BRANCHING PROCESSES WITH DETERIORATING RANDOM ENVIRONMENTS PETER JAGERS AND LU ZHUNWEI Abstract. In this paper, a BPRE (branching process in random environ-
ments) is presented, in which a population development in deteriorating environments is described. Some primary results on process growth and extinction probability are shown. At last, two simple numerical examples that attempt to simulate the probability model are given to help understand its mathematical framework.
1. Introduction Consider a Galton-Watson branching process fZng in random environments = fn g. Here n is measurable with respect to Bn (Z0 ; : : : ; Zn ) for every n = 0; 1; 2; : : : and is called the environment of nth generation or season. Let n (s), s 2 [0; 1] be the pgf (probability generating function) according to which every individual of the nth generation reproduces independently given Bn. Clearly, n is also determined by Bn and therefore dependent on Z0 ; : : : ; Zn . The dependence is assumed to be measurable. We denote by j = 0j (1) the average number of the o spring per individual in the j th generation. Again, j is random and measurable with respect to Bj . Noting that Bn Bn+1, we can see that in general the random environment fn g is no longer iid (independent and identically distributed) and not even stationary. Now, we suppose Z0 = z0 > 1 and assume j # a:s: (j ! 1), where is a positive random variable or, in particular cases, a positive constant. What can we say about fZn g? In a simpli ed manner the above mathematical framework describes population development in deteriorating environments. The environment of a season is in uenced by earlier population history but may also have independent exogenous components. But due to depletion of resourses or increasing pollution mean reproduction decreases. 2. Further description of fZng Now let us put the assumption j # a:s:(j ! 1) aside for the moment. Then in our mathematical model here, fZng is a branching process with a fBn g adapted random environment = fn g. By the fZn g we can understand the following evolving population of particles. Assume that we start with Z0 particles and denote the population size at time n by Zn. The transition from Zn to Zn+1 takes place as below. All the Zn members 1991 Mathematics Subject Classi cation. 60J80, 60J10. Key words and phrases. branching process; fBn g adapted random environment. Lu Zhunwei thanks China Scholarship Council very much for her support. . 1
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PETER JAGERS AND LU ZHUNWEI
of the nth generation live for a unit of time, after which each of them splits into a random number of (n + 1)th generation particles independently according to the same o spring distribution with pgf n (s), where n (s) is chosen at random from a collection of pgf's according to the information from Bn , which is a ected by the size of Z0 ; : : : ; Zn and maybe other factors due to the hypothesis Bn (Z0 ; : : : ; Zn). Let fXj(n) g, j = 1; 2; : : : be an iid random variables sequence with the same pgf n (s). Then fZn g can be expressed by the recursion formula as
Z0 = z0 ; Zn+1 =
Z X n
j =1
Xj(n ) ; (n 0):
We can deduce the transition probabilities given Bn as
P fZn+1 = j jZn = i; Bng =
(P
n1 +:::+ni =j 0j
Qi
1 (nk ) k=1 nk ! n (0)
i 1; j 0; i = 0; j 0;
which is an i-fold convolution of the distribution law fpk (n ) = k1! (kn) (0)g, k = 0; 1; : : : corresponding to the pgf n (s). By taking expectation, we get the transition probabilities of fZng as
(P
Q i 1 (nk ) n1 +:::+ni =j E [ k=1 nk ! n (0)] 0j
i 1; j 0; i = 0; j 0: Furthermore, noting that Bn gives us Zn and the environment n , we obtain that P fZn+1 = j jZn = ig =
E [sZn+1 ] = E [E [sZn+1 jBn ]] = E [E [sX1
(n )
+:::+XZ(nn ) jB ]] = E [ (s)]Zn ; (n 0) n n
where we use the same sign E all di erent of expectations according to context. Since the environment n of the nth generation includes the information about Z0 ; : : : ; Zn and 0 ; : : : ; n?1 , it may well happen that E [sZn jBn ] = [ n (s)]Zn 6= [ ( (: : : ( n (s)) : : : ))]z ; (n 1) i.e. the pgf of Zn can no longer in general be expressed by some expectation for an iteration of random pgf's. Clearly, in our model, the fZng is di erent from that of the either Smith-Wilkinson BPRE model or the Athreya-Karlin BPRE model, in which the random environment is an iid random variables sequence or a stationary ergodic process (see [1],[2],[3]). It is more complicated here. In particular cases, if fZng is a Markov chain, then it degenerates to population-size-dependent branching process or population-size-dependent branching proces with random environments (see [4],[5]). +1
0
1
0
3. Process growth
Q ?1 i , in which the products Write Wn = Znn , Wn = z0 QnZ?n1 and Yn = z0 in=0 Qn?1 in W and Qn?1 i i=0in Y i are de ned by 1 as n = 0. Then we have n n i=0 i=0 i
Theorem 1. fWng is a non-negative martingale with respect to fBng, and 0 limn!1 Wn = W < +1 a:s: exists.
BRANCHING PROCESSES WITH DETERIORATING RANDOM ENVIRONMENTS
3
Proof. Wn 0 for all n 0 is obvious by its de nition. That fWn g is a martingale with respect to fBng follows by
P
n Xj(n) Zn+1 jB ] = E [ ZjQ=1 Q jB ] = E [ n+1 n z0 ni=0 i n z0 ni=0 i jBn ] = z ZQnnn = QZn?n1 = Wn (n 0): 0 i=0 i z0 i=0 i Noting that E [W0 ] = 1 for all n 0, by the martingale convergence theorem, it must yield that 0 limn!1 Wn = W < +1 a:s: exists. Theorem 2. fWng is a non-negative submartingale with respect to fBng, and 0 limn!1 Wn = W +1 is well de ned on the set fW 6= 0g [ fY 6= +1g. Proof. Since Wn = Znn 0 and j 1 a:s: (j 0) by the assumption of j # a:s:, and noting that Zn Zn Zn+1 E [Wn+1 jBn ] = E [ n+1 jBn ] = n n n = Wn a:s: (n 0); we see that fWn g is a non-negative submartingale with respect to fBng. From the j assumption z0 > 1 and 1 a:s: (j 0), we know that Yn+1 Yn z0 a:s: for all n 0. Then z0 limn!1 Yn = Y +1 must exist. Therefore, since Wn = Wn Yn , 0 limn!1 Wn = W = W Y +1 is well de ned unless W = 0 and Y = +1 as required. Remark. 1. However, even on the set fW = 0g\fY = +1g, i.e. when limn!1 Wn = 0 and limn!1 Yn = +1, obviously, sometimes 0 limn!1 Wn = W +1 still is well de ned. And then if Wn = O(Yn?1 ) a:s:, we still have 0 nlim !1 Wn = nlim !1 Wn Yn = W < +1 a:s: otherwise W = +1 a:s:. 2. Since fWn g is a non-negative submartingale with respect to fBng, if sup E [Wn ] = nlim !1 E [Wn ] < +1;
E [W
n1
then by the submartingale convergence theorem, 0 limn!1 Wn = W < +1 a:s: exists. Therefore, we know that 0 W < +1 a:s:, if (i) limn!1 E [Wn ] < +1, or (ii) Y < +1, or (iii) limn!1 Yn = +1, but the limit limn!1 Wn exists and Wn = O(Yn?1 ) a:s:. Otherwise, either W = +1 a:s: or the limit limn!1 Wn does no exist. Furthermore, we have Theorem 3. On the set fP1j=0 (j ? ) < +1g [ fZn ! 0g, 0 limn!1 Wn = W < +1 a:s: exists. P ( ? ) < +1 a:s: is equivalent to Q1 j Proof. First, we know that 1 j j =0 Q j=0 j 0 for all n 0 since > 0. Therefore, Wn+N = Znn NN = 0 for all n 0, which implies that, on the set fZn ! 0g, limn!1 Wn = 0 and the proof is complete. Corollary 1. If there existXa constant > 0 and a non-negative random variable X , such that (j ? ) j a:s: for all suciently large j , then limn!1 Wn = W < +1 a:s:. P ( ?) < Proof. Since (j ?) j X a:s: for all suciently large j , we have 1 j =0 j +1 a:s: and the assertion follows Theorem 3. + +
1+
1+
4. About the extinction probability Let q = P fZn ! 0g be the extinction probability of fZng. We know that q = P fZn = 0; for some ng = P f[1 n=1 fZn = 0gg Z n = nlim !1 P fZn = 0g = nlim !1 E [s ]s=0 : First, let us look at the case of < 1 a:s:. Clearly, if limn!1 Wn = W < +1 is well de ned, then lim Z = lim n lim W = 0 W = 0 a:s:; n!1 n n!1 n!1 n
i.e. q = P fZn ! 0g = 1. The following theorem does not use information about fWn g. Theorem 4. If < 1 a:s:, then q = 1. Proof. Since j # < 1 a:s:, we can de ne a stopping time by n0 = minfj : j < 1g: Then n n < 1 for all n 0: Conditionally upon Bn , E [Zn jBn ] = Zn and E [Zn +1 jBn ] = Zn n . If E [Zn +n jBn ] Zn nn , then E [Zn +n+1 jBn ] = E [E [Zn +n+1 jBn +n ]jBn ] = E [Zn +n n n jBn ] n E [Zn +n jBn ] Zn nn+1 : By induction over n, thus E [Zn +n jBn ] Zn nn for all n 0. Letting n ! 1, we get E [Zn +n jBn ] ! 0, which together with Fatou's lemma implies Zn +n ! 0 a:s:. Clearly Zn ! 0 if and only if Zn +n ! 0. Therefore, we obtain Zn ! 0 a:s:, i.e. q = P fZn ! 0g = 1. The proof is complete. Secondly, let us look at the case of > 1 a:s:. Clearly, if 0 < limn!1 Wn = W +1 is well de ned, noting that Wn = Znn and > 1 a:s:, then we have n nlim !1 Zn = nlim !1 nlim !1 Wn = +1 a:s:: Thus q = 0 < 1 holds. However, if we strengthen the condition j for 8j 0 to X X pk (j ) pk ( ) for 8i; j 0; () 0+
0
0
0
0
0
0
0+ 0
0
0
0
0
0
0
0
0
0
0
0
0
0
k>i
0
0
0
0
0
0
0
0
0
0
0
k>i
0
BRANCHING PROCESSES WITH DETERIORATING RANDOM ENVIRONMENTS
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where fpk ( )g is a random probability distribution law dependent on some random D and assume that E [? log(1 ? p ( ))] < +1, environment with expectation = 0 then as Theorem 4, the following theorem also does not use information about Wn . Theorem 5. Under the assumption above, if 1 a:s: and P f > 1g > 0, then q < 1. Proof. From the assumption together with the result of [1], we know that there is D for 8n 0 and thus an iid random environment = fn g, such that n = 1 X
and
k=1
X k>i
D ; for 8n 0; kpk (n ) = n = D
pk (j )
X k>i
pk (j ); for 8i; j 0:
Further choose > 1 so that P f > g > 0. Then (i) E [log j ] = E [log ] E [log( ^ )] P f > g > 0 for 8j 0; and (ii) E [? log(1 ? p0 (j ))] < +1; for 8j 0; so that the supercriticality holds. In other words, the reproduction in the environment j is dominated by that in environment j stochastically for any j 0 so that if fZn g is a BPRE with initial population size Z0 = z0 = Z0 living in the random environment = fn g, then it must be supercritical, and D
Hence
Zn Zn ; for 8n 0: q = P fZn ! 0g = nlim !1 P fZn = 0g = nlim !1 P fZn 0g nlim !1 P fZn 0g = nlim !1 P fZn = 0g = q < 1;
as required. At the last, let us look at the case of = 1 a:s:. In order to avoid trivialities and additional absorbing states, we shall assume that 0 < p0 (n ) + p1 (n ) < 1 for all n 0. Then we have Theorem 6. If = 1 a:s:, P1j=0 (j ? 1) < +1 a:s: and there is a constant c > 0 such that p0 (k ) > c for all k from some k0 > 0 onwards, then q = 1. Proof. First, noting that 0 < c < p0 (k ) < 1, then for any constant x > 0, we always can take = cx > 0 such that P fZn ! 0jBk g (p0 (k ))Zk cx = ; on fZk xg: Therefore, by Theorem 2 in [6], we have P fZn ! 0 or Zn ! +1g = 1: () P 1 Moreover, since = 1 a:s: ) Wn = Zn a:s:, noting that j=0 (j ? 1) < +1 a:s:, by Theorem 3, we know that 0 limn!1 Zn = Z < +1 a:s: exists, which together with () tells us q = P fZn ! 0g = 1 as required.
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PETER JAGERS AND LU ZHUNWEI
It is easily to obtain that Corollary 2. If = 1 a:s: and there exist a non-negative random variable X and constants ; c > 0, such that j ? 1 j X for all suciently large j and p0 (k ) > c for all k from some k0 > 0 onwards respectively, then q = 1. Corollary 3. If there exists some J > 0 such that j = 1 a:s: for all j J , then q = 1. We conjecture that if 1 a:s: with P f > 1g > 0, then q < 1 is probably always true. 1+
5. Two numerical examples In order to explain our probability model and help to understand its mathematical framework we provide two simple numerical examples here.
Example 1. In this example, we consider a population development in some environment, where the total amount of resource consumption will determine the reproduction. We can persume that the total amount of resource consumption is proportional to the accumulated number of the individuals who lived before or at time n and with the decrease of the amount of available resource the reproducing size will be reduced. In our BPRE mathematical model we can suppose that the random reproducing pgf in nth generation has the form n (s) = Z ++Zn (s), i.e. n = Z0 + + Zn , and , the collection of pgf's is = f m (s); m = 0; 1; 2; : : : g. Clearly, fn g here is non-decreasing, since Zn is non-negative. By the de nition of deteriorating environment, we should assume that m # as m ! 1. One can easily de ne m (s) in various forms for meeting di erent cases discussed above in Sections 3 and 4 of the paper and construct simple counterexamples to illustrate that the pgf of Zn can not be expressed by iteration of pgf's of ; ; : : : ; n? . 0
0
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1
Example 2. In this example, we intend to try to establish another BPRE model, where the environment deteriorates more slowly. Here we suppose that the resource can be expanded within certain limits in the initial period but can not resuscitate, which can describe that a group of individuals emigrate a closed place such as an isolated island. We hope that in our model if the population size keeps in some range then its corresponding reprodution shall be unchanged. Therefore, we can use the the number of population increases minus the number of decreases to set up a threshold restraining the development of the population. A concrete way of doing this as below: Let K 2 Z+, c 2 R+ , = f k (s)g, k = 1; 2; : : : , where k (s) is the pgf of Poisson distribution with parameter c + k1 , i.e. k (s) = e?(c+ k )(1?s) . Given Z0 = z0 > 1, take = 1 . Further, if n = j , then if jfi : Zi ? Zi?1 > 0; i = 1; 2; : : : ; ngj ? jfi : Zi ? Zi?1 0; i = 1; 2; : : : ; ngj > K; where jfgj means the number of elements in the corresponding set fg, then we take n = j+1 , otherwise n = j . 1
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+1
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BRANCHING PROCESSES WITH DETERIORATING RANDOM ENVIRONMENTS
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Here Bn = (Z0 ; : : : ; Zn ), = fn g is adapted to fBn g, and is not iid and not stationary. Let = c. Then we can see that lim = lim 0 (1) = klim (c + k1 ) = c = ; j !1 j k!1 k !1
and j is monotonically decreasing to as we required. Moreover, we can establish a more complicated model, in which n not only depends upon Z0 ; Z1 ; : : : ; Zn but also upon a given random variable n . To the end, on the base of above example, we add a pgf set = f k (s)g, k = 1; 2; : : : , in which k (s) is the pgf of Poisson distribution with parameter c + k1 , and an iid random variable sequence fn g with the distribution law as P fn = cg = p, P fn = c g = 1 ? p (0 < p < 1) for n = 0; 1; 2; : : : . We start according to value of 0 to choose = 1 or 1 . In case n = j or j , if jfi : Zi ? Zi?1 > 0; i = 1; 2; : : : ; ngj ? jfi : Zi ? Zi?1 0; i = 1; 2; : : : ; ngj > K; we choose n = j+1 or j+1 at random according to the value of n+1 , otherwise we let n = j or j as the same as n . Here Bn = (Z0 ; : : : ; Zn; n ) (Z0 ; : : : ; Zn ) and the pgf collection is [ . We still have j # a:s: (j ! 1), and where is a random variable with the distribution law as P f = cg = p, P f = c g = 1 ? p. Clearly, we can use the results in Sections 3 and 4 to deal with these mathematical models. 0
+1
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References [1]. W.L. SMITH, W.E. WILKINSON, On branching processes in random environments. Ann. Math. Statist., 40 (1969), 814-827. [2]. K.B. ATHREYA, S. KARLIN, On branching processes with random environments, I: extinction probabilities. Ann. Math. Statist., 42 (1971), 1499-1520. [3]. K.B. ATHREYA, P.E. NEY, Branching Processes. Springer-Verlag Berlin Heidelberg New York. (1972), 249-256. [4]. F.C. KLEBANER, On population-size-dependent branching processes. Adv. Appl. Prob., 16 (1984), 30-55. [5]. HANXING WANG, Extinction of population-size-dependent branching processes in random environments. J. Appl. Prob., 36 (1999), 146-154. [6]. P. JAGERS, Stabilities and instabilities in population dynamics. J. Appl. Prob., 29 (1992), 770-780. Peter Jagers, Department of Mathematical Statistics, Chalmers University of Technology and Goteborg University, SE-412 96 Goteborg, Sweden
E-mail address :
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Lu Zhunwei, Department of Mathematics, Taiyuan University of Technology, 030024, Taiyuan,Shanxi province,The People's Republic of China
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