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Branching Processes Markovian Branching Processes Markovian Binary Trees Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Branching processes Sophie Hautphenne University of Melbourne & EPFL Stochastic Modelling Meets Phylogenetics 16{18 November 2015 1 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Introduction Branching processes Branching processes are stochastic processes describing the dynamics of a population of individuals which reproduce and die independently, according to some specific probability distributions. Branching processes have numerous applications in population biology and phylogenetics 2 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Introduction Branching processes There are many types of branching processes : Discrete time (Galton-Watson branching processes) Continuous time, with exponential lifetime distributions (Markovian branching process), or general lifetime distributions (age-dependent, Bellman-Harris branching process) Single type, or multitype (with finitely or 1ly many types) Individuals reproduction rules may depend on the actual size of the population (population size-dependent branching process) Branching processes can undergo catastrophes or live in a random environment ... 3 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Outline 1 Introduction 2 Galton-Watson branching processes 3 Markovian branching processes 4 Markovian binary trees 4 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees 1 Introduction 2 Galton-Watson branching processes 3 Markovian branching processes 4 Markovian binary trees 5 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees The Galton-Watson branching process Time is discrete and represents successive generations Each individual has a unit lifetime, at the end of which it might give birth to one or more offsprings simultaneously The offspring distribution is described by a random variable ξ taking non-negative integer values with corresponding probabilities pk = P[ξ = k]; k ≥ 0: All individuals behave independently of each other 6 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees The Galton-Watson branching process A realisation of a GW process through 3 generations starting with a single individual at generation 0 : 0 1 2 3 Generation n 7 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees The Galton-Watson branching process The population size at generation n is denoted by Zn We have the branching process equation Zn−1 X Zn = ξi ; n ≥ 1; i=1 where ξ1; ξ2;::: are i.i.d. copies of ξ. The process fZn; n ≥ 0g is a discrete-time Markov chain on the state space f0; 1; 2; 3;:::g where state 0 is absorbing and all other states are transient. 8 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Population size distribution The probability generating function (p.g.f.) of ξ is 1 X P(s) := E[sξ] = P[ξ = k]sk ; s 2 [0; 1]: k=0 If Z0 = 1, then P(s) corresponds to the p.g.f of Z1. Let Pn(s) denote the p.g.f. of Zn, 1 X k Pn(s) := P[Zn = k]s : k=0 9 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Composition of probability generating functions Define the random variable SN as N X SN := Xi ; i=1 where Xi are i.i.d. with p.g.f. GX (s), and N is an independent + random variable taking values in Z , with p.g.f. GN (s). Then the p.g.f. of SN is given by GSN = GN (GX (s)): 10 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Population size distribution Recall that for all n ≥ 1, Zn−1 X Zn = ξi i=1 where ξ1; ξ2;::: are i.i.d. copies of ξ. Proposition Conditionally on Z0 = 1, the p.g.f. of Zn satisfies Pn(s) = Pn−1(P(s)) = P(P(::: P(s))) | {z } n = P(Pn−1(s)); n ≥ 1; with P1(s) = P(s). 11 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Examples of progeny distributions Binary case : 0 w.p. 1=3; ξ = 2 w.p. 2=3 X 1 2 ! P(s) = P[ξ = k]sk = + s2 3 3 k Geometric case : ξ ∼ Geom(p), p P[ξ = k] = (1 − p)k p ! P(s) = 1 − (1 − p)s Poisson case : ξ ∼ Poisson(λ), P[ξ = k] = e−λλk =k! ! P(s) = eλ(s−1) 12 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Mean progeny and criticality The mean progeny is m := E[ξ] = P0(1−) 2 (0; 1): Corollary The mean population size at generation n, conditional on Z0 = 1, is given by E 0 n Mn := [Zn j Z0 = 1] = Pn(1−) = m : m < 1 : subcritical case, Mn & 0 as n ! 1 m = 1 : critical case Mn = 1 for all n m > 1 : supercritical case, Mn % 1 as n ! 1 13 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Extinction probability Let qn = P[Zn = 0] = Pn(0) be the probability that the nth generation is empty. If Z0 = 1, then q0 = 0 and Pn(s) = P(Pn−1(s)) ) qn = P(qn−1); n ≥ 1: The probability of extinction of the branching process is q = lim P[Zn = 0] = P[ lim Zn = 0] ) q = P(q) n!1 n!1 Theorem The extinction probability q is the minimal nonnegative solution of the fixed point equation s = P(s): 14 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Computation of q In practice, when s = P(s) cannot be solved explicitly, q is obtained as the limit of the sequence qn computed using the functional iteration qn = P(qn−1) with q0 = 0. Examples : Binary case : s = (1=3) + (2=3)s2 ! q = 1=2 p Geometric case : s = ! q = min(p=(1 − p); 1), 1 − (1 − p)s i.e. 1 if p ≥ 1=2 q = p=(1 − p) if p < 1=2 Poisson case : s = eλ(s−1) ! the functional iteration is useful here ! 15 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees A closer look at the progeny generating function P(s) is an increasing, convex function such that P(1) = 1 ! P(s) has at most two fixed points in [0; 1] : P(s) P(s) 6 6 1 1 r r p0 p0 r - - q = 1sq 1 s m = P0(1) ≤ 1 m = P0(1) > 1 ! P(s) has a fixed point q < 1 if and only if m > 1 16 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Extinction criterion Proposition (Extinction criterion) q < 1 , m > 1 Recall in the supercritical case, m > 1 ) q < 1 in the critical case, m = 1 ) q = 1 in the subcritical case, m > 1 ) q = 1 17 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Dichotomy : extinction or explosion ! Theorem Regardless the value of m, any state k 6= 0 is transient, so that lim P[Zn = k] = 0 for any k = 1; 2;:::. n!1 Moreover, P[Zn ! 0] + P[Zn ! 1] = 1: | {z } | {z } q 1−q In the supercritical case, conditionally on non-extinction, limn!1 Zn = +1 a.s. Can we tell more about the growth rate of Zn ? 18 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Limiting behaviour in the supercritical case 1 < m < 1 n Let Wn = Zn=m . fWn; n ≥ 0g is a martingale with E[Wn] = 1 for all n, so it converges a.s. to a nonnegative random variable W , Zn n W := lim ! Zn ∼ W m n!1 mn Theorem (Kesten-Stigum) Either P[W = 0] = q or P[W = 0] = 1. The following are equivalent : (i) E[ξ log+ ξ] < 1 (ii) P[W = 0] = q 1 (iii) Wn converges in mean (L ) (iv) E[W ] = 1. 19 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Quasi-stationary distribution in the subcritical case m < 1 Theorem (Yaglom) For each j = 1; 2;:::; lim P[Zn = j j Zn 6= 0] = bj n!1 P exists, and j bj = 1. P k Moreover, the p.g.f. G(s) = k bk s satisfies the equation G(P(s)) = m G(s) + 1 − m: In addition, the vector b = (b1; b2;:::) satisfies bQ = m b; where Q is the truncated probability transition matrix of the GW restricted to the transient states. 20 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees The critical case m = 1 Theorem (Kesten-Ney-Spitzer) Assume σ2 :=Var(ξ) < 1. Then we have Kolmogorov's estimate 2 lim n P[Zn 6= 0] = : n!1 σ2 Yaglom's universal limit law 2 lim P[Zn=n ≥ x j Zn 6= 0] = exp(−2x/σ ); x > 0: n!1 21 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Multitype Galton-Watson branching process Suppose now there are r > 1 types of individuals, each type having its own reproduction law. Example with r = 2 : 0 1 2 3 Generation n 22 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Multitype Galton-Watson branching process Population size vector : Zn = (Zn1; Zn2;:::; Znr ); n 2 N, where Zni : # of individuals of type i at the nth generation r fZng is an r-dimensional Markov process with state space N and an absorbing state 0 = (0; 0;:::; 0)T Progeny distribution : pij : j = (j1; j2;:::; jr ), where pij = probability that a type i gives birth to j1 children of type1, j2 children of type2,. , jr children of type r: 23 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Multitype Galton-Watson branching processes Progeny generating vector P(s) = (P1(s); P2(s);:::; Pr (s)), where Pi (s) is the progeny generating function of an individual of type i r X j X Y jk Pi (s) = pij s = pij sk ; si 2 [0; 1] j2Nr j2Nr k=1 Mean progeny matrix M with elements @Pi (s) Mij = @sj s=1 = expected number of direct offsprings of type j born to a parent of type i Irreducible branching process ≡ M is irreducible 24 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Extinction probability As usual, we assume that the process starts with a single individual > Extinction probability vector q = (q1; q2;:::; qr ) , with h i qi = P lim jZnj = 0 '0 = i n!1 h i q = P lim jZnj = 0 '0 ; n!1 where '0 is the type of the first individual in generation 0.
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