# Extinction Moment for a Branching Tree Evolution with Birth Rate and Death Rate Both Depending on Age

Authors: Xi Hu; Yun-Zhi Yan; Zhong-Tuan Zheng; Hong-Yan Li; Hong-Yan Zhao
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, a branching tree evolution is established, in which the birth rate and the death rate are both dependent on node’s age. The extinction probability and the t-pre-extinction (extinct before timet) probability are studied, by which the distribution of the extinction moment can be given. The analytical formula and the approximation algorithm for the distribution of extinction moment are given; furthermore, the analytical formula and the approximation algorithm of extinction probability are given, and a necessary and sufficient condition of extinction with probability 1 is given. It is the first time to study the distribution of extinction time for the branching process with birth rate and the death rate both depending on node’s age, and the results will do great help in the theory of branching process. It is expected to be applied in the fields of biology, genetics, medicine, epidemiology, demography, nuclear physics, actuarial mathematics, algorithm, and data structures, etc.


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## Body

## 1. Introduction

The classical biological reproduction model G-W branching process [1] has been extended to different biological reproduction models, such as branching processes in random environments [2–4] and branching population evolution models [5–11]. The age-dependent branching process was introduced by Bellman and Harris [6]. In branching models, the population extinction problem is one of the primary research contents. Many problems in branching models related to the population extinction are studied, but the distribution of extinction moment is hardly involved. In this paper, a branching tree evolution is established, in which the birth rate and death rate are both dependent on node’s age. The extinction probability and the t-pre-extinction (extinct before time t) probability are studied, by which the distribution of the extinction moment can be given.The paper is organized as follows. The model is described and the existence theorem is presented inSection 2. In Section 3, the extinction probability is studied, and the analytical formula and the approximation algorithm of the extinction probability are given. A necessary and sufficient condition of extinction with probability 1 is also given. In Section 4, the t-pre-extinction probability is studied, the iterative integral equation with unique solution is established, which is satisfied by the t-pre-extinction probability, and the analytical formula and the approximation algorithm of t-pre-extinction probability are given. The stochastic order of extinction moment is studied in Section 5. The conclusions are presented in Section 6.


## 2. Description and Existence Theorem for the Model

In this paper, based on the mechanism of asexual reproduction of biological population, a continuous time random graph evolution is constructed, in which a node’s birth rate and death rate are both dependent on the node’s age.Given a population is composed of biological individuals (nodes). The evolution of the population is based on the following basic assumptions:(1)
All nodes in the population are homogeneous and mutually independent(2)
The node’s death rate in the population is a nonnegative functionα⋅ dependent on the node’s age, such that ∫0∞αtdt=+∞(3)
The node’s birth rate in the population is a nonnegative functionβ⋅ dependent on the node’s age(4)
Conditioned under a node being alive, the node’s reproduction behaviors in the future are conditional independent(5)
Conditioned under a node being alive, the node’s death is conditional independent with the node’s reproduction(6)
At initial timet=0, there is only one initial node in the population (this condition is not essential, only for convenience of presentation)(7)
In addition to the initial node, each of other nodes in the population has only one parent nodeBased on the above assumptions, the branching tree evolution is described as follows.Given a nodei in the population, its age is s at time t. For a sufficiently small period Δt>0, conditioned under node i being alive at time t, the conditional probability for node i being dead in the period t,t+Δt is αsΔt+oΔt, the conditional probability for node i producing one child node in the period t,t+Δt is βsΔt+oΔt, and the conditional probability for node i producing more than one child node in the period t,t+Δt is oΔt.In the population, if nodej is a child of node i, then there is a directed link from node i to node j. When at least one of the parent and child dies, the link between them is a virtual (dotted) line, and the dead node is called a virtual node. Otherwise, it is called a real node, and so on. At time t≥0, all nodes (real and virtual) and directed links (real and virtual) construct a directed random tree, denoted by Gt⋅. And thus, the process of reproduction is an evolution of random trees, denoted by Gt⋅t≥0. As the evolution is characterized by the birth rate β⋅ and the death rate α⋅, therefore, the model is referred to as “branching tree evolution with birth rate and death rate both depending on age,” denoted by Gtβ⋅,α⋅t≥0.According to the definition of the model,∀t≥0, the number of offspring born in period 0,t is finite, and no more than one offspring will be born at the same time. Therefore, the initial node and all its offspring nodes can be ordered as 1,2,…,n,… according to the order of birth time.∀n≥1, denote(1)n⟶=1,2,…,n,f⟶n=f1,f2,…,fn:f1=0,1≤fk≤k−1,2≤k≤n,i⟶n=i1,i2,…,in:ik∈0,1,1≤k≤n,b⟶n=b1,b2,…,bn:bi∈R+,1≤i≤n,and0=b1<b2<⋅⋅⋅<bn,where n⟶ is the vector of the labeled nodes; f⟶n is the vector of the adjacency relation (parent-child relation) between nodes: f1=0 means that the initial node has no parent node. fk=j,1≤j≤k−1 indicates that node j is the parent of node k, 2≤k≤n; i⟶n is the vector of node’s alive-death status: ik=1 denotes that node k is alive and ik=0 denotes that node k is dead, 1≤k≤n; and b⟶n is the birth time vector: b1=0 represents there is an initial node at time t=0, bj is the time when node j is born, 2≤j≤n, and b1<b2<⋯<bn implies that no more than one node is born at the same time.∀n≥1, denote(2)F¯n=f⟶n,E¯n=i⟶n,B¯n=b⟶n,C3×n=n⟶,f⟶n,i⟶nT,Sn3=n⟶×F¯n×E¯n,C4×n=n⟶,f⟶n,i⟶n,b⟶nT,Sn=n⟶×F¯n×E¯n×B¯n,S=∪n=1∞Sn.∀C4×n=n⟶,f⟶n,i⟶n,b⟶nT, denote IC4×n=k:ik=1. ∀k∈IC4×n, denote (3)DkC4×n=C4×n+1=n+1⟶,f⟶n+1,r⟶n+1,b⟶n+1T:b⟶n+1=b⟶n,bn+1;f⟶n+1=f⟶n,fn+1,fn+1=k;rj≤ij,1≤j≤n.∀C4×n=n⟶,f⟶n,i⟶n,b⟶n, and bn≤s<t, define the function fnks,t,C4×n+1|C4×n on Sn+1. ∀C4×n+1=n+1⟶,f⟶n+1,r⟶n+1,b⟶n+1,(4)fnks,t,C4×n+1|C4×n=I˜DkC4×nC4×n+1∏j∈IC4×nj≠k,rj=0∫ste−∫s−bjy−bjβuduαye−∫syαududy×∏j∈IC4×nj≠k,rj=1e−∫s−bjt−bjβudue−∫stαudu×βbn+1−bkg1t,bk,rkg2t,bn+1,rn+1,where I˜A⋅ is a indicative function, and(5)g1t,bk,rk=e−∫stαudue−∫s−bjt−bjβudu,rk=1,e−∫s−bkbn+1−bkβudu∫bn+1te−∫bn+1−bky−bkβuduαye−∫syαududy,rk=0,g2t,bk,rn+1=e−∫bn+1tαudue−∫bn+1tβudu,rn+1=1,∫bn+1te−∫bn+1yβuduαye−∫bn+1yαududy,rn+1=0.∀C4×n=n⟶,f⟶n,i⟶n,b⟶n, and bn≤s<t, define the function fns,t,C4×n+1|C4×n on Sn+1:(6)fns,t,C4×n+1|C4×n=∑k∈IC4×nfnks,t,C4×n+1|C4×n.Letdbn+1 be a Lebesgue measure on s,t, for a given b⟶n, and bn≤s, then δb⟶n×dbn+1 is a measure on B¯n+1,ℬB¯n+1. Let μn+1⋅ be a count measure on Sn+13,ℬSn+13, denote vn+1⋅=μn+1×δb⟶n×dbn+1⋅, and then vn+1⋅ is a measure on Sn+1,ℬSn+1. Define ∀Dn+1∈ℬSn+1,(7)Qns,t,Dn+1|C4×n=∫Dn+1fns,t,C4×n+1|C4×nvn+1dC4×n+1.Then,∀0≤s<t, C4×n∈Sn, Qns,t,⋅|C4×n is a measure on Sn+1,ℬSn+1, ∀0≤s<t, Dn+1∈ℬSn+1, Qns,t,Dn+1|⋅ is a measurable function on Sn,ℬSn.Let(8)A2×n=n⟶f⟶n,B2×n=i⟶nb⟶n,C4×n=A2×nB2×n.LetNt be the number of nodes in the random branching tree Gtβ⋅,α⋅, then Gtβ⋅,α⋅ can be expressed by a 2×Nt matrix, i.e.,(9)Gtβ⋅,α⋅=A2×Nt.The birth time and the alive-death status of theNt nodes in Gtβ⋅,α⋅ can be expressed by the 2×Nt matrix B2×Nt, and denote(10)Xt=A2×NtB2×Nt=C4×Nt.We have the following theorem.Theorem (existence)Gtβ⋅,α⋅t≥0 is the marginal process of the nonhomogeneous Markov process Xtt≥0 in the state space S, where the transfer function of Xtt≥0 is ∀D∈ℬS,0≤s<t,n≥1,C4×n=n⟶,f⟶n,i⟶n,b⟶nT∈Sn, here Dm∈ℬSm(11)PXt∈D|Xs=C4×n=∑m=1∞PXt∈D,Nt=m|Xs=C4×n=∑m=1∞PXt∈Dm|Xt=C4×n=PXt∈Dn|Xs=C4×n+∑m=n+1∞PXt∈Dm|Xs=C4×n,where(12)PXt∈Dn|Xs=C4×n=∑r⟶n∈D^n∏j∈IC4×nrj=0∫ste−λy−bj−λs−bjαe−αy−sdy×∏j∈IC4×nrj=1e−λt−bj−λs−bje−αt−s,(13)PXt∈Dn+k|Xs=C4×n=PXt∈Dn+k|Xs=C4×n=∫Sn+1⋯∫Sn+k−1∫Dn+kQns,bn+1,dC4×n+1|C4×n⋯×Qn+k−2bn+k−2,bn+k−1,dC4×n+k−1|C4×n+k−2×Qn+k−1bn+k−1,t,dC4×n+k|C4×n+k−1,k≥1,where D^n=r⟶n:n⟶,f⟶n,r⟶n,b⟶n∈Dn. Substituting (12) and (13) into (11), the transfer function of Xtt≥0 is obtained.It is not difficult to prove thatS is a Borel subset of the separable complete distance space R∞, and the existence theorem of Xtt≥0 can be proved by the existence theorem of Markov process. Gtβ⋅,α⋅t≥0 is a marginal process of Xtt≥0, and thus, the existence of Gtβ⋅,α⋅t≥0 is proved.


## 3. The Extinction Probability

Define(14)Tω=inft>0:population extinction in the period0,t;Pt=PTω≤t,0≤t<∞;P∞=PTω<∞.Tω is called the extinction moment, at which the population extinct. The probability Pt is called t-pre-extinction probability, which is the probability of the population extinct before time t, and the probability P∞ is called extinction probability. The distribution of extinction moment Tω is given by t-pre-extinction probability and PTω=∞=1−PTω<∞=1−P∞. If P∞=1, i.e., PTω=∞=0. Then, Tω is a real-valued random variable, so the t-pre-extinction probability Pt,t≥0 is the distribution function of Tω.In this section, the extinction probability for the branching tree evolutionGtβ⋅,α⋅t≥0 is studied, the analytical formula and the approximation algorithm of extinction probability are given, and a necessary and sufficient condition of extinction with probability 1 is also given.Letηt be the number of nodes that are alive in the population at time t, then Pt=Pηt=0,0≤t<∞. Obviously, ηs=0⊆ηt=0 when s≤t, and P∞=limt⟶∞Pηt=0.It is obvious thatPt has the following properties:(1)
∀0<s<t,0=P0<Ps<Pt<P∞(2)
Pt is continuous on 0,∞Lemma 1.
Given a nodei in the population, its lifespan is Y, and then Y has the probability density function:(15)fYt=αte−∫0tαudu,t≥0,0,t<0,where α⋅ is the node’s death rate.
It is easy to prove Lemma1.Theorem 1.
The following iterative integral equation is satisfied by the t-pre-extinction probabilityPt,0<t<∞(16)Pt=∫0te−∫0s1−Pt−uβudu⋅αse−∫0sαududs,0<t<∞,where β⋅ is the node’s birth rate and α⋅ is the node’s death rate.Proof.
∀0<t<∞,0<Pt<1, Let Y be the node’s lifespan, then we get(17)Pt=Pηt=0=Pηt=0,Y≤t=∫0tPηt=0|Y=s⋅αse−∫0sαududs.
In the following, we first calculatePηt=0|Y=s. Equally divide the interval 0,s into n intervals, denote δ=s/n, and let Ak be the random event: the initial node produces a child node in the period kδ,k+1δ and the offspring of this child node extinct before time t or the initial node does not produce a child node in the period kδ,k+1δ, 0≤k≤n−1.
Whenδ is sufficiently small, the probability that the initial node does not produce a child node in the period kδ,k+1δ is 1−βkδ⋅δ+oδ; the probability of producing more than one child node is oδ; the probability of producing one child node is βkδ⋅δ+oδ, and this child node’s offspring extinct before time t with probability Pt−kδ,0≤k≤n−1, so(18)PAk|Y=s=βkδ⋅δ⋅Pt−kδ+1−βkδ⋅δ+oδ,0≤k≤n−1.
Noting the independent assumptions of the model, we have(19)Pηt=0|Y=s=limδ⟶0P∩k=0n−1Ak|Y=s=limδ⟶0∏k=0n−1PAk|Y=s=limδ⟶0∏k=0n−1βkδ⋅δ⋅Pt−kδ+1−βkδ⋅δ+oδ=limδ⟶0∏k=0n−11−βkδ⋅1−Pt−kδ⋅δ+oδ.
Then,(20)lnPηt=0|Y=s=limδ⟶0∑k=0n−1ln1−βkδ⋅1−Pt−kδ⋅δ+oδ=−limδ⟶0∑k=0n−1βkδ⋅1−Pt−kδ⋅δ+oδ.
That is(21)Pηt=0|Y=s=e−∫0s1−Pt−uβudu.
So(22)Pt=∫0te−∫0s1−Pt−uβudu⋅αse−∫0sαududs.
Thus, the theorem is proved.
Denote(23)gx=∫0∞e−∫0s1−xβudu⋅αse−∫0sαududs,0≤x≤1.Lemma 2.
The extinction probabilityP∞ is a solution of the equation(24)x=gx,0≤x≤1.Proof.
0<t0<t<∞,(25)Pt−gP∞=∫0te−∫0s1−Pt−uβudu⋅αse−∫0sαududs−∫0∞e−∫0s1−P∞βudu⋅αse−∫0sαududs≤∫0t0e−∫0s1−Pt−uβudu⋅αse−∫0sαududs−∫0t0e−∫0s1−P∞βudu⋅αse−∫0sαududs+∫t0te−∫0s1−Pt−uβudu⋅αse−∫0sαududs−∫t0te−∫0s1−P∞βudu⋅αse−∫0sαududs+∫t∞e−∫0s1−Pt−uβudu⋅αse−∫0sαududs,∀ε>0,∃t0,0<t0<t, and let t be large enough, such that(26)∫0t0e−∫0s1−Pt−uβudu⋅αse−∫0sαududs−∫0t0e−∫0s1−P∞βudu⋅αse−∫0sαududs<ε3,∫t0te−∫0s1−Pt−uβudu⋅αse−∫0sαududs−∫t0te−∫0s1−P∞βudu⋅αse−∫0sαududs<ε3,∫t∞e−∫0s1−P∞βudu⋅αse−∫0sαududs<ε3,i.e., limt⟶∞Pt=gP∞, which imply P∞=gP∞.
Thus, Lemma2 is proved.
It is easy to prove that the functiongx has the following properties.Lemma 3.
(1)
g0>0,g1=1(2)
gx is increasing on 0,1(3)
gx is a strictly concave function on 0,1Theorem 2.
The extinction probabilityP∞ is the smallest solution of the equation(27)x=gx,0≤x≤1.Proof.
By Lemma 3,gx is a strictly concave function on 0,1, and thus, g˜x=gx−x is also a strictly concave function on 0,1. It is easy to see that any strictly concave function has at most two different roots in its definition domain; hence, g˜x=0 has at most two different solutions on 0,1, one of which is x=1. Let x=q be the smallest solution of the equation x=gx.(1)
Ifq=1, since q=1 is the smallest solution of the equation, then the equation has no solution in 0,1. But, by Lemma 2, P∞ is the solution of the equation, infer that P∞=1, i.e., P∞ is the smallest solution of the equation x=gx.(2)
0<q<1, let x such that q<x<1, since q is the unique solution of the equation x=gx in 0,1, it is easy to see that x>gx.
It is easy to prove∀t>0,Pt≤q. In fact, suppose contrarily ∃t>0, such that Pt>q, then(28)Pt>gPt.
Noting thatPt is increasing, then(29)Pt=∫0te−∫0s1−Pt−uβudu⋅αse−∫0sαududs≤∫0te−∫0s1−Ptβudu⋅αse−∫0sαududs≤∫0∞e−∫0s1−Ptβudu⋅αse−∫0sαududs=gPt.
This contradictsPt>gPt, so the above assumption is not true. By Lemma 2, P∞ is the solution of the equation, i.e., P∞=gP∞≤q. By the fact that q is the smallest solution of the equation, imply that P∞=q. So P∞ is the smallest solution of the equation. Therefore, the theorem is proved.Corollary 1.
Ifβ⋅=β>0, then(30)P∞=αβ∧1.
Furthermore,P∞=1⇔α≥β.Proof.
Ifβ⋅=β>0, then(31)gx=∫0∞e−1−x∫0sβudu⋅αe−αsds=∫0∞αe−1−xβ+αsds=α1−xβ+α.
Letgx=x, that is,(32)α1−xβ+α=x,βx2−β+αx+α=0.
Obviously,Δ=β+α2−4βα=β−α2≥0; the two roots of the above equation are as follows:(33)x1,2=β+α±β−α2β.
Then,(34)P∞=αβ∧1,thus,(35)P∞=1⇔α≥β.
The proof is completed.
As a consequence of Theorem2, a sufficient condition for P∞=1 is given.Corollary 2.
∀s>0, if ∫0sβudu≤∫0sαudu, then P∞=1.Proof.
According to the assumptions, there is(36)gx=∫0∞e−1−x∫0sβudu⋅αse−∫0sαududs≥∫0∞e−1−x∫0sαudu⋅αse−∫0sαududs=12−x∫0∞2−xαse−2−x∫0sαududs=12−x.
If0<x≤1 is a solution of the equation x=gx, then x satisfies(37)gx=x≥12−x.
Thus,x2−2x+1≤0, i.e., x−12≤0, so x=1, deduce P∞=1.
Corollary2 shows that when the death rate is greater than the birth rate, the population is certainly extinct, which is intuitive.Corollary 3.
LetGtβ1⋅,α⋅t≥0 and Gtβ2⋅,α⋅t≥0 be two branching tree evolutions with different birth rates and the same death rate. The corresponding extinction probabilities are denoted by P1∞ and P2∞, respectively. If β1u≥β2u,u≥0, then P1∞≤P2∞.Proof.
∀0≤x≤1, gix=∫0∞e−1−x∫0sβiudu⋅αse−∫0sαududs,i=1,2.
Ifβ1u≥β2u,u≥0, then by the definition of gix,i=1,2, it is easy to see that g1x≤g2x,0≤x≤1, and g1x−x≤g2x−x, 0≤x≤1.
Sincegix−x is a continuous function with at least one smallest root on 0,1(i=1,2), and 0<g10≤g20, therefore, the smallest root of g1x−x on 0,1 is less than or equal to the smallest root of g2x−x on 0,1, and by Theorem 2, there is(38)P1∞≤P2∞.
The proof is completed.
Corollary3 implies that, for two models with the same death rate, the higher the birth rate is, the lower the extinction probability is, which is intuitive.Corollary 4.
LetGtβ⋅,α1⋅t≥0 and Gtβ⋅α2⋅t≥0 be two branching tree evolutions with different death rates but with the same birth rate. The corresponding extinction probabilities are denoted by P1∞ and P2∞, respectively, ∀s≥0, if ∫0sα1udu≤∫0sα2udu, then P1∞≤P2∞.Proof.
DenoteF¯is=1−Fis=e−∫0sαiudu,i=1,2, ∀s≥0, if ∫0sα1udu≤∫0sα2udu, then F¯1s≥F¯2s.
So∫0∞AsdF1s≤∫0∞AsdF2s for any decreasing function A⋅.
The corresponding functions togx are denoted by g1x and g2x, respectively. Noting that e−1−x∫0sβudu is a decreasing function with s, so by the definition of(39)gix=∫0∞e−1−x∫0sβududFis,i=1,2.imply g1x≤g2x,0≤x≤1, then g1x−x≤g2x−x, 0≤x≤1,
Sincegix−x is a continuous function with at least one smallest root on 0,1, i=1,2, and 0<g10≤g20, therefore, the smallest root of g1x−x on 0,1 is less than or equal to the smallest root of g2x−x on 0,1, and by Theorem 2,(40)P1∞≤P2∞.
Corollary4 shows that, for two models with the same birth rate, the randomly longer the lifespan is, the smaller the extinction probability is, which is intuitive.Theorem 3.
P∞=1⇔g′x<1,0<x<1.Proof.
(1)
Sufficiency: assumeg′x<1,0<x<1, let g˜x=gx−x,0≤x≤1, then g′˜x=g′x−1<0,0<x<1; i.e., g˜x is the decreasing function on 0,1. Noting that g˜0=g0>0,g˜1=g1−1=0, thus x=1 is the smallest root of g˜x on 0,1, and by Theorem 2, we get P∞=1.(2)
Necessity: assumeP∞=1, because gx is a strictly concave function on 0,1, g˜x is also a strictly concave function on 0,1, and g˜0=g0>0. In addition, by Theorem 2 and the assumptions, it is obvious that x=1 is the smallest root of g˜x on 0,1, so g˜x is decreasing on 0,1. Hence, g′˜x<0, that is, g′x<1,0<x<1.
Thus, the theorem is proved.
Forgx=∫0∞e−1−x∫0sβudu⋅αse−∫0sαududs,0≤x≤1 introduced above, noting that ∀0≤x≤1, 0<gx≤1, denote(41)g0x=x,0≤x≤1,g1x=gx,gnx=ggn−1x,n≥2,i.e., gn⋅ is the n times iteration of g⋅, and then, there is the following conclusion.Theorem 4.
∀0≤x<1, there is limn⟶∞gnx=P∞.Proof.
(1)
IfP∞=1, by g0>0 and Theorem 2, we have ∀0≤x<1, x<gx. For the increasing property of g⋅, so gnx<gn+1x,n≥1, in addition, g⋅ is continuous, then(42)q=limn⟶∞gnx=limn⟶∞ggn−1x=glimn⟶∞gn−1x,i.e.,q=gq, and then, we can get q=1 by Theorem 2, i.e., limn⟶∞gnx=P∞.(2)
If0<P∞<1, ∀0≤x<P∞, then x<gx, and gnx<gn+1x,n≥1; thus,(43)q=limn⟶∞gnx=limn⟶∞ggn−1x=glimn⟶∞gn−1x=gq.By Theorem2, q=P∞, i.e., limn⟶∞gnx=P∞.(3)
If0<P∞<1, ∀P∞≤x<1, then x≥gx, and gnx≥gn+1x,n≥1, and thus,(44)q=limn⟶∞gnx=limn⟶∞ggn−1x=glimn⟶∞gn−1x=gq.Noting that the equationx=gx has no root on the interval P∞,1, so q=P∞, i.e.,(45)limn⟶∞gnx=P∞.
Note: the significance of Theorem4 is obvious. It gives a numerical method to calculate the asymptotic value of extinction probability. For any initial value x00≤x0<1, iteration value gnx0 is the asymptotic value of the extinction probability P∞.


## 4. Thet-Pre-Extinction Probability

In this section, the analytic formula and the approximation algorithm of t-pre-extinction probability are given, and the iterative integral equation with unique solution is established, which is satisfied by the t-pre-extinction probability.Lett>0, n≥1, denote Δn=t/2n. Divide the interval 0,t equally into 2n intervals kΔn,k+1Δn,k=0,1,2,…,2n−1. Step function is defined as follows:(46)Hns=0,0≤s≤Δn,HnΔn,Δn<s≤2Δn,⋯⋯HnkΔn,kΔn<s≤k+1Δn,⋯⋯Hnt−Δn,t−Δn<s≤t,where(47)HnΔn=∫0Δne−∫0s1−HnΔn−uβudu⋅αse−∫0sαududs;HnkΔn=∫0kΔne−∫0s1−HnkΔn−uβudu⋅αse−∫0sαududs,k=1,2,…,2n−1.We always assume that the birth rate functionβ⋅ is bounded in any finite interval; denote(48)β=sup0≤u≤tβu.Theorem 5.
(1)
∀n≥1, Hn⋅ is nondecreasing on 0,t(2)
Hn⋅n≥1 is a monotonic increasing sequence of functions(3)
∀t≥0, limn⟶∞Hnt=PtProof.
(1)
To proveHn⋅ is a nondecreasing function on 0,t because(49)HnΔn=∫0Δne−∫0s1−HnΔn−uβudu⋅αse−∫0sαududs≥∫0Δnαse−∫0sαududs>0.So,Hn⋅ is nondecreasing on 0,2Δn; suppose inductively that Hn⋅ is nondecreasing on 0,kΔn, then(50)HnkΔn=∫0kΔne−∫0s1−HnkΔn−uβudu⋅αse−∫0sαududs≥∫0k−1Δne−∫0s1−HnkΔn−uβudu⋅αse−∫0sαududs≥∫0k−1Δne−∫0s1−Hnk−1Δn−uβudu⋅αse−∫0sαududs=Hnk−1Δn.Thus,Hn⋅ is nondecreasing on 0,k+1Δn. It is deduced by mathematical induction that Hn⋅ is nondecreasing on 0,t.(2)
∀n≥1, to prove Hn⋅≤Hn+1⋅ on 0,kΔn, k=1,2,…,2n.(51)0≤s≤Δn,Hns=0,0≤s≤Δn2=Δn+1,Hn+1s=0,Δn2<s≤Δn=2Δn+1,Hn+1s=Hn+1Δn+1>0.Hence,Hn⋅≤Hn+1⋅ on 0,Δn. Suppose inductively that Hn⋅≤Hn+1⋅ on 0,kΔn, by the definitions of Hn⋅ and Hn+1⋅, we have(52)Hns=HnkΔn,kΔn<s≤k+1Δn,Hn+1s=Hn+12kΔn+1,2kΔn+1=kΔn<s≤kΔn+Δn+1,Hn+1s=Hn+12k+1Δn+1,2k+1Δn+1<s≤k+1Δn,where(53)Hn+12k+1Δn+1≥Hn+12kΔn+1=Hn+1kΔn=∫0kΔne−∫0s1−Hn+1kΔn−uβudu⋅αse−∫0sαududs≥∫0kΔne−∫0s1−HnkΔn−uβudu⋅αse−∫0sαududs=HnkΔn.Thus, it can be proved by mathematical induction thatHns≤Hn+1s, s∈0,t,n≥1, i.e., Hn⋅n≥1 is a monotonic increasing sequence of functions.(3)
For simplicity, denoteΔ=Δn, H⋅=Hn⋅. Because P⋅ is uniform continuous on 0,t, so ∀ε>0, when n is sufficiently large, that is Δ=Δn sufficiently small.(54)Pu−Pv<ε,u−v<Δ.
The following conclusion can be deduced by mathematical induction:(55)0≤PkΔ−HkΔ≤kεβΔ+oεβΔ,k=1,2,…,2n.
It is easy to prove thatHkΔ≤PkΔ, k=1,2,…,2n.(56)PΔ−HΔ=∫0Δe−∫0s1−PΔ−uβudu−e−∫0s1−HΔ−uβudu⋅αse−∫0sαududs=∫0Δe∫0sPΔ−uβudu−1⋅e−∫0sβudu⋅αse−∫0sαududs≤∫0ΔεβΔ+0εβΔ⋅αse−∫0sαududs≤εβΔ+oεβΔ,P2Δ−H2Δ≤∫02Δe∫0sP2Δ−u−H2Δ−uβudu−1⋅αse−∫0sαududs=∫0Δe∫0sP2Δ−u−H2Δ−uβudu−1⋅αse−∫0sαududs+∫Δ2Δe∫0sP2Δ−u−H2Δ−uβudu−1⋅αse−∫0sαududs.
When0≤s≤Δ,(57)∫0sP2Δ−u−H2Δ−uβudu≤∫0sP2Δ−u−PΔ+PΔ−HΔβudu≤εβΔ+oεβΔ≤2εβΔ+o2εβΔ.
WhenΔ≤s≤2Δ,(58)∫0sP2Δ−u−H2Δ−uβudu=∫0ΔP2Δ−u−H2Δ−uβudu+∫ΔsP2Δ−u−H2Δ−uβudu≤εβΔ+oεβΔ+∫ΔsP2Δ−u−PΔ+PΔ−HΔβudu≤2εβΔ+o2εβΔ.
So,(59)P2Δ−H2Δ≤∫0Δ2εβΔ+o2εβΔ⋅αse−∫0sαududs+∫Δ2Δ2εβΔ+o2εβΔ⋅αse−∫0sαududs=∫02Δ2εβΔ+o2εβΔ⋅αse−∫0sαududs≤2εβΔ+o2εβΔ.
Suppose inductively thatPjΔ−HjΔ≤jεβΔ+ojεβΔ,1≤j≤k−1.
Then,(60)PkΔ−HkΔ≤∫0kΔe∫0sPkΔ−u−HkΔ−uβudu−1⋅αse−∫0sαududs=∑j=0k−1∫jΔj+1Δe∫0sPkΔ−u−HkΔ−uβudu−1⋅αse−∫0sαududs.
Whenk−1Δ≤s≤kΔ,(61)∫0sPkΔ−u−HkΔ−uβudu=∑j=0k−2∫jΔj+1ΔPkΔ−u−HkΔ−uβudu+∫k−1ΔsPkΔ−u−HkΔ−uβudu,∀0≤j≤k−2,(62)∫jΔj+1ΔPkΔ−u−HkΔ−uβudu=∫jΔj+1ΔPkΔ−u−Pk−j−1Δ+Pk−j−1Δ−Hk−j−1Δβudu<εβΔ+oεβΔ,∫k−1ΔsPkΔ−u−HkΔ−uβudu≤εβΔ+oεβΔ.
So(63)∫0sPkΔ−u−HkΔ−uβudu≤kεβΔ+okεβΔ.
It is obvious that whens1≤s2, there is(64)∫0s1PkΔ−u−HkΔ−uβudu≤∫0s2PkΔ−u−HkΔ−uβudu.
So(65)PkΔ−HkΔ≤∑j=0k−1∫jΔj+1ΔkεβΔ+okεβΔ⋅αse−∫0sαududs=∫0kΔkεβΔ+okεβΔ⋅αse−∫0sαududs≤kεβΔ+okεβΔ.
It is proved deductively that0≤PkΔ−HkΔ≤kεβΔ+okεβΔ, k=1,2,…,2n.
Especially,0≤Pt−Ht≤εβt+oεβt.
That is,(66)limn⟶∞Hnt=Pt.
The proof is complete.Theorem 6.
The t-pre-extinction probabilityPt is the unique solution of the iterative integral equation in Theorem 1.Proof.
According to Theorem1, the t-pre-extinction probability is the solution of the iterative integral equation in Theorem 1, and it is not difficult to deduce by Theorem 5 that the solution of the iterative integral equation is unique. Therefore, the t-pre-extinction probability is the unique solution of the iterative integral equation. The theorem is proved.


## 5. The Stochastic Order of Extinction Moment

IfP∞=1, then the extinction moment Tω is a real-valued random variable, and the t-pre-extinction probability Pt is the distribution function of Tω. In this section, we study the stochastic order of the extinction moment for different branching tree evolutions.Theorem 7.
(1)
LetP1∞ and T1ω be, respectively, the extinction probability and extinction moment for the branching tree evolutions Gtβ1⋅,α⋅t≥0; let P2∞ and T2ω be, respectively, the extinction probability and extinction moment for the branching tree evolutions Gtβ2⋅,α⋅t≥0. If P1∞=1, P2∞=1, and β1⋅≤β2⋅, then T1ω is stochastically smaller than T2ω, that is T1ω≤s.t.T2ω.(2)
LetP˜1∞ and T˜1ω be, respectively, the extinction probability and extinction moment for the branching tree evolutions Gtβ⋅,α1⋅t≥0; let P˜2∞ and T˜2ω be, respectively, the extinction probability and extinction moment for the branching tree evolutions Gtβ⋅,α2⋅t≥0. If P˜1∞=1, P˜2∞=1, and ∀t>0, ∫0tα1sds≤∫0tα2sds, then T˜2ω is stochastically smaller than T˜1ω, that is T˜2ω≤s.t.T˜1ω.Proof.
(1)
Corresponding to the branching tree evolutionGtβ1⋅,α⋅t≥0 and Gtβ2⋅,α⋅t≥0, similarly to Theorem 5, define the step function series as Hn1⋅n≥1 and Hn2⋅n≥1, respectively.By the hypothesis ofβ1⋅≤β2⋅ and the definition of Hni⋅,i=1,2, applying the mathematical induction, it is easy to prove that Hn1kΔn≥Hn2kΔn, 1≤k≤2n, n≥1, and thus Hn1⋅≥Hn2⋅. By Theorem 5,(67)P1t=PT1ω≤t=limn⟶∞Hn1t,t≥0,P2t=PT2ω≤t=limn⟶∞Hn2t,t≥0.Thus,P1t≥P2t, t≥0.BecauseT1ω≤s.t.T2ω⇔P1t≥P2t,t≥0, so T1ω≤s.t.T2ω. Thus, (1) is proved.(2)
Corresponding to the branching tree evolutionGtβ⋅,α1⋅t≥0 and Gtβ⋅,α2⋅t≥0, similarly to Theorem 5, define the step function series as H˜n1⋅n≥1 and H˜n2⋅n≥1, respectively, and denote Fit=1−e−∫0tαiudu,t≥0,i=1,2; then F1t≤F2t, 0≤t<∞. Denote Dis=I0,k△nse−∫0s1−H˜nik△n−uβudu,s⟶0,i=1,2; then Dis is a decreasing function of s. Applying the mathematical induction, we have(68)H˜n1k△n=∫0k△ne−∫0s1−H˜nik△n−uβududF1s=∫0∞D1sdF1s≤∫0∞D1sdF2s≤∫0∞D2sdF2s=H˜n2k△n,1≤k≤2n.
Thus,H˜n1⋅≤H˜n2⋅, and by Theorem 5,(69)P˜1t=PT˜1ω≤t=limn⟶∞H˜n1t,t≥0,P˜2t=PT˜2ω≤t=limn⟶∞H˜n2t,t≥0.
Then,P˜1t≤P˜2t, so T˜1ω≥s.t.T˜2ω. Thus, (2) is proved and theorem is proved.


## 6. Conclusions

This paper addresses an important problem in the field of branching process. The extinction probability and the t-pre-extinction probability are studied by constructing a branching tree evolution model in which the birth rate and the death rate are both dependent on node’s age. The analytical formula and the approximation algorithm for the distribution of extinction moment are given; furthermore, the analytical formula and the approximation algorithm of extinction probability are given, and a necessary and sufficient condition of extinction with probability 1 is given.Due to publishing constraints, only the population extinction is studied, the graph-topological properties and the age structure of nodes will be studied in subsequent papers.



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