General branching processes are defined through individuals

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1 On the path to etinction Peter Jagers*, Fima C. Klebaner, and Serik Sagitov* *Department of Mathematical Sciences, Chalmers University of Technology, SE- 9 Gothenburg, Sweden; and School of Mathematical Sciences, Monash University, Clayton, Victoria 8, Australia Communicated by David Co, University of Oford, Oford, United Kingdom, January, 7 (received for review September, ) Populations can die out in many ways. We investigate one basic form of etinction, stable or intrinsic etinction, caused by individuals on the average not being able to replace themselves through reproduction. The archetypical such population is a subcritical branching process, i.e., a population of independent, aseually reproducing individuals, for which the epected number of progeny per individual is less than one. The main purpose is to uncover a fundamental pattern of nature. Mathematically, this emerges in large systems, in our case subcritical populations, starting from a large number,, of individuals. First we describe the behavior of the time to etinction T: as grows to infinity, it behaves like the logarithm of, divided by r, where r is the absolute value of the Malthusian parameter. We give a more precise description in terms of etreme value distributions. Then we study population size partway (or u-way) to etinction, i.e., at times ut, for < u <, e.g., u gives halfway to etinction. (Note that mathematically this is no stopping time.) If the population starts from individuals, then for large, the proper scaling for the population size at time ut is into the power u. Normed by this factor, the population u-way to etinction approaches a process, which involves constants that are determined by life span and reproduction distributions, and a random variable that follows the classical Gumbel distribution in the continuous time case. In the Markov case, where an eplicit representation can be deduced, we also find a description of the behavior immediately before etinction. General branching processes are defined through individuals who give birth independently of each other. We limit ourselves to single-type such populations, all individuals having the same reproduction and, more generally life-path, distribution. Benchmark cases of such processes are the simple Galton Watson, birth-and-death, and Markov branching processes. Even though such Markovian structures dominate the probabilistic literature, more general processes (allowing aging, various distributions of life spans and fertility periods, as well as repeated litters of varying sizes) are relevant for biological modelling (). A presentation from a mathematical point of view can be found in ref. (multi-type populations) or ref. (single-type processes). A multi-type framework would have been still more general but also far less accessible. The not-so-mathematical reader will undoubtedly note that even this presentation is demanding at certain points. However, conclusions should be clear, and they, rather than the work to derive them, form the message of this paper: intrinsic stable etinction follows a simple and beautiful pattern. The purpose is thus general understanding, rather than paving the way for specific applications. For those, multi-type theory may be useful, but at the bitter end they will tend to need custom-made models of narrow relevance. A (single-type) branching process is subcritical if the epected number m of children per individual is less than one. Let (a) denote the epected number of children by age a. Clearly, m (). In the lattice case, births can occur only at multiples of some time unit, e.g., yearly. For simplicity, our focus is on the nonlattice case. We consider so-called Malthusian processes. These are defined by the requirement that there eists a number, the Malthusian parameter, such that ˆ e a da. This is always fulfilled in the critical (m ) and ercritical ( m ), and under very mild conditions (always met with in the real world) also in subcritical cases. Then, and since will play an important role, we shall write r. In the case of Galton Watson processes, r ln m. Let Z t be a shorthand for the number of individuals alive at time t, provided the population started from individuals at time, taken as newborn then, for preciseness. Z t without a suffi, is Z t, the case of one ancestor. If L denotes the life span distribution, any nonlattice Malthusian process will then satisfy Z t e ru Ludu ue ru du e rt, t (ref., p. ). We write C for the proportionality constant in this, i.e., [Z t ] Ce t Ce rt and recall that in the lattice case the enumerator integral is replaced by a sum. (To the nonmathematical reader, the importance of this is not the special formula for C, but rather the fact that the constant can be calculated from demographic facts.) For Galton Watson and Markov branching processes C, and epressions are eact, [Z t ] m t for integer t in the Galton Watson case. The epected population size thus has the same asymptotic form in all three cases,,,. For variances, though, asymptotics take different forms in the sub- and ercritical cases. Assume that m and that Malthusianness is strengthened to a second order property: let denote the reproduction point process, so that [], and write ˆ e a da; it is required that [ˆ() ]. Then, VarZ t Be rt, for some constant B. This can be checked from the convolution equation holding for second moments (ref., p. ). In the more often studied ercritical case, the leading term is proportional to e t. We impose second order Malthusianness throughout and assume that the life span distribution L and reproduction measure satisfy Author contributions: P.J. designed research; P.J., F.C.K., and S.S. performed research; P.J., F.C.K., and S.S. contributed new reagentsanalytic tools; and P.J., F.C.K., and S.S. wrote the paper. The authors declare no conflict of interest. Freely available online through the PNAS open access option. To whom correspondence should be addressed. jagers@chalmers.se. 7 by The National Academy of Sciences of the USA PNAS April, 7 vol. no. 7

2 e rt tldt e and rt tdt, [] respectively. As a consequence, the so called log -condition holds, guaranteeing the Kolmogorov and Yaglom theorems on survival chances and conditional population size given nonetinction (see ref., p. 7, for the case of age-dependent processes and ref., p. 9, and ref., for general processes). In the nonlattice formulation, they tell us that for some c, Z t ce rt, t [] lim Z t kz t b k, k,,,... [] t eist, k b k, and b kb k Cc. [] k It is worth noting that these two theorems do not require the full strength of branching process independence assumptions (see ref. for population-size-dependent branching). Even though we shall not embark upon the subtleties of measuring the population in other ways than by just counting its individuals, it turns out that the proof of our results is considerably facilitated by use of counting with a particular random characteristic (ref., p. 7 ff.). The characteristic we shall use will be one that records all events to come in an individual s life and daughter process. It is thus not individual like the ones treated in op. cit. but still well behaved (7). The first and second moment results quoted above for processes just counting the number of individuals alive, clearly etend to processes counted with characteristics such that e rt [(t)] is directly Riemann integrable, and so do the Kolmogorov-Yaglom theorems (). The Time to Etinction Now consider processes with ancestors, Z t. By the time to etinction, satisfies T inft ; Z t T t T t Z t c t e rt for some c t tending to c, ast. As a consequence, T [] T tdt ln ln c r, [] where Euler s. To verify note that for any fied number v, v On the other hand, v v T tdt v,. T tdt c t e v rt c t e rt k dt, k where tv c t c for any, given that v is sufficiently large. Thus it remains to observe that as kv ce rt ce rt k dt r k k cerv k ln ln c v o, r where o(), denotes a remainder term, vanishing as. Etreme value theory handily delivers distributional forms of this, though again distinction should be made between continuous-time and lattice cases. In the former, Kolmogorov s theorem yields the eponential tail that implies a classical Gumbel limit (ref. 8, p. 7). Indeed, for each initial number, define by Then, as, T ln ln c r. [7] y epe y, y. [8] This etends Pakes s (9) result for Markov branching processes. In Between Dawn and Demise Let Z ut denote the population size at some time ut, u, between its inception at size Z and etinction. We consider nonlattice processes, and to ease eposition, write t ln, so that T t r, by7 and 8. Since [Z t ] Ce rt,ast, also u [Z u(t t)/r] Ce ut,, and it is natural to guess that u provides the right norming and that u Z ut C u b u e u in distribution, as the initial population size, at least for fied u. A first check of this conjecture can be made by simulation. Consider a binary splitting Markov branching process, i.e., birth-and-death process, with the probability p.7 for zero and p. for two offspring, and epected life span. For birth-and-death, as pointed out, C and further b can be shown to equal p (p ), which is. in the present case. Fig. displays the simulation results. The two upper panels show that the process Z ut, u displays much more of variation than does Z t. The lower panels indicate that the amount of variation is just right to allow nontrivial normalisation. They also point at convergence towards a process with a rather constant mean and a variance increasing with u. The epectation and variance of the proposed limiting population size are C u b u e u C u b u u, VarC u b u e u C u b u u u, u, in terms of the classical Gamma function. Thus, for bc close to one (it cannot be smaller), the Gamma function eerts a strong influence on the behavior of epected size, and since () () the overall picture is that of a fairly constant mean. For large bc it increases practically eponentially. This may seem intriguing, but is eplained by the norming through u. The limiting variance always increases with u, fromtob. Fig. shows these epectations and standard deviations for different values of b assuming C. The situation for b. thus seems to fit simulations. 8 Jagers et al.

3 8 (Z, <t<t) t 8 (Z ut, <u<)....8 ( u Z ut, <u<) ( u Z ut, <u<) Fig.. Markov branching with p.7, p., life epectancy. (Upper) Forty simulations with initial number,, population size on the vertical ais, time (Left) and proportion of time u to etinction (Right) horizontally. (Lower) Forty simulations of normed size with, (Left) and, (Right), proportion u of time to etinction on the horizontal ais. The Path-to-Etinction Theorem and Its Proof Our proof of the convergence follows a three-step programme, which requires acquaintance with weak convergence theory (). The reader interested in the basic pattern of etinction, rather than mathematical technique is thus advised to jump to Theorem.. For fied u, write (t) u Z u(t t)/r and prove that t Ce ut, t, as, the arrow indicating weak convergence in the Lindvall Skorohod topology etended to the whole real line.. Consider at the random argument : ln c : ln (Cb), where is Gumbel by 7 and 8, and check conditions for ( ) Ce u C u b u e u in distribution for fied u b b= b. b Fig.. Mean (upper line) and standard deviation (lower line) of the limit process b u e u for b.,.,, u on the horizontal ais. The upper right panel shows simulations of (.) u e u, to be compared with the simulations in Fig.. Jagers et al. PNAS April, 7 vol. no. 9

4 . Study the asymptotics of ( )(u) u Z ut as a function of u varying inside the unit interval. Step. We already noted that t u Z utt/r u Ce utt Ce ut,. Similarly, Var t Var u Z utt/r u B u e ut. We can conclude that for any fied t, (t) Ce ut in mean square. The finite dimensional convergence t, t,..., t k Ce ut, Ce ut,...,ce utk is obvious. Turning to tightness, we note that for any v and a tv t a u tut v/r Z t a u tut v/r Z t a. But the total progeny Y of a population from one newborn ancestor clearly dominates the maimum of the numbers ever simultaneously present, and the sum of the maima majorises the remum of the sum. Since [Y] ( m), it follows that Z ut v/r m] u Ce uv m. Z t tut v/r We can conclude that tv t a Ka for some constant K. Now consider the population counted with a characteristic (ref., p. 7) that for any h gives to each individual present the value of Y Y an indicator of dying the net hr time units plus the total number of progeny of the individual born within the same net time span of length hr. Denote the process thus counted by D h (t). Clearly, for any t, tsth Z uts/r Z utt/r D h ut tr. Write hr, and denote the total progeny within t time units stemming from one newborn individual by Y t. Then, t Lt Lt t t Y tu du. For nonlattice processes it follows that, as t, D h t e rt e rt Lt Ltdt e rt t t Y tu dudt te rt dt e rt Oh, h. If we assume that reproduction is L -Lipschitz in the sense that the second moment of the number of births between ages t and t h, v h (t), satisfies v h te rt dt Oh, h, [9] then a similar, but prohibitive, analysis of the convolution equation for second moments, shows that also D h t e rt Oh. This Lipschitz-property is broadly satisfied, e.g., if the number of children in short intervals is bounded. Indeed, if B is a bound, then v h (t) B((t h) (t), so that such qualities of the reproduction function transfer. By Chebyshev s inequality, therefore, for a suitable K, tsth s t uts/r tsthz D h ut tr u Z utt/r u D h ut tr D h ut tr u u Kh VarD h ut tr u Kh e ut K u h for h little enough and K some constant. Tightness follows along the lines of the Corollaries in ref. (p. 8 and p. ). The process (t) thus has the weak nonrandom continuous limit Ce ut, t, as. The modification from there is that those corollaries concern processes on the unit interval, whereas our processes are defined on the whole line, with the natural Lindvall Skorohod topology of convergence on all finite subintervals. Step. In the nonlattice case, the pair (, ) is also weakly convergent and by the continuity of the eponential limit function and the Gumbel limit, d u Z ut ln Cb O C u b u e u, in distribution, as claimed (see ref., p., or ref., Section.). Step. The preceding two steps can now be repeated for any linear combination i ui Z ui T yielding the finite dimensional distributional convergence fd u Z ut,uo C u b u e u,u, [] where is standard Gumbel. We summarize the situation for nonlattice populations: Jagers et al.

5 Theorem. Consider subcritical Malthusian general, single-type, nonlattice branching processes with finite reproduction variances. Assume conditions and 9. Then the finite dimensional distributional convergence holds. It remains to note the discontinuity at the endpoints of the unit interval in that u Z ut equals at u andatu. The former disappears in the Markov case, where age distribution does not matter and C. The latter mirrors the drastic fall of population size from u close to but tou, at which we proceed to have a closer look. Markov Branching Processes on the Eve of Etinction Markov branching processes, i.e., splitting processes (children are only born at their mother s death) where individuals have eponentially distributed life spans, have been much used in mathematical biology, even though they are of limited biological relevance, since eponential life spans imply the absence of aging. Being nonlattice, Markov branching processes are covered by our general results on etinction time and path to etinction. However, they also allow a direct approach. The latter is mathematically interesting, using a conditioning argument at a nonstopping time. For the Markov case, it recovers Theorem under slightly relaed conditions, and adds new knowledge because it renders it possible to study the process immediately before etinction, viewing the discontinuity from the left, as u, through a magnifying glass, as it were (Theorem ). Now, let Z t be a Markov branching process in which particles have eponentially distributed lifetimes with parameter a. Then Z t e rt, eactly. Hence, as pointed out, C ; also r a( m), as any tetbook would tell you. Further, consider a function v(t) t. A total probability argument yields Z vt y Z vt Z vt Z vt Z vt y, T dt y, T dt yt dtz vt y yt y vt dt. In words, the last equation holds because, conditionally on Z v(t) y, the probability of the original population, initiated by ancestors, dying out at dt is the same as the probability that a population of y individuals at v(t) dies out after time t v(t). Note that this argument applies quite generally to Markov processes. What is special in the branching case is the simple relation T y t T t y, valid for any positive integer y and helpful in calculating an integral representation of the probability generating function s Z vt. Analysis of the latter for the case v(t) ut, u yields Theorem for the Markov branching case, as mentioned under somewhat less restrictive conditions: the finite reproduction moment can be replaced by the log condition k log kp k, where p k is the probability of splitting into k children. In the present contet, it is more interesting that the case v(t) t u, u t, yields distributional convergence of Z T u,as, the limiting process Y u, thus displaying the properties of etinct populations on the eve of their disappearance. Indeed, write y, y,,... for the unique solution (up to multiplicative factors) of the system y Z t y, y,,..., the stationary measure of the process (ref., p. ). Then, the following can be proved to hold: Theorem. Consider a subcritical Markov branching process Z t starting from individuals, having parameters a and {p k } satisfying the log condition, as above. Then, as, Z T u Y u, in distribution for fied u, and finite-dimensionally. The limiting process is Markov. It starts from Y, stays at any position y,,,... an eponentially distributed time, with parameter ay, and then jumps to a position z with probability z z p y yz. y In the long run, it increases eponentially: as u, Y u Xe ru in distribution. Here X is an eponentially distributed random variable, with mean b. We thank a referee for taking readability seriously. This work has been ported by the Swedish and Australian Research Councils.. Haccou P, Jagers P, Vatutin V () Branching Processes: Variation, Growth, and Etinction of Populations (Cambridge Univ Press, Cambridge, UK).. Jagers P (989) Stoch Proc Appl :8.. Jagers P (97) Branching Processes with Biological Applications (Wiley, New York).. Athreya K, Ney P (97) Branching Processes (Springer, Berlin).. Green PJ (977) J Appl Prob :.. Gosselin F () Ann Appl Prob : Jagers P, Nerman O (98) J Appl Prob : Leadbetter MR, Lindgren G, Rootzén H (98) Etremes and Related Properties of Random Sequences and Processes (Springer, New York). 9. Pakes AG (989) Adv Appl Prob : 9.. Billingsley P (999) Convergence of Probability Measures (Wiley, New York), nd Ed.. Silvestrov DS () Limit Theorems for Randomly Stopped Stochastic Processes (Springer, London).. Harris TE (9) The Theory of Branching Processes (Springer, Berlin). Jagers et al. PNAS April, 7 vol. no.