and subcriticality are, respectively, E(ln m t ) > 0, E(ln
m t ) = 0, and E(ln m t ) < 0, under additional assumptions (Smith and Wilkinson
1969; Athreya and Karlin 1971). Thus, provided E(ln m t ) < 0, the BP RE certainly
ultimately goes extinct and has a QSD and an asymptotic growth rate. Results
about more general environmental processes were obtained by Athreya and Karlin
(1971) but under somewhat more stringent conditions that make these extensions
of little help in biology. For instance, the time reversibility (or exchangeability in
the terms of Athreya and Karlin 1971) required is not met when the environment
process is an autoregressive moving average (ARMA; e.g., Box and Jenkins
1970) process.
Closer to Realism
Mathematical results are now available for BPs that combine density dependence,
environmental variability, and several kinds of individuals (Gosselin 1998b). For
instance, sufficient conditions for certainty of ultimate extinction and quasistationarity to hold are that (1) the density dependence and the random environment considered are such that, in some way, the mean number of offspring per
individual tends to zero when population size tends to infinity; and (2) the random
environment does not have any temporal autocorrelation (for more details and
further results, see Gosselin 1998b). However, spatial autocorrelation of the
random environment (as found in, e.g., Hanski and Woiwod 1993) can be
incorporated.
Discussion
We discuss the relevance of BPs as extinction models, first in terms of structure
and second in terms of theoretical results available. We then discuss their use as
PVA extinction models and the further work needed in this direction.