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Fr´ ed´ eric Gosselin and Jean-Dominique Lebreton
two stochastic components: the fecundity in newborn females per female, with
expected value f, and the survival probability from birth until age 1, p. One would
then have m = s + fp.
General Behavior of BGW BP
Each individual is replaced, on the average, by m individuals. From the conditional expectation formula E(Z t | Z t−1 ) = m Z t−1 , one gets
E(Z t ) = mE(Z t−1 )
(13.5)
Hence, expected population size varies exponentially:
E(Z t ) = m
t E(Z 0 )
(13.6)
If m > 1, the expected population size diverges over time, but Equation (13.5) tells
us that extinction is still possible. Indeed, if m > 1, the process goes extinct with a
non-null probability r and diverges toward infinity with probability 1 − r > 0; this
case is called supercritical.
However, when m ≤ 1, ultimate extinction is certain, because, from Equation
(13.6), the population size cannot diverge to infinity with a positive probability.
The so-called critical case, m = 1, is unrealistic in practice (Lebreton 1981). The
case m < 1 is called subcritical.
At first sight, the supercritical case is the most attractive, as a natural stochastic
counterpart of exponential growth, the oldest model of population growth (Malthus 1798). However, the emphasis in PVA is on regulation, which tends to keep
population size away from infinity. We see below that realistic forms of density
dependence imply the certainty of (ultimate) extinction. This qualitative reasoning, which we also discuss below, gives a special interest to the subcritical case.
Quasi-Stationarity of Subcritical BGW BP
Because in subcritical BGW BPs extinction is certain, we are naturally led to
study the behavior of the process before extinction. Let us consider a small initial
population size (Z 0 > 0). If at least some of the Z 0 individuals have survived and/or
have given birth to new individuals, the population is not extinct at the next time
step (Z 1 > 0). As long as this goes on (i.e., as long as extinction has not taken
place), we obtain a series of positive random variables Z 1 , Z 2 , . . . . The key result
is that, in subcritical BGW BPs, the probability distribution at time t of a population size conditioned on nonextinction converges, when t tends to infinity, to a
probability distribution (b k ) k∈N* (Fig. 13.2), irrespective of the initial population
size (e.g., Seneta and Vere-Jones 1966; Jagers 1975).
The probability distribution (b k ) k∈N* is called the quasi-stationary distribution
(QSD) because
1. It is stationary in the sense that the probability distribution of Z t converges to it
irrespective of the initial value (i.e., the value of Z 0 > 0).
Fr´ ed´ eric Gosselin and Jean-Dominique Lebreton
two stochastic components: the fecundity in newborn females per female, with
expected value f, and the survival probability from birth until age 1, p. One would
then have m = s + fp.
General Behavior of BGW BP
Each individual is replaced, on the average, by m individuals. From the conditional expectation formula E(Z t | Z t−1 ) = m Z t−1 , one gets
E(Z t ) = mE(Z t−1 )
(13.5)
Hence, expected population size varies exponentially:
E(Z t ) = m
t E(Z 0 )
(13.6)
If m > 1, the expected population size diverges over time, but Equation (13.5) tells
us that extinction is still possible. Indeed, if m > 1, the process goes extinct with a
non-null probability r and diverges toward infinity with probability 1 − r > 0; this
case is called supercritical.
However, when m ≤ 1, ultimate extinction is certain, because, from Equation
(13.6), the population size cannot diverge to infinity with a positive probability.
The so-called critical case, m = 1, is unrealistic in practice (Lebreton 1981). The
case m < 1 is called subcritical.
At first sight, the supercritical case is the most attractive, as a natural stochastic
counterpart of exponential growth, the oldest model of population growth (Malthus 1798). However, the emphasis in PVA is on regulation, which tends to keep
population size away from infinity. We see below that realistic forms of density
dependence imply the certainty of (ultimate) extinction. This qualitative reasoning, which we also discuss below, gives a special interest to the subcritical case.
Quasi-Stationarity of Subcritical BGW BP
Because in subcritical BGW BPs extinction is certain, we are naturally led to
study the behavior of the process before extinction. Let us consider a small initial
population size (Z 0 > 0). If at least some of the Z 0 individuals have survived and/or
have given birth to new individuals, the population is not extinct at the next time
step (Z 1 > 0). As long as this goes on (i.e., as long as extinction has not taken
place), we obtain a series of positive random variables Z 1 , Z 2 , . . . . The key result
is that, in subcritical BGW BPs, the probability distribution at time t of a population size conditioned on nonextinction converges, when t tends to infinity, to a
probability distribution (b k ) k∈N* (Fig. 13.2), irrespective of the initial population
size (e.g., Seneta and Vere-Jones 1966; Jagers 1975).
The probability distribution (b k ) k∈N* is called the quasi-stationary distribution
(QSD) because
1. It is stationary in the sense that the probability distribution of Z t converges to it
irrespective of the initial value (i.e., the value of Z 0 > 0).