13. Potential of Branching Processes as a Modeling Tool
201
graphic performances. They have four characteristics particularly relevant to
PVA:
1. The population size in the model takes only non-negative integer values. As a
consequence, contrary to models with real-valued population sizes, extinction
is defined unambiguously as reaching a population size equal to 0.
2. BPs are fundamentally stochastic: the population size at time t, Z t , is obtained
stochastically from that at time t − 1, Z t−1 , by transition probabilities. As a
consequence, even conditional on a particular value of Z t−1 , Z t is an integer
valued random variable. Thus, the process cannot be reduced to a deterministic
process plus white noise.
3. Individuals are considered explicitly in a BP: the transition probabilities depend on the particular demographic rules retained when building the model,
and these demographic rules are defined at the individual’s level. The core
property of a BP in this respect is the branching property, according to which
individuals reproduce and die independently of each other;
4. Although the properties above are presented for the simple case of a single type
of individual, BPs can be generalized to account for several types of individuals (e.g., age classes, spatial cells), environmental variability, and density
dependence.
BPs as Infinite Markov Chains with an Absorbing State
Technically, for the simple case of a single type of individuals, a BP is a Markov
chain (i.e., Z t depends on the past only through the previous time t − 1, indeed only
through the value of Z t−1 ). The state space of the Markov chain is made up of all
possible values of Z t (i.e., all non-negative integers). We call such a chain, which
has an infinite denumerable number of states, an infinite Markov chain. Markov
chains that are commonly used in ecology (e.g., for modeling succession) (e.g.,
Facelli and Pickett 1990) or for simple population models (e.g., Verboom et al.
1991; Day and Possingham 1995) have only a finite number of states and, as such,
are called finite Markov chains.
In accordance with the above definition of extinction, in a BP, 0 is an absorbing
state (i.e., once the population is extinct, it remains so). BPs are thus specific
infinite Markov chains with an absorbing state. In a finite Markov chain with an
absorbing state, under mild conditions, absorption is certain ultimately:
Pr ( lim
t→∞
Z t = 0 ) = lim
t→∞
Pr(Z t = 0) = 1
(13.1)
A striking difference is that this is no more the case for an infinite Markov chain
with an absorbing state. In the case of BP, divergence to infinity can occur as well
as extinction; indeed, under mild conditions,
Pr ( lim
t→∞
Z t = 0 ) + Pr ( lim
t→∞
Z t = ∞ ) = 1
(13.2)
This is so because, when there is one absorbing state, the ultimate probability of
presence in any other state tends toward 0 (i.e., all other states are transient). In a
finite Markov chain, absorption is then certain. In an infinite Markov chain, the
201
graphic performances. They have four characteristics particularly relevant to
PVA:
1. The population size in the model takes only non-negative integer values. As a
consequence, contrary to models with real-valued population sizes, extinction
is defined unambiguously as reaching a population size equal to 0.
2. BPs are fundamentally stochastic: the population size at time t, Z t , is obtained
stochastically from that at time t − 1, Z t−1 , by transition probabilities. As a
consequence, even conditional on a particular value of Z t−1 , Z t is an integer
valued random variable. Thus, the process cannot be reduced to a deterministic
process plus white noise.
3. Individuals are considered explicitly in a BP: the transition probabilities depend on the particular demographic rules retained when building the model,
and these demographic rules are defined at the individual’s level. The core
property of a BP in this respect is the branching property, according to which
individuals reproduce and die independently of each other;
4. Although the properties above are presented for the simple case of a single type
of individual, BPs can be generalized to account for several types of individuals (e.g., age classes, spatial cells), environmental variability, and density
dependence.
BPs as Infinite Markov Chains with an Absorbing State
Technically, for the simple case of a single type of individuals, a BP is a Markov
chain (i.e., Z t depends on the past only through the previous time t − 1, indeed only
through the value of Z t−1 ). The state space of the Markov chain is made up of all
possible values of Z t (i.e., all non-negative integers). We call such a chain, which
has an infinite denumerable number of states, an infinite Markov chain. Markov
chains that are commonly used in ecology (e.g., for modeling succession) (e.g.,
Facelli and Pickett 1990) or for simple population models (e.g., Verboom et al.
1991; Day and Possingham 1995) have only a finite number of states and, as such,
are called finite Markov chains.
In accordance with the above definition of extinction, in a BP, 0 is an absorbing
state (i.e., once the population is extinct, it remains so). BPs are thus specific
infinite Markov chains with an absorbing state. In a finite Markov chain with an
absorbing state, under mild conditions, absorption is certain ultimately:
Pr ( lim
t→∞
Z t = 0 ) = lim
t→∞
Pr(Z t = 0) = 1
(13.1)
A striking difference is that this is no more the case for an infinite Markov chain
with an absorbing state. In the case of BP, divergence to infinity can occur as well
as extinction; indeed, under mild conditions,
Pr ( lim
t→∞
Z t = 0 ) + Pr ( lim
t→∞
Z t = ∞ ) = 1
(13.2)
This is so because, when there is one absorbing state, the ultimate probability of
presence in any other state tends toward 0 (i.e., all other states are transient). In a
finite Markov chain, absorption is then certain. In an infinite Markov chain, the