214
Fr´ ed´ eric Gosselin and Jean-Dominique Lebreton
Relevant Structure for Extinction Models . . .
BPs have several features that make them attractive as extinction models. The first
is the representation of population sizes as integers, which induces a clear definition of extinction, as reaching population size 0.
The second is the branching property, according to which population size at any
time is obtained as the sum of independent integer random variables representing
the individuals’ contributions. With a cost associated to the restrictive assumption
of independence (discussed later), the use of a sum of random individual contributions has at least two practical advantages: first, the individual contributions can
be written easily in terms of demographic parameters (see further details below);
second, demographic stochasticity is represented in a canonical way via the randomness of the individual contributions, which is particularly relevant at least for
small or spatially subdivided populations in which demographic stochasticity
cannot be neglected (Chesson 1978; Mode and Pickens 1986; Gabriel and B¨ urger
1992).
Although the branching property may seem restrictive, the independence stated
is conditional on the state of the full system at a given time. This makes it possible
to consider the effect on the individual contributions of the abiotic environment,
of population size, and even the population size of interacting species (see an
example in Lebreton 1990). If the independence hypothesis seems yet not acceptable, a splitting of the time scale may make it acceptable (Gosselin 1998b):
compensatory mortality in a quarry species during the hunting season could, for
instance, be represented by a series of density-dependent monthly mortalities.
A third feature of BPs that makes them suitable in practice is their flexibility.
Density dependence, environmental variability, and different types of individuals
(according to their age, sex, or discrete spatial location) can be considered, simultaneously if needed. It may, however, be difficult to distinguish whether a model
combining these different demographic features is a BP. Actually, a necessary and
sufficient condition for a model to be a BP is that it is in discrete time, relies on
integer numbers, and satisfies the following statement: it is conditional on the
state of the population (i.e., the number of individuals in each type) at time t − 1,
conditional on the random environment at time t − 1, and conditional on any event
concerning the further past, the number of offspring in the different types at the
next time step t of each individual in the population at time t − 1 must be drawn at
random, independently from offspring random numbers of other individuals, according to a probability distribution that may depend solely on the type of the
individual, the state of the population, and the random environment at time t − 1.
Altogether BPs appear as fully stochastic extinction models, contrary to models
obtained by adding real valued white or pink noise to a deterministic equation.
. . . with Relevant Results Available . . .
On the basis of the relevant structure of BPs, one should expect the theoretical
results to be relevant, in particular when realism is enhanced by considering
Fr´ ed´ eric Gosselin and Jean-Dominique Lebreton
Relevant Structure for Extinction Models . . .
BPs have several features that make them attractive as extinction models. The first
is the representation of population sizes as integers, which induces a clear definition of extinction, as reaching population size 0.
The second is the branching property, according to which population size at any
time is obtained as the sum of independent integer random variables representing
the individuals’ contributions. With a cost associated to the restrictive assumption
of independence (discussed later), the use of a sum of random individual contributions has at least two practical advantages: first, the individual contributions can
be written easily in terms of demographic parameters (see further details below);
second, demographic stochasticity is represented in a canonical way via the randomness of the individual contributions, which is particularly relevant at least for
small or spatially subdivided populations in which demographic stochasticity
cannot be neglected (Chesson 1978; Mode and Pickens 1986; Gabriel and B¨ urger
1992).
Although the branching property may seem restrictive, the independence stated
is conditional on the state of the full system at a given time. This makes it possible
to consider the effect on the individual contributions of the abiotic environment,
of population size, and even the population size of interacting species (see an
example in Lebreton 1990). If the independence hypothesis seems yet not acceptable, a splitting of the time scale may make it acceptable (Gosselin 1998b):
compensatory mortality in a quarry species during the hunting season could, for
instance, be represented by a series of density-dependent monthly mortalities.
A third feature of BPs that makes them suitable in practice is their flexibility.
Density dependence, environmental variability, and different types of individuals
(according to their age, sex, or discrete spatial location) can be considered, simultaneously if needed. It may, however, be difficult to distinguish whether a model
combining these different demographic features is a BP. Actually, a necessary and
sufficient condition for a model to be a BP is that it is in discrete time, relies on
integer numbers, and satisfies the following statement: it is conditional on the
state of the population (i.e., the number of individuals in each type) at time t − 1,
conditional on the random environment at time t − 1, and conditional on any event
concerning the further past, the number of offspring in the different types at the
next time step t of each individual in the population at time t − 1 must be drawn at
random, independently from offspring random numbers of other individuals, according to a probability distribution that may depend solely on the type of the
individual, the state of the population, and the random environment at time t − 1.
Altogether BPs appear as fully stochastic extinction models, contrary to models
obtained by adding real valued white or pink noise to a deterministic equation.
. . . with Relevant Results Available . . .
On the basis of the relevant structure of BPs, one should expect the theoretical
results to be relevant, in particular when realism is enhanced by considering