13. Potential of Branching Processes as a Modeling Tool
223
Yule GU (1924) A mathematical theory of evolution based on the conclusions of Dr JC
Willis, FRS. Philosophical Transactions of the Royal Society of London B 213:21–87
Appendix 1. Notation and Abbreviations
In this chapter, we denote by N = {0, 1, 2, . . .} and N* = {1, 2, . . .} the set of nonnegative and positive integers, respectively. Time t takes discrete values, in N.
We denote by Pr(A) the probability of the event A and by E(X) the expectation of the random variable X (if it is defined). If Pr(B) > 0, we denote by
Pr(A | B) =
Pr(A ∩ B)
Pr(B)
the conditional probability of A with respect to B. When
both X and Y are random variables, we denote by E(X | Y) the conditional expectation of X relative to Y, a classical notion in probability theory.
Z t denotes the random value of population size at time t. Note that either the
branching process (BP) has only one type of individuals and Z t takes non-negative
integer values or the BP has at least two types of individuals (i.e., the BP is
multitype [MT]) and Z t is a random vector, whose components are non-negative
integers representing the number of individuals in each type. We also used X t,j to
denote the offspring number at the next time t + 1 of the jth individual in the
population at time t. When the BP is density dependent, we denote by (i) X t,j the
offspring number at the next time t + 1 of the jth individual in a population of size i
at time t. We then denote by m = E(X t,j ) and (i) m = E( (i) X t,j ) the associated expectations. In a BP in random environment, m t is the (random) mean offspring number
at time t conditional on the value of the environmental process ζ t at time t. The
equivalent quantity in a MT BP is a matrix, that we denoted by M = (m α,β ) 1≤α,β≤d ,
whose dominant eigenvalue, µ, describes the asymptotic behavior of the BP. We
denote by (b k ) k∈N* the quasi-stationary distribution (QSD) of a BGW BP and by µ
the asymptotic growth rate of a BP.
Finally, we used the following abbreviations in this chapter:
• PVA, population viability analysis
• BP, (discrete-time) branching process
• BGW BP, Bienaym´ e-Galton-Watson branching process
• QSD, quasi-stationary distribution
• MT BP, multitype Bienaym´ e-Galton-Watson branching process
• DD BP, is density-dependent Bienaym´ e-Galton-Watson branching process
• BP RE, is Bienaym´ e-Galton-Watson branching process in random environment
Appendix 2. Basic Results on Markov Chains with a
Denumerable Number of States
Let (Y t ) t ∈ N be a sequence of random variables with values in N. This sequence of
random variables is an homogeneous Markov chain if for every t in N and (i t+1 , i t )
in N,
223
Yule GU (1924) A mathematical theory of evolution based on the conclusions of Dr JC
Willis, FRS. Philosophical Transactions of the Royal Society of London B 213:21–87
Appendix 1. Notation and Abbreviations
In this chapter, we denote by N = {0, 1, 2, . . .} and N* = {1, 2, . . .} the set of nonnegative and positive integers, respectively. Time t takes discrete values, in N.
We denote by Pr(A) the probability of the event A and by E(X) the expectation of the random variable X (if it is defined). If Pr(B) > 0, we denote by
Pr(A | B) =
Pr(A ∩ B)
Pr(B)
the conditional probability of A with respect to B. When
both X and Y are random variables, we denote by E(X | Y) the conditional expectation of X relative to Y, a classical notion in probability theory.
Z t denotes the random value of population size at time t. Note that either the
branching process (BP) has only one type of individuals and Z t takes non-negative
integer values or the BP has at least two types of individuals (i.e., the BP is
multitype [MT]) and Z t is a random vector, whose components are non-negative
integers representing the number of individuals in each type. We also used X t,j to
denote the offspring number at the next time t + 1 of the jth individual in the
population at time t. When the BP is density dependent, we denote by (i) X t,j the
offspring number at the next time t + 1 of the jth individual in a population of size i
at time t. We then denote by m = E(X t,j ) and (i) m = E( (i) X t,j ) the associated expectations. In a BP in random environment, m t is the (random) mean offspring number
at time t conditional on the value of the environmental process ζ t at time t. The
equivalent quantity in a MT BP is a matrix, that we denoted by M = (m α,β ) 1≤α,β≤d ,
whose dominant eigenvalue, µ, describes the asymptotic behavior of the BP. We
denote by (b k ) k∈N* the quasi-stationary distribution (QSD) of a BGW BP and by µ
the asymptotic growth rate of a BP.
Finally, we used the following abbreviations in this chapter:
• PVA, population viability analysis
• BP, (discrete-time) branching process
• BGW BP, Bienaym´ e-Galton-Watson branching process
• QSD, quasi-stationary distribution
• MT BP, multitype Bienaym´ e-Galton-Watson branching process
• DD BP, is density-dependent Bienaym´ e-Galton-Watson branching process
• BP RE, is Bienaym´ e-Galton-Watson branching process in random environment
Appendix 2. Basic Results on Markov Chains with a
Denumerable Number of States
Let (Y t ) t ∈ N be a sequence of random variables with values in N. This sequence of
random variables is an homogeneous Markov chain if for every t in N and (i t+1 , i t )
in N,