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Fr´ ed´ eric Gosselin and Jean-Dominique Lebreton
Thus, far from invalidating or challenging past PVA simulations, our theoretical results will help (1) to render explicit some underlying qualitative behaviors at
work in simulation models (cf. White Stork example); (2) to frame the simulation
(e.g., by illuminating which quantities should be studied, which estimation procedures should be used, . . . ); and (3) to compare different PVA, as a result of the
similar theoretical framework.
Further Steps Required to Use BPs More Efficiently in PVA
Any further progress will first require that some conjectures are solved. The most
urgent one concerns the situation in which there is both a finite number of types
and a random environment, the existence of a QSD under more general conditions
than above. Results about a random environment with temporal autocorrelation,
as studied by simulation by Mode and Jacobson (1987a, b), are particularly
critical.
Additional technical tools would also be needed for the use of BPs in PVA
simulations. For instance, a procedure that would reflect when the asymptotic
behavior (see above) is reached with a given precision, would be very useful, as
well as an accurate procedure for the estimation of the asymptotic growth rate.
More globally, the user may need to have a clearer picture of the assumptions,
typical simulations, and theoretical results of different kinds of stochastic extinction models. Other kinds of stochastic population models can be clustered roughly
into stochastic difference equation models (DeAngelis 1976), birth and death
processes (Cohen 1969; Wissel 1989; Wissel and St¨ ocker 1991; Wissel and
Zaschke 1994), diffusion approximations (Lande and Orzack 1988; Dennis et al.
1991), and stochastic patch-occupancy metapopulation models (Gyllenberg and
Silvestrov 1994; Hanski 1994; Day and Possingham 1995; Gosselin 1998c) or
other finite state Markov chains (Verboom et al. 1991). A first comparison between different continuous time models was performed by Durrett and Levin
(1994).
We complement this analysis by three remarks. First, BPs and diffusion approximations lead to qualitatively different results, because diffusion approximations
do not induce an asymptotically geometric probability distribution of the time to
extinction (Lande and Orzack 1988; Dennis et al. 1991). Second, continuous time
and discrete state homogeneous Markov chains are, when considered at regular
discrete times, discrete-time Markov chains (e.g., Reddingius 1971). Thus, the
framework presented here and that studied, for example, in the work of Gosselin
(1998d) embrace many continuous-time stochastic frameworks (e.g., Wissel
1989; Wissel and St¨ ocker 1991; Mangle and Tier 1993; Wissel and Zaschke
1994). Third, stochastic patch-occupancy models and other finite absorbing
Markov chain population models have the same theoretical properties as subcritical BPs (i.e., certain extinction and quasi-stationarity; cf. Verboom et al. 1991;
Day and Possingham 1995; Gosselin 1998c).
Fr´ ed´ eric Gosselin and Jean-Dominique Lebreton
Thus, far from invalidating or challenging past PVA simulations, our theoretical results will help (1) to render explicit some underlying qualitative behaviors at
work in simulation models (cf. White Stork example); (2) to frame the simulation
(e.g., by illuminating which quantities should be studied, which estimation procedures should be used, . . . ); and (3) to compare different PVA, as a result of the
similar theoretical framework.
Further Steps Required to Use BPs More Efficiently in PVA
Any further progress will first require that some conjectures are solved. The most
urgent one concerns the situation in which there is both a finite number of types
and a random environment, the existence of a QSD under more general conditions
than above. Results about a random environment with temporal autocorrelation,
as studied by simulation by Mode and Jacobson (1987a, b), are particularly
critical.
Additional technical tools would also be needed for the use of BPs in PVA
simulations. For instance, a procedure that would reflect when the asymptotic
behavior (see above) is reached with a given precision, would be very useful, as
well as an accurate procedure for the estimation of the asymptotic growth rate.
More globally, the user may need to have a clearer picture of the assumptions,
typical simulations, and theoretical results of different kinds of stochastic extinction models. Other kinds of stochastic population models can be clustered roughly
into stochastic difference equation models (DeAngelis 1976), birth and death
processes (Cohen 1969; Wissel 1989; Wissel and St¨ ocker 1991; Wissel and
Zaschke 1994), diffusion approximations (Lande and Orzack 1988; Dennis et al.
1991), and stochastic patch-occupancy metapopulation models (Gyllenberg and
Silvestrov 1994; Hanski 1994; Day and Possingham 1995; Gosselin 1998c) or
other finite state Markov chains (Verboom et al. 1991). A first comparison between different continuous time models was performed by Durrett and Levin
(1994).
We complement this analysis by three remarks. First, BPs and diffusion approximations lead to qualitatively different results, because diffusion approximations
do not induce an asymptotically geometric probability distribution of the time to
extinction (Lande and Orzack 1988; Dennis et al. 1991). Second, continuous time
and discrete state homogeneous Markov chains are, when considered at regular
discrete times, discrete-time Markov chains (e.g., Reddingius 1971). Thus, the
framework presented here and that studied, for example, in the work of Gosselin
(1998d) embrace many continuous-time stochastic frameworks (e.g., Wissel
1989; Wissel and St¨ ocker 1991; Mangle and Tier 1993; Wissel and Zaschke
1994). Third, stochastic patch-occupancy models and other finite absorbing
Markov chain population models have the same theoretical properties as subcritical BPs (i.e., certain extinction and quasi-stationarity; cf. Verboom et al. 1991;
Day and Possingham 1995; Gosselin 1998c).