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Multitype population size-dependent branching processes with dependent offspring
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Statistics & Probability Letters 70 (2004) 145–154 www.elsevier.com/locate/stapro
Multitype population size-dependent branching processes with dependent offspring$ Miguel Gonza´lez, Rodrigo Martı´ nez, Manuel Mota Facultad de Ciencias, Department of Mathematics, University of Extremadura, Badajoz 06071, Spain Received 5 November 2003; received in revised form 4 June 2004 Available online 21 September 2004
Abstract In this paper, the multitype population size-dependent branching process with dependent offspring is introduced and the probability of extinction of such a process is investigated. Sufficient conditions for both almost sure extinction and survival of the process are shown to depend on the Perron–Frobenius eigenvalue of adequate limit matrices. Finally, theoretical results are illustrated with an example. r 2004 Elsevier B.V. All rights reserved. Keywords: Multitype population size-dependent branching process; Dependent offspring
1. Introduction In branching process theory, it is usual to assume that each individual or particle reproduces independently of the rest and according to the same probability law. Though it seems to be a very strong assumption for biological populations or dynamical systems, the fact of being independent and identically distributed plays a fundamental role in some of the mathematical tools used in the study of branching processes. However several authors have considered the possibility of removing some of these conditions (e.g. see Cohn and Klebaner, 1986; Klebaner, 1994; Jagers, $ Research supported by the Ministerio de Ciencia y Tecnologı´ a and the FEDER through the Plan Nacional de Investigacio´n Cientı´ fica, Desarrollo e Innovacio´n Tecnolo´gica, Grant BFM2003-06074. Corresponding author. E-mail address: [email protected] (R. Martı´ nez).
0167-7152/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2004.08.012
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1999). On the lines of this research, we propose a multitype branching model in which the reproduction of different individuals does not follow necessarily the same law and, in addition, independence assumption is avoided. More specifically, in Section 2 we introduce the multitype population size-dependent branching process (MPDP) with dependent offspring and study some basic properties of this model. In Section 3 we investigate the probability of extinction for such a model. In that section we provide the main results of the paper where sufficient conditions for both, survival and almost sure extinction of the process are established. Finally in Section 4, as illustration, an example is considered and its evolution in particular situations is simulated.
2. The probability model The MPDP in discrete time has been widely studied (e.g. see Klebaner, 1989a,b; 1991). All these works suppose the individuals of a generation give birth to their descendants independently of each other and according to the same probability distribution for each type. Now we introduce a model generalizing the MPDP by removing the independence assumption and allowing individuals coexisting inside a generation, even those of the same type, to reproduce with different offspring distribution. Let us provide the mathematical definition of such a process. Definition 1. Let fX i;j;n ðzÞ : z 2 Nm 0 ; i ¼ 1; . . . ; m; j ¼ 1; . . . ; zi ; n ¼ 0; 1; . . .g be a sequence of mdimensional random vectors with non-negative integer valued coordinates and verifying the following hypotheses: (i) For every fixed z 2 Nm 0 ; the random vectors of the sequence fX ðz; nÞgnX0 defined by X ðz; nÞ :¼ ðX 1;1;n ðzÞ; . . . ; X 1;z1 ;n ðzÞ; . . . ; X m;1;n ðzÞ; . . . ; X m;zm ;n ðzÞÞ
(1)
are independent and identically distributed. (ii) For any z; z¯ 2 Nm ¯ 2 N such that na¯n; the random vectors X ðz; nÞ and X ð¯z; n¯ Þ are 0 and n; n independent. The sequence of m-dimensional random vectors fZðnÞgnX0 ; defined in the recursive way: Zð0Þ ¼ z;
z 2 Nm 0;
Zðn þ 1Þ ¼
m Z i ðnÞ X X
X i;j;n ðZðnÞÞ;
nX0
(2)
i¼1 j¼1
with
P0 1
¼ 0; i.e. the null vector, is called MPDP with dependent offspring.
Intuitively, each coordinate of the vector ZðnÞ represents the number of individuals of the corresponding type coexisting in the population in generation n. The m-dimensional vector X i;j;n ðzÞ denotes the offspring produced by the jth individual of type i in the nth generation provided that the number of individuals of the different types at this moment is given by the vector z. Remark 2. Discrete time population size-dependent models in the literature are particular cases of the MPDP with dependent offspring. In this sense a special mention deserves the branching model introduced by Cohn and Klebaner (1986).
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In the next result some basic properties of the process fZðnÞgnX0 are established. Proposition 3. The MPDP with dependent offspring fZðnÞgnX0 is a homogeneous Markov chain whose state space is a subset of Nm 0 : Moreover, the state 0 is absorbing, and if PðX ðz; 0Þ ¼ 0Þ40 for za0; then the state z is transient. Proof. The first statement is a direct consequence of (2). From the same equation can be deduced that the state 0 is absorbing. Now, let za0 be such that PðX ðz; 0Þ ¼ 0Þ40: Then PðZðnÞ ¼ z
for some njZð0Þ ¼ zÞp1 PðZð1Þ ¼ 0jZð0Þ ¼ zÞ ¼ 1 PðX ðz; 0Þ ¼ 0Þo1
and consequently the state z is transient. & Corollary 4. Let fZðnÞgnX0 be a MPDP with dependent offspring. Suppose that PðX ðz; 0Þ ¼ 0Þ40 for all za0; then PðZðnÞ ! 0Þ þ PðkZðnÞk ! 1Þ ¼ 1
(3)
with k k an arbitrary norm in Rm : Proof. Since za0 is transient, Markov chains theory assures that PðZðnÞ ¼ z i:o:Þ ¼ 0: On the other hand, if fZðnÞðoÞgnX0 converges neither 0 nor infinity, then there exists a positive real number M such that o2
1 [ 1 \
f0okZðnÞkpMg ¼ f0okZðnÞkpM i:o:g:
n0 ¼1 n¼n0
But f0okZðnÞkpM i:o:g ¼
[
fZðnÞ ¼ z i:o:g
0okzkpM
and therefore, Pð0okZðnÞkpM i:o:Þ ¼ 0: & Let MðzÞ :¼ ðmij ðzÞÞ1pi;jpm be the matrix whose coefficients are defined by 8P zi i;l;0 > < l¼1 E½X j ðzÞ if zi 40; mij ðzÞ :¼ i; j 2 f1; . . . ; mg: zi > : 0 if zi ¼ 0; Intuitively, mij ðzÞ can be interpreted as the average rate of type j descendants produced by each individual of type i when the total number of progenitors of the different types coexisting in the population is given by the vector z. It is immediate that E½Zðn þ 1ÞjZðnÞ ¼ ZðnÞMðZðnÞÞ a:s: Note that if the components of vector X ðz; nÞ defined in (1) corresponding to the offspring originated by individuals of type i (i.e. X i;l;n ðzÞ; l ¼ 1; . . . ; zi ) are identically distributed, then ðzÞ for all j 2 f1; . . . ; mg: mij ðzÞ ¼ E½X i;1;0 j
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3. The extinction problem The problem of extinction has been widely studied in population size-dependent processes both in the single type (see Klebaner, 1984) and multitype cases (see Klebaner, 1989a, 1991). In this section, we investigate such a problem for a MPDP with dependent offspring. Definition 5. Let fZðnÞgnX0 be a MPDP with dependent offspring. The extinction probability of fZðnÞgnX0 given that Zð0Þ ¼ z 2 Nm 0 is the probability qz :¼ PðZðnÞ ! 0jZð0Þ ¼ zÞ: The main objective of this paper is to determine sufficient conditions for the existence of a positive probability of survival for the process, i.e. qz o1; and sufficient conditions for the almost sure extinction of the process, i.e. qz ¼ 1: In this sense it is convenient to realize that if m 2 Rm þ; that is all the coordinates of m are positive real numbers, then fZðnÞ ! 0g ¼ fZðnÞm ! 0g: Some processes of the form fZðnÞmgnX0 ; with m 2 Rm þ ; will play a fundamental role in our investigation. 3.1. Survival of the process Firstly, we search sufficient conditions for the non-extinction of a MPDP with dependent offspring. Actually, we will show that under adequate assumptions the process not only has a positive probability of survival, but also of growing indefinitely, i.e. PðkZðnÞk ! 1Þ40: Note that, initially, we do not assume that relation (3) holds. In what follows, it will be necessary to define the matrix M :¼ ðmij Þ1pi;jpm by mij :¼ lim inf
kzk!1:zi a0
mij ðzÞ i; j 2 f1; . . . ; mg:
If M is irreducible, then there exists a positive eigenvalue, r ; called the Perron–Frobenius eigenvalue, with an associated right eigenvector, m ; with positive coordinates also (see Seneta, 1981 for details). Now we can provide the following result: Theorem 6. Let fZðnÞgnX0 be a MPDP with dependent offspring such that M is irreducible. If r 41 and there exists a constant dX0 such that E½jZðn þ 1Þm zMðzÞm j1þd jZðnÞ ¼ z ¼ Oðkzkd Þ; then qz o1 for kzk large enough. Proof. Since r 41 and its associated eigenvalue m has positive coordinates, there exist constants r41 and N 0 40 such that, for kzk4N 0 ; zMðzÞm Xrzm :
(4)
So we just have to prove that if kzk is large enough PðZðnÞm ! 1jZð0Þ ¼ zÞ40 or more specifically that there exist N40 and 0oor 1 such that if kzk4N; then ! 1 \ fZðn þ 1Þm 4ðr ÞZðnÞm gjZð0Þ ¼ z 40: P n¼0
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Let 0oor 1; applying Eq. (4) for kzk4N 0 40; zMðzÞm Xr : zm
(5)
Now denote, for simplicity, An :¼ fZðn þ 1Þm 4ðr ÞZðnÞm g;
n ¼ 0; 1; . . . :
Taking into account (5) and Markov inequality, if kzk4N 0 ; for every dX0; we have Zðn þ 1Þm c PðAn jZðnÞ ¼ zÞ ¼ P pr jZðnÞ ¼ z ZðnÞm Zðn þ 1Þm ZðnÞMðZðnÞÞm p jZðnÞ ¼ z pP ZðnÞm ZðnÞm
Zðn þ 1Þm ZðnÞMðZðnÞÞm
pP
XjZðnÞ ¼ z ZðnÞm p
E½jZðn þ 1Þm ZðnÞMðZðnÞÞm j1þd jZðnÞ ¼ z 1þd ðzm Þ1þd
:
By hypothesis, there exists dX0 verifying E½jZðn þ 1Þm ZðnÞMðZðnÞÞm j1þd jZðnÞ ¼ z ¼ Oðkzkd Þ: For this value of d it is possible to find constants C 0 ; N 1 40 such that if kzk4N 1 ; then E½jZðn þ 1Þm ZðnÞMðZðnÞÞm j1þd jZðnÞ ¼ zpC 0 ðzm Þd : Thus, if kzk4 maxfN 0 ; N 1 g and being C :¼ 1 d C 0 ; we get PðAcn jZðnÞ ¼ zÞp
C : zm
(6)
On the other hand, if kX1;
! !
n\ k k 1 \ Y
P An jZð0Þ ¼ z ¼ PðA0 jZð0Þ ¼ zÞ P An A \ fZð0Þ ¼ zg :
l¼0 l n¼0 n¼1 Tn 1 m For every z~ 2 Nm 0 ; define Tthat fBz~ gz~2N0 is a Tn 1 Bz~ :¼ l¼0 Al \ fZð0Þ ¼ zg \ fZðnÞ ¼ z~g: It is obvious A \ fZð0Þ ¼ partition of the set l¼0 Al \ fZð0Þ ¼ zg and also that ZðnÞm 4zm ðr Þn on n 1 l l¼0 zg: Consequently,
!
n 1
\ P An Al \ fZð0Þ ¼ zg ¼ PðAn j[z~ Bz~ ÞX inf PðAn jBz~ Þ
l¼0 z~m 4zm ðr Þn X
inf
z~m 4zm ðr Þn
PðAn jZðnÞ ¼ z~Þ:
ð7Þ
Since z~m 4zm ðr Þn and r 41; and using the equivalence of the norms in Rm ; there exists another constant N 2 40 so that if kzk4N 2 ; then k~zk4 maxfN 0 ; N 1 g: If
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kzk4N :¼ maxfN 0 ; N 1 ; N 2 g; taking into account (6) and (7), we obtain
!
n 1 C
\ P An Al \ fZð0Þ ¼ zg X1
l¼0 ðzm Þðr Þn and therefore, if we choose NXC we have, for kzk4N; ! ! 1 k 1 \ \ Y P An jZð0Þ ¼ z ¼ lim P An jZð0Þ ¼ z X 1 k!1
n¼0
This product is positive, because
n¼0
P1
n¼0 ðr
n¼0
C : zm ðr Þn
Þ o þ 1; which completes the proof. & n
Remark 7. In the particular case of the MPDP, Klebaner (1991) obtains, using other techniques, a result similar to the previous theorem. He assumes to exist some moments of an upper order than those required for the proof of Theorem 6. In Theorem 6 we have dealt with the case r 41: Nevertheless, survival is also possible if r ¼ 1: In fact, suppose that for every i; j 2 f1; . . . ; mg; there exists mij :¼ limkzk!1:zi a0 mij ðzÞ: Now M ¼ ðmij Þ1pi;jpm with r and m as defined above. Let us call s2 ðzÞ :¼ Var½Zðn þ 1Þm jZðnÞ ¼ z;
(8)
assumed to be finite for all z 2 Nm 0: The process fZðnÞm gnX0 can be written in the form Zðn þ 1Þm ¼ ZðnÞm þ gðZðnÞÞ þ xnþ1
(9)
with gðzÞ :¼ zMðzÞm zm for every z 2 Nm 0 and xnþ1 :¼ Zðn þ 1Þm ZðnÞMðZðnÞÞm : The stochastic difference equation given by (9) has been widely investigated in Kersting (1986). Following this methodology we may obtain the next result: Theorem 8. Let fZðnÞgnX0 be a MPDP with dependent offspring such that M is irreducible and for every C40; there exists nX0 for which PðZðnÞm 4CjZð0Þ ¼ zð0Þ Þ40: Suppose further that for some 0odp1 and some a40; at least one of the next equalities holds: E½jxnþ1 j2þd jZðnÞ ¼ z ¼ oðgðzÞðzm Þ1þd =ðlogðzm ÞÞ1þa Þ; E½jxnþ1 j2þd jZðnÞ ¼ z ¼ oððzm Þd s2 ðzÞ=ðlogðzm ÞÞ1þa Þ: ÞgðzÞ 41; then qzð0Þ o1: If r ¼ 1 and lim inf kzk!1 2ðzm s2 ðzÞ
We omit the proof that follows similar steps to that of Theorem 3 in Klebaner (1989b). 3.2. Almost sure extinction In the next paragraphs, we will focus our attention in determining sufficient conditions for the almost sure extinction of the MPDP with dependent offspring for which (3) holds. To this end, we assume to be under hypotheses of Corollary 4. Let M :¼ ðm ¯ ij Þ1pi;jpm be a matrix with entries m ¯ ij :¼ lim sup mij ðzÞ kzk!1
i; j 2 f1; . . . ; mg:
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Suppose M irreducible, with Perron–Frobenius eigenvalue and associated right eigenvector, r and m ; respectively. Since m has positive coordinates, it is immediate that (3) is equivalent to PðZðnÞm ! 0Þ þ PðZðnÞm ! 1Þ ¼ 1:
(10)
Now, we can formulate the following result: Theorem 9. Let fZðnÞgnX0 be a MPDP with dependent offspring such that M is irreducible. If one of the next conditions holds: (i) r o1; (ii) r ¼ 1 and, for every i; j 2 f1; . . . ; mg; mij ð¯zÞpm ¯ ij for k¯zk large enough, then qz ¼ 1 for every z 2 Nm 0: Proof. According to (10) it suffices to prove that PðZðnÞm ! 1jZð0Þ ¼ zÞ ¼ 0: Given A40; ! 1 X \ PðZðnÞm ! 1jZð0Þ ¼ zÞp P fZðnÞm XAgjZð0Þ ¼ z l¼0
nXl
1 X P min ZðnÞm XAjZð0Þ ¼ z ¼ ¼
l¼0 1 X
nXl
X
pðlÞ z~z P
l¼0 z~m XA
min ZðnÞm XAjZðlÞ ¼ z~
n4l
with pðlÞ z~z :¼ PðZðlÞ ¼ z~jZð0Þ ¼ zÞ: So it only remains to show for some A that, if zm XA; then P min ZðnÞm XAjZð0Þ ¼ z ¼ 0: n40
We define the truncated process fZ ðnÞgnX0 by taking Z ð0Þ :¼ Zð0Þ and for nX0 0 if Z ðnÞm oA; Z ðn þ 1Þ :¼ Zðn þ 1Þ if Z ðnÞm XA: From this definition it is easy to verify that fZ ðnÞgnX0 is also a homogeneous Markov chain and for every pair of vectors z; z such that zm ; z m XA: PðZðn þ 1Þ ¼ z jZðnÞ ¼ zÞ ¼ PðZ ðn þ 1Þ ¼ z jZ ðnÞ ¼ zÞ;
for all nX0
and therefore P min ZðnÞm XAjZð0Þ ¼ z ¼ P min Z ðnÞm XAjZ ð0Þ ¼ z n40
n40
¼ 1 PðZ ðnÞm ! 0jZ ð0Þ ¼ zÞ:
ð11Þ
Moreover, if either (i) or (ii) holds and choosing A large enough, then, for every z 2 Nm 0; E½Z ðn þ 1Þm jZ ðnÞ ¼ zpzm and consequently fZ ðnÞm gnX0 is a non-negative supermartingale with respect to the sequence of s-algebras fFn gnX0 ; defined by Fn :¼ sðZ ð0Þ; . . . ; Z ðnÞÞ: Making use of the Martingale
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Convergence Theorem, we get that fZ ðnÞm gnX0 converges almost surely to a finite limit that is forced to be null. Taking into account (11) the proof is complete. & Remark 10. In the cases covered by Theorem 9(ii), the hypotheses of Klebaner (1991) for MPDP are improved by removing the conditions on higher order moments. If r ¼ 1; we can find another set of conditions guaranteeing the almost sure extinction of the process. Suppose that for every i; j 2 f1; . . . ; mg; there exists mij ¼ limkzk!1:zi a0 mij ðzÞ: Now, M ¼ ðmij Þ1pi;jpm with r and m as defined above. In this situation r ¼ r and m ¼ m ; so (8) can be rewritten as s2 ðzÞ ¼ Var½Zðn þ 1Þm jZðnÞ ¼ z and the process fZðnÞm gnX0 can be decomposed in a stochastic difference equation like (9). Making use again of the methodology of Kersting (1986) and more specifically the application of this theory due to Klebaner (see Klebaner, 1991), we can establish the following theorem whose proof is avoided: Theorem 11. Let fZðnÞgnX0 be a MPDP with dependent offspring such that M is irreducible. Suppose further that for some 0odp1 either E½jxnþ1 j2þd jZðnÞ ¼ z ¼ oððzm Þ1þd gðzÞÞ or E½jxnþ1 j2þd jZðnÞ ¼ z ¼ oððzm Þd s2 ðzÞÞ holds. ÞgðzÞ o1; then qz ¼ 1 for every z 2 Nm If r ¼ 1 and lim supkzk!1 2ðzm 0: s2 ðzÞ
4. Illustrative example We finish this paper with an example illustrating the possible behaviour of a MPDP with dependent offspring along the generations. Let us denote 1 as the vector whose components are all equal to 1 and tðzÞ :¼ X ðz; 0Þ1; the total offspring whose parents belong to a generation of size z. We suppose the distribution of the random vector X ðz; 0Þ conditioned to tðzÞ ¼ t is multinomial with size parameter t and vector of probabilities ðmz1Þ 1 1: For z 2 Nm 0 and i 2 f1; . . . ; mg; if zi a0; then mij ðzÞ ¼ E½tðzÞ=ðmz1Þ for every j 2 f1; . . . ; mg: Let us consider, for za0; tðzÞ distributed according to a binomial law with parameters aðz1 þ bÞ and p, being 0opo1 and a; b integers with a40 and bX 1: Then E½tðzÞ ¼ apðz1 þ bÞ and consequently mij ðzÞ ¼
ap apb þ ; m mz1
i; j 2 f1; . . . ; mg
with mij ¼
lim
kzk!1:zi a0
mij ðzÞ ¼
ap ; m
i; j 2 f1; . . . ; mg:
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With these entries the limit matrix is irreducible with Perron–Frobenius eigenvalue r ¼ r ¼ ap and right eigenvector associated m ¼ m ¼ 1: Therefore, we can consider four different situations: (i) If poa 1 ; then r o1 and, from Theorem 9(i), qz ¼ 1 for every z 2 N0 : (ii) If p4a 1 ; then r 41: Furthermore, E½ðZðn þ 1Þ1 zMðzÞ1Þ2 jZðnÞ ¼ z ¼ Var½tðzÞ ¼ OðkzkÞ so hypotheses of Theorem 6 are verified with d ¼ 1 and qz o1 if kzk is large enough. We have simulated such a process where m ¼ 3; Zð0Þ ¼ ð1; 2; 3Þ; a ¼ 2; b ¼ 1 and p ¼ 0:55; so that r ¼ 1:1: We observe what seems to be, not only the survival of process, but also an exponential growth. The evolution of the population simulated (left) and its logarithm (right) are given in Fig. 1: (iii) If p ¼ a 1 and bp0; then r ¼ 1 and mij ðzÞpmij for all z 2 Nm 0 and i; j 2 f1; . . . ; mg: So hypotheses of Theorem 9(ii) are verified and qz ¼ 1 for every z 2 Nm 0: (iv) If p ¼ a 1 and b40; then r ¼ r ¼ 1 but mij ðzÞ4mij for all non-null z 2 Nm 0 and i; j 2 f1; . . . ; mg: In this case, with the notation of (9), gðzÞ ¼ b; E½jxnþ1 j3 jZðnÞ ¼ z ¼ E½jtðzÞ E½tðzÞj3 ¼ Oððz1Þ3=2 Þ ¼ oððz1Þ2 =ðlogðz1ÞÞ3=2 Þ and lim inf kzk!1
2ðz1ÞgðzÞ 2b ¼ 42: Var½tðzÞ 1 p
Therefore, hypotheses of Theorem 8 are verified with d ¼ 1; a ¼ 0:5 and consequently qz o1 for every z 2 Nm 0: We have simulated the last two cases up to the 400th generation taking m ¼ 3 and Zð0Þ ¼ ð1; 2; 3Þ: The case (iii) with parameters a ¼ 2; b ¼ 0 and p ¼ 0:5 and case (iv) with a ¼ 2; b ¼ 1 and p ¼ 0:5: Though both processes can be classified as ‘‘critical’’, all the paths corresponding to case (iii) become extinct, whereas some of the paths of case (iv) seem to have a linear growth. In 1000
7
800
6
600
5
400
4 3
200
2
0 0
10
20
30
40
50
60
Fig. 1.
0
10
20
30
40
50
60
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350
150
300 250
100
200 150 50
100 50
0
0 0
100
200
300
400
0
100
200
300
400
Fig. 2.
Fig. 2 we show the evolution of one of the processes with distribution given by (iii) (left graphic) and one (perhaps!) non-extinct process with distribution given by (iv) (right graphic).
References Cohn, H., Klebaner, F., 1986. Geometric rate of growth in Markov chains with applications to population-sizedependent models with dependent offspring. Stochastic Ann. Appl. 4, 283–307. Jagers, P., 1999. Branching processes with dependence but homogeneous growth. Ann. Appl. Probab. 9, 1160–1174. Kersting, G., 1986. On recurrence and transience of growth models. J. Appl. Probab. 23, 614–625. Klebaner, F., 1984. On population size dependent branching processes. Adv. Appl. Probab. 16, 30–55. Klebaner, F., 1989a. Geometric growth in near-supercritical population size dependent multitype Galton–Watson processes. Ann. Probab. 17, 1466–1477. Klebaner, F., 1989b. Linear growth in near-critical population-size-dependent multitype Galton–Watson processes. J. Appl. Probab. 26, 431–445. Klebaner, F., 1991. Asymptotic behaviour of near-critical multitype branching processes. J. Appl. Probab. 28, 512–519. Klebaner, F., 1994. Asymptotic behaviour of Markov population processes with asymptotically linear rate of change. J. Appl. Probab. 31, 614–625. Seneta, E., 1981. Non-negative Matrices and Markov Chains. Springer, Berlin.
Statistics & Probability Letters 70 (2004) 145–154 www.elsevier.com/locate/stapro
Multitype population size-dependent branching processes with dependent offspring$ Miguel Gonza´lez, Rodrigo Martı´ nez, Manuel Mota Facultad de Ciencias, Department of Mathematics, University of Extremadura, Badajoz 06071, Spain Received 5 November 2003; received in revised form 4 June 2004 Available online 21 September 2004
Abstract In this paper, the multitype population size-dependent branching process with dependent offspring is introduced and the probability of extinction of such a process is investigated. Sufficient conditions for both almost sure extinction and survival of the process are shown to depend on the Perron–Frobenius eigenvalue of adequate limit matrices. Finally, theoretical results are illustrated with an example. r 2004 Elsevier B.V. All rights reserved. Keywords: Multitype population size-dependent branching process; Dependent offspring
1. Introduction In branching process theory, it is usual to assume that each individual or particle reproduces independently of the rest and according to the same probability law. Though it seems to be a very strong assumption for biological populations or dynamical systems, the fact of being independent and identically distributed plays a fundamental role in some of the mathematical tools used in the study of branching processes. However several authors have considered the possibility of removing some of these conditions (e.g. see Cohn and Klebaner, 1986; Klebaner, 1994; Jagers, $ Research supported by the Ministerio de Ciencia y Tecnologı´ a and the FEDER through the Plan Nacional de Investigacio´n Cientı´ fica, Desarrollo e Innovacio´n Tecnolo´gica, Grant BFM2003-06074. Corresponding author. E-mail address: [email protected] (R. Martı´ nez).
0167-7152/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2004.08.012
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1999). On the lines of this research, we propose a multitype branching model in which the reproduction of different individuals does not follow necessarily the same law and, in addition, independence assumption is avoided. More specifically, in Section 2 we introduce the multitype population size-dependent branching process (MPDP) with dependent offspring and study some basic properties of this model. In Section 3 we investigate the probability of extinction for such a model. In that section we provide the main results of the paper where sufficient conditions for both, survival and almost sure extinction of the process are established. Finally in Section 4, as illustration, an example is considered and its evolution in particular situations is simulated.
2. The probability model The MPDP in discrete time has been widely studied (e.g. see Klebaner, 1989a,b; 1991). All these works suppose the individuals of a generation give birth to their descendants independently of each other and according to the same probability distribution for each type. Now we introduce a model generalizing the MPDP by removing the independence assumption and allowing individuals coexisting inside a generation, even those of the same type, to reproduce with different offspring distribution. Let us provide the mathematical definition of such a process. Definition 1. Let fX i;j;n ðzÞ : z 2 Nm 0 ; i ¼ 1; . . . ; m; j ¼ 1; . . . ; zi ; n ¼ 0; 1; . . .g be a sequence of mdimensional random vectors with non-negative integer valued coordinates and verifying the following hypotheses: (i) For every fixed z 2 Nm 0 ; the random vectors of the sequence fX ðz; nÞgnX0 defined by X ðz; nÞ :¼ ðX 1;1;n ðzÞ; . . . ; X 1;z1 ;n ðzÞ; . . . ; X m;1;n ðzÞ; . . . ; X m;zm ;n ðzÞÞ
(1)
are independent and identically distributed. (ii) For any z; z¯ 2 Nm ¯ 2 N such that na¯n; the random vectors X ðz; nÞ and X ð¯z; n¯ Þ are 0 and n; n independent. The sequence of m-dimensional random vectors fZðnÞgnX0 ; defined in the recursive way: Zð0Þ ¼ z;
z 2 Nm 0;
Zðn þ 1Þ ¼
m Z i ðnÞ X X
X i;j;n ðZðnÞÞ;
nX0
(2)
i¼1 j¼1
with
P0 1
¼ 0; i.e. the null vector, is called MPDP with dependent offspring.
Intuitively, each coordinate of the vector ZðnÞ represents the number of individuals of the corresponding type coexisting in the population in generation n. The m-dimensional vector X i;j;n ðzÞ denotes the offspring produced by the jth individual of type i in the nth generation provided that the number of individuals of the different types at this moment is given by the vector z. Remark 2. Discrete time population size-dependent models in the literature are particular cases of the MPDP with dependent offspring. In this sense a special mention deserves the branching model introduced by Cohn and Klebaner (1986).
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In the next result some basic properties of the process fZðnÞgnX0 are established. Proposition 3. The MPDP with dependent offspring fZðnÞgnX0 is a homogeneous Markov chain whose state space is a subset of Nm 0 : Moreover, the state 0 is absorbing, and if PðX ðz; 0Þ ¼ 0Þ40 for za0; then the state z is transient. Proof. The first statement is a direct consequence of (2). From the same equation can be deduced that the state 0 is absorbing. Now, let za0 be such that PðX ðz; 0Þ ¼ 0Þ40: Then PðZðnÞ ¼ z
for some njZð0Þ ¼ zÞp1 PðZð1Þ ¼ 0jZð0Þ ¼ zÞ ¼ 1 PðX ðz; 0Þ ¼ 0Þo1
and consequently the state z is transient. & Corollary 4. Let fZðnÞgnX0 be a MPDP with dependent offspring. Suppose that PðX ðz; 0Þ ¼ 0Þ40 for all za0; then PðZðnÞ ! 0Þ þ PðkZðnÞk ! 1Þ ¼ 1
(3)
with k k an arbitrary norm in Rm : Proof. Since za0 is transient, Markov chains theory assures that PðZðnÞ ¼ z i:o:Þ ¼ 0: On the other hand, if fZðnÞðoÞgnX0 converges neither 0 nor infinity, then there exists a positive real number M such that o2
1 [ 1 \
f0okZðnÞkpMg ¼ f0okZðnÞkpM i:o:g:
n0 ¼1 n¼n0
But f0okZðnÞkpM i:o:g ¼
[
fZðnÞ ¼ z i:o:g
0okzkpM
and therefore, Pð0okZðnÞkpM i:o:Þ ¼ 0: & Let MðzÞ :¼ ðmij ðzÞÞ1pi;jpm be the matrix whose coefficients are defined by 8P zi i;l;0 > < l¼1 E½X j ðzÞ if zi 40; mij ðzÞ :¼ i; j 2 f1; . . . ; mg: zi > : 0 if zi ¼ 0; Intuitively, mij ðzÞ can be interpreted as the average rate of type j descendants produced by each individual of type i when the total number of progenitors of the different types coexisting in the population is given by the vector z. It is immediate that E½Zðn þ 1ÞjZðnÞ ¼ ZðnÞMðZðnÞÞ a:s: Note that if the components of vector X ðz; nÞ defined in (1) corresponding to the offspring originated by individuals of type i (i.e. X i;l;n ðzÞ; l ¼ 1; . . . ; zi ) are identically distributed, then ðzÞ for all j 2 f1; . . . ; mg: mij ðzÞ ¼ E½X i;1;0 j
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3. The extinction problem The problem of extinction has been widely studied in population size-dependent processes both in the single type (see Klebaner, 1984) and multitype cases (see Klebaner, 1989a, 1991). In this section, we investigate such a problem for a MPDP with dependent offspring. Definition 5. Let fZðnÞgnX0 be a MPDP with dependent offspring. The extinction probability of fZðnÞgnX0 given that Zð0Þ ¼ z 2 Nm 0 is the probability qz :¼ PðZðnÞ ! 0jZð0Þ ¼ zÞ: The main objective of this paper is to determine sufficient conditions for the existence of a positive probability of survival for the process, i.e. qz o1; and sufficient conditions for the almost sure extinction of the process, i.e. qz ¼ 1: In this sense it is convenient to realize that if m 2 Rm þ; that is all the coordinates of m are positive real numbers, then fZðnÞ ! 0g ¼ fZðnÞm ! 0g: Some processes of the form fZðnÞmgnX0 ; with m 2 Rm þ ; will play a fundamental role in our investigation. 3.1. Survival of the process Firstly, we search sufficient conditions for the non-extinction of a MPDP with dependent offspring. Actually, we will show that under adequate assumptions the process not only has a positive probability of survival, but also of growing indefinitely, i.e. PðkZðnÞk ! 1Þ40: Note that, initially, we do not assume that relation (3) holds. In what follows, it will be necessary to define the matrix M :¼ ðmij Þ1pi;jpm by mij :¼ lim inf
kzk!1:zi a0
mij ðzÞ i; j 2 f1; . . . ; mg:
If M is irreducible, then there exists a positive eigenvalue, r ; called the Perron–Frobenius eigenvalue, with an associated right eigenvector, m ; with positive coordinates also (see Seneta, 1981 for details). Now we can provide the following result: Theorem 6. Let fZðnÞgnX0 be a MPDP with dependent offspring such that M is irreducible. If r 41 and there exists a constant dX0 such that E½jZðn þ 1Þm zMðzÞm j1þd jZðnÞ ¼ z ¼ Oðkzkd Þ; then qz o1 for kzk large enough. Proof. Since r 41 and its associated eigenvalue m has positive coordinates, there exist constants r41 and N 0 40 such that, for kzk4N 0 ; zMðzÞm Xrzm :
(4)
So we just have to prove that if kzk is large enough PðZðnÞm ! 1jZð0Þ ¼ zÞ40 or more specifically that there exist N40 and 0oor 1 such that if kzk4N; then ! 1 \ fZðn þ 1Þm 4ðr ÞZðnÞm gjZð0Þ ¼ z 40: P n¼0
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Let 0oor 1; applying Eq. (4) for kzk4N 0 40; zMðzÞm Xr : zm
(5)
Now denote, for simplicity, An :¼ fZðn þ 1Þm 4ðr ÞZðnÞm g;
n ¼ 0; 1; . . . :
Taking into account (5) and Markov inequality, if kzk4N 0 ; for every dX0; we have Zðn þ 1Þm c PðAn jZðnÞ ¼ zÞ ¼ P pr jZðnÞ ¼ z ZðnÞm Zðn þ 1Þm ZðnÞMðZðnÞÞm p jZðnÞ ¼ z pP ZðnÞm ZðnÞm
Zðn þ 1Þm ZðnÞMðZðnÞÞm
pP
XjZðnÞ ¼ z ZðnÞm p
E½jZðn þ 1Þm ZðnÞMðZðnÞÞm j1þd jZðnÞ ¼ z 1þd ðzm Þ1þd
:
By hypothesis, there exists dX0 verifying E½jZðn þ 1Þm ZðnÞMðZðnÞÞm j1þd jZðnÞ ¼ z ¼ Oðkzkd Þ: For this value of d it is possible to find constants C 0 ; N 1 40 such that if kzk4N 1 ; then E½jZðn þ 1Þm ZðnÞMðZðnÞÞm j1þd jZðnÞ ¼ zpC 0 ðzm Þd : Thus, if kzk4 maxfN 0 ; N 1 g and being C :¼ 1 d C 0 ; we get PðAcn jZðnÞ ¼ zÞp
C : zm
(6)
On the other hand, if kX1;
! !
n\ k k 1 \ Y
P An jZð0Þ ¼ z ¼ PðA0 jZð0Þ ¼ zÞ P An A \ fZð0Þ ¼ zg :
l¼0 l n¼0 n¼1 Tn 1 m For every z~ 2 Nm 0 ; define Tthat fBz~ gz~2N0 is a Tn 1 Bz~ :¼ l¼0 Al \ fZð0Þ ¼ zg \ fZðnÞ ¼ z~g: It is obvious A \ fZð0Þ ¼ partition of the set l¼0 Al \ fZð0Þ ¼ zg and also that ZðnÞm 4zm ðr Þn on n 1 l l¼0 zg: Consequently,
!
n 1
\ P An Al \ fZð0Þ ¼ zg ¼ PðAn j[z~ Bz~ ÞX inf PðAn jBz~ Þ
l¼0 z~m 4zm ðr Þn X
inf
z~m 4zm ðr Þn
PðAn jZðnÞ ¼ z~Þ:
ð7Þ
Since z~m 4zm ðr Þn and r 41; and using the equivalence of the norms in Rm ; there exists another constant N 2 40 so that if kzk4N 2 ; then k~zk4 maxfN 0 ; N 1 g: If
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kzk4N :¼ maxfN 0 ; N 1 ; N 2 g; taking into account (6) and (7), we obtain
!
n 1 C
\ P An Al \ fZð0Þ ¼ zg X1
l¼0 ðzm Þðr Þn and therefore, if we choose NXC we have, for kzk4N; ! ! 1 k 1 \ \ Y P An jZð0Þ ¼ z ¼ lim P An jZð0Þ ¼ z X 1 k!1
n¼0
This product is positive, because
n¼0
P1
n¼0 ðr
n¼0
C : zm ðr Þn
Þ o þ 1; which completes the proof. & n
Remark 7. In the particular case of the MPDP, Klebaner (1991) obtains, using other techniques, a result similar to the previous theorem. He assumes to exist some moments of an upper order than those required for the proof of Theorem 6. In Theorem 6 we have dealt with the case r 41: Nevertheless, survival is also possible if r ¼ 1: In fact, suppose that for every i; j 2 f1; . . . ; mg; there exists mij :¼ limkzk!1:zi a0 mij ðzÞ: Now M ¼ ðmij Þ1pi;jpm with r and m as defined above. Let us call s2 ðzÞ :¼ Var½Zðn þ 1Þm jZðnÞ ¼ z;
(8)
assumed to be finite for all z 2 Nm 0: The process fZðnÞm gnX0 can be written in the form Zðn þ 1Þm ¼ ZðnÞm þ gðZðnÞÞ þ xnþ1
(9)
with gðzÞ :¼ zMðzÞm zm for every z 2 Nm 0 and xnþ1 :¼ Zðn þ 1Þm ZðnÞMðZðnÞÞm : The stochastic difference equation given by (9) has been widely investigated in Kersting (1986). Following this methodology we may obtain the next result: Theorem 8. Let fZðnÞgnX0 be a MPDP with dependent offspring such that M is irreducible and for every C40; there exists nX0 for which PðZðnÞm 4CjZð0Þ ¼ zð0Þ Þ40: Suppose further that for some 0odp1 and some a40; at least one of the next equalities holds: E½jxnþ1 j2þd jZðnÞ ¼ z ¼ oðgðzÞðzm Þ1þd =ðlogðzm ÞÞ1þa Þ; E½jxnþ1 j2þd jZðnÞ ¼ z ¼ oððzm Þd s2 ðzÞ=ðlogðzm ÞÞ1þa Þ: ÞgðzÞ 41; then qzð0Þ o1: If r ¼ 1 and lim inf kzk!1 2ðzm s2 ðzÞ
We omit the proof that follows similar steps to that of Theorem 3 in Klebaner (1989b). 3.2. Almost sure extinction In the next paragraphs, we will focus our attention in determining sufficient conditions for the almost sure extinction of the MPDP with dependent offspring for which (3) holds. To this end, we assume to be under hypotheses of Corollary 4. Let M :¼ ðm ¯ ij Þ1pi;jpm be a matrix with entries m ¯ ij :¼ lim sup mij ðzÞ kzk!1
i; j 2 f1; . . . ; mg:
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Suppose M irreducible, with Perron–Frobenius eigenvalue and associated right eigenvector, r and m ; respectively. Since m has positive coordinates, it is immediate that (3) is equivalent to PðZðnÞm ! 0Þ þ PðZðnÞm ! 1Þ ¼ 1:
(10)
Now, we can formulate the following result: Theorem 9. Let fZðnÞgnX0 be a MPDP with dependent offspring such that M is irreducible. If one of the next conditions holds: (i) r o1; (ii) r ¼ 1 and, for every i; j 2 f1; . . . ; mg; mij ð¯zÞpm ¯ ij for k¯zk large enough, then qz ¼ 1 for every z 2 Nm 0: Proof. According to (10) it suffices to prove that PðZðnÞm ! 1jZð0Þ ¼ zÞ ¼ 0: Given A40; ! 1 X \ PðZðnÞm ! 1jZð0Þ ¼ zÞp P fZðnÞm XAgjZð0Þ ¼ z l¼0
nXl
1 X P min ZðnÞm XAjZð0Þ ¼ z ¼ ¼
l¼0 1 X
nXl
X
pðlÞ z~z P
l¼0 z~m XA
min ZðnÞm XAjZðlÞ ¼ z~
n4l
with pðlÞ z~z :¼ PðZðlÞ ¼ z~jZð0Þ ¼ zÞ: So it only remains to show for some A that, if zm XA; then P min ZðnÞm XAjZð0Þ ¼ z ¼ 0: n40
We define the truncated process fZ ðnÞgnX0 by taking Z ð0Þ :¼ Zð0Þ and for nX0 0 if Z ðnÞm oA; Z ðn þ 1Þ :¼ Zðn þ 1Þ if Z ðnÞm XA: From this definition it is easy to verify that fZ ðnÞgnX0 is also a homogeneous Markov chain and for every pair of vectors z; z such that zm ; z m XA: PðZðn þ 1Þ ¼ z jZðnÞ ¼ zÞ ¼ PðZ ðn þ 1Þ ¼ z jZ ðnÞ ¼ zÞ;
for all nX0
and therefore P min ZðnÞm XAjZð0Þ ¼ z ¼ P min Z ðnÞm XAjZ ð0Þ ¼ z n40
n40
¼ 1 PðZ ðnÞm ! 0jZ ð0Þ ¼ zÞ:
ð11Þ
Moreover, if either (i) or (ii) holds and choosing A large enough, then, for every z 2 Nm 0; E½Z ðn þ 1Þm jZ ðnÞ ¼ zpzm and consequently fZ ðnÞm gnX0 is a non-negative supermartingale with respect to the sequence of s-algebras fFn gnX0 ; defined by Fn :¼ sðZ ð0Þ; . . . ; Z ðnÞÞ: Making use of the Martingale
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Convergence Theorem, we get that fZ ðnÞm gnX0 converges almost surely to a finite limit that is forced to be null. Taking into account (11) the proof is complete. & Remark 10. In the cases covered by Theorem 9(ii), the hypotheses of Klebaner (1991) for MPDP are improved by removing the conditions on higher order moments. If r ¼ 1; we can find another set of conditions guaranteeing the almost sure extinction of the process. Suppose that for every i; j 2 f1; . . . ; mg; there exists mij ¼ limkzk!1:zi a0 mij ðzÞ: Now, M ¼ ðmij Þ1pi;jpm with r and m as defined above. In this situation r ¼ r and m ¼ m ; so (8) can be rewritten as s2 ðzÞ ¼ Var½Zðn þ 1Þm jZðnÞ ¼ z and the process fZðnÞm gnX0 can be decomposed in a stochastic difference equation like (9). Making use again of the methodology of Kersting (1986) and more specifically the application of this theory due to Klebaner (see Klebaner, 1991), we can establish the following theorem whose proof is avoided: Theorem 11. Let fZðnÞgnX0 be a MPDP with dependent offspring such that M is irreducible. Suppose further that for some 0odp1 either E½jxnþ1 j2þd jZðnÞ ¼ z ¼ oððzm Þ1þd gðzÞÞ or E½jxnþ1 j2þd jZðnÞ ¼ z ¼ oððzm Þd s2 ðzÞÞ holds. ÞgðzÞ o1; then qz ¼ 1 for every z 2 Nm If r ¼ 1 and lim supkzk!1 2ðzm 0: s2 ðzÞ
4. Illustrative example We finish this paper with an example illustrating the possible behaviour of a MPDP with dependent offspring along the generations. Let us denote 1 as the vector whose components are all equal to 1 and tðzÞ :¼ X ðz; 0Þ1; the total offspring whose parents belong to a generation of size z. We suppose the distribution of the random vector X ðz; 0Þ conditioned to tðzÞ ¼ t is multinomial with size parameter t and vector of probabilities ðmz1Þ 1 1: For z 2 Nm 0 and i 2 f1; . . . ; mg; if zi a0; then mij ðzÞ ¼ E½tðzÞ=ðmz1Þ for every j 2 f1; . . . ; mg: Let us consider, for za0; tðzÞ distributed according to a binomial law with parameters aðz1 þ bÞ and p, being 0opo1 and a; b integers with a40 and bX 1: Then E½tðzÞ ¼ apðz1 þ bÞ and consequently mij ðzÞ ¼
ap apb þ ; m mz1
i; j 2 f1; . . . ; mg
with mij ¼
lim
kzk!1:zi a0
mij ðzÞ ¼
ap ; m
i; j 2 f1; . . . ; mg:
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With these entries the limit matrix is irreducible with Perron–Frobenius eigenvalue r ¼ r ¼ ap and right eigenvector associated m ¼ m ¼ 1: Therefore, we can consider four different situations: (i) If poa 1 ; then r o1 and, from Theorem 9(i), qz ¼ 1 for every z 2 N0 : (ii) If p4a 1 ; then r 41: Furthermore, E½ðZðn þ 1Þ1 zMðzÞ1Þ2 jZðnÞ ¼ z ¼ Var½tðzÞ ¼ OðkzkÞ so hypotheses of Theorem 6 are verified with d ¼ 1 and qz o1 if kzk is large enough. We have simulated such a process where m ¼ 3; Zð0Þ ¼ ð1; 2; 3Þ; a ¼ 2; b ¼ 1 and p ¼ 0:55; so that r ¼ 1:1: We observe what seems to be, not only the survival of process, but also an exponential growth. The evolution of the population simulated (left) and its logarithm (right) are given in Fig. 1: (iii) If p ¼ a 1 and bp0; then r ¼ 1 and mij ðzÞpmij for all z 2 Nm 0 and i; j 2 f1; . . . ; mg: So hypotheses of Theorem 9(ii) are verified and qz ¼ 1 for every z 2 Nm 0: (iv) If p ¼ a 1 and b40; then r ¼ r ¼ 1 but mij ðzÞ4mij for all non-null z 2 Nm 0 and i; j 2 f1; . . . ; mg: In this case, with the notation of (9), gðzÞ ¼ b; E½jxnþ1 j3 jZðnÞ ¼ z ¼ E½jtðzÞ E½tðzÞj3 ¼ Oððz1Þ3=2 Þ ¼ oððz1Þ2 =ðlogðz1ÞÞ3=2 Þ and lim inf kzk!1
2ðz1ÞgðzÞ 2b ¼ 42: Var½tðzÞ 1 p
Therefore, hypotheses of Theorem 8 are verified with d ¼ 1; a ¼ 0:5 and consequently qz o1 for every z 2 Nm 0: We have simulated the last two cases up to the 400th generation taking m ¼ 3 and Zð0Þ ¼ ð1; 2; 3Þ: The case (iii) with parameters a ¼ 2; b ¼ 0 and p ¼ 0:5 and case (iv) with a ¼ 2; b ¼ 1 and p ¼ 0:5: Though both processes can be classified as ‘‘critical’’, all the paths corresponding to case (iii) become extinct, whereas some of the paths of case (iv) seem to have a linear growth. In 1000
7
800
6
600
5
400
4 3
200
2
0 0
10
20
30
40
50
60
Fig. 1.
0
10
20
30
40
50
60
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350
150
300 250
100
200 150 50
100 50
0
0 0
100
200
300
400
0
100
200
300
400
Fig. 2.
Fig. 2 we show the evolution of one of the processes with distribution given by (iii) (left graphic) and one (perhaps!) non-extinct process with distribution given by (iv) (right graphic).
References Cohn, H., Klebaner, F., 1986. Geometric rate of growth in Markov chains with applications to population-sizedependent models with dependent offspring. Stochastic Ann. Appl. 4, 283–307. Jagers, P., 1999. Branching processes with dependence but homogeneous growth. Ann. Appl. Probab. 9, 1160–1174. Kersting, G., 1986. On recurrence and transience of growth models. J. Appl. Probab. 23, 614–625. Klebaner, F., 1984. On population size dependent branching processes. Adv. Appl. Probab. 16, 30–55. Klebaner, F., 1989a. Geometric growth in near-supercritical population size dependent multitype Galton–Watson processes. Ann. Probab. 17, 1466–1477. Klebaner, F., 1989b. Linear growth in near-critical population-size-dependent multitype Galton–Watson processes. J. Appl. Probab. 26, 431–445. Klebaner, F., 1991. Asymptotic behaviour of near-critical multitype branching processes. J. Appl. Probab. 28, 512–519. Klebaner, F., 1994. Asymptotic behaviour of Markov population processes with asymptotically linear rate of change. J. Appl. Probab. 31, 614–625. Seneta, E., 1981. Non-negative Matrices and Markov Chains. Springer, Berlin.