On the path to extinction Peter Jagers*†, Fima C. Klebaner‡, and Serik Sagitov* *Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden; and ‡School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia Communicated by David Cox, University of Oxford, Oxford, United Kingdom, January 4, 2007 (received for review September 15, 2006)
G
eneral 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 (1). A presentation from a mathematical point of view can be found in ref. 2 (multi-type populations) or ref. 3 (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 extinction 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 expected number m of children per individual is less than one. Let (a) denote the expected 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 exists a number ␣, the Malthusian parameter, such that www.pnas.org兾cgi兾doi兾10.1073兾pnas.0610816104
ˆ 共␣兲 ⫽
冕
⬁
e⫺␣a共da兲 ⫽ 1.
This is always fulfilled in the critical (m ⫽ 1) and supercritical (1 ⬍ m ⬍ ⬁), and under very mild conditions (always met with in the real world) also in subcritical cases. Then, ␣ ⬍ 0 and since ⫺␣ will play an important role, we shall write r ⫽ ⫺ ␣ ⬎ 0. In the case of Galton–Watson processes, r ⫽ 兩ln m兩. Let Z xt be a shorthand for the number of individuals alive at time t, provided the population started from x individuals at time 0, taken as newborn then, for preciseness. Z t without a suffix, is Z 1t , the case of one ancestor. If L denotes the life span distribution, any nonlattice Malthusian process will then satisfy ⺕关Zt兴 ⬃
兰⬁0 eru共1 ⫺ L共u兲兲du 兰⬁0 ueru共du兲
e⫺rt, t 3 ⬁
(ref. 3, p. 156). We write C for the proportionality constant in this, i.e., ⺕[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 ⫽ 1, and expressions are exact, ⺕[Z t] ⫽ m t for integer t in the Galton–Watson case. The expected population size thus has the same asymptotic form in all three cases, ␣ ⬎, ⫽, ⬍ 0. For variances, though, asymptotics take different forms in the sub- and supercritical cases. Assume that m ⬍ 1 and that Malthusianness is strengthened to a second order property: let denote the reproduction point process, so that ⫽ ⺕[], and write ˆ共␣兲 ⫽
冕
⬁
e⫺␣a共da兲;
0
it is required that ⺕[ˆ(␣)2] ⬍ ⬁. Then, Var关Z t兴 ⬃ Be ⫺rt, for some constant B. This can be checked from the convolution equation holding for second moments (ref. 3, p. 136). In the more often studied supercritical case, the leading term is proportional to e 2␣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 reagents兾analytic 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. E-mail:
[email protected].
© 2007 by The National Academy of Sciences of the USA
PNAS 兩 April 10, 2007 兩 vol. 104 兩 no. 15 兩 6107– 6111
APPLIED MATHEMATICS
0
POPULATION BIOLOGY
Populations can die out in many ways. We investigate one basic form of extinction, stable or intrinsic extinction, 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, asexually reproducing individuals, for which the expected 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, x, of individuals. First we describe the behavior of the time to extinction T: as x grows to infinity, it behaves like the logarithm of x, divided by r, where r is the absolute value of the Malthusian parameter. We give a more precise description in terms of extreme value distributions. Then we study population size partway (or u-way) to extinction, i.e., at times uT, for 0 < u < 1, e.g., u ⴝ 1兾2 gives halfway to extinction. (Note that mathematically this is no stopping time.) If the population starts from x individuals, then for large x, the proper scaling for the population size at time uT is x into the power u ⴚ 1. Normed by this factor, the population u-way to extinction 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 explicit representation can be deduced, we also find a description of the behavior immediately before extinction.
冕
⬁
erttL共dt兲 ⬍ ⬁ and
0
冕
⬁
e rtt 共dt兲 ⬍ ⬁,
[1]
0
⺠共Zt ⬎ 0兲 ⬃ ce⫺rt, t 3 ⬁
[2]
for some 0 ⬍ c ⬍ 1, lim ⺠共Z t ⫽ k兩Z t ⬎ 0兲 ⫽ b k , k ⫽ 1, 2, 3, . . .
[3]
t3⬁
b⫽
冘
kbk ⫽ C兾c.
[4]
k
It is worth noting that these two theorems do not require the full strength of branching process independence assumptions (see ref. 6 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. 3, p. 167 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 extend to processes counted with characteristics such that ert⺕[(t)] is directly Riemann integrable, and so do the Kolmogorov-Yaglom theorems (5). The Time to Extinction Now consider processes with x ancestors, Z xt . By 2 the time to extinction, Tx ⫽ inf兵t ⱖ 0; Z xt ⫽ 0其 satisfies ⺠共Tx ⬎ t兲 ⫽ 1 ⫺ ⺠共T1 ⱕ t兲x ⫽ 1 ⫺ ⺠共Zt ⫽ 0兲x ⫽ 1 ⫺ 共1 ⫺ ct e⫺rt兲x [5] for some c t tending to c, as t 3 ⬁. As a consequence, ⺕关Tx兴 ⫽
冕
⬁
⺠共Tx ⬎ t兲dt ⫽ 共ln x ⫹ ln c ⫹ ␥ x兲兾r,
0
where ␥ x 3 Euler’s ␥. To verify 6 note that for any fixed number v, 0 ⬍ v ⬍ ⬁
冕
v
⺠共Tx ⬎ t兲dt 3 v, x 3 ⬁.
0
On the other hand,
冕
⬁
v
⺠共Tx ⬎ t兲dt ⫽
冕
⬁
v
冘
x⫺1
ct e⫺rt
冘冕
x⫺1
respectively. As a consequence, the so called x log x-condition holds, guaranteeing the Kolmogorov and Yaglom theorems on survival chances and conditional population size given nonextinction (see ref. 4, p. 170, for the case of age-dependent processes and ref. 3, p. 159, and ref. 5, for general processes). In the nonlattice formulation, they tell us that
exist, 兺 k b k ⫽ 1, and
where suptⱖv 兩c t ⫺ c兩 ⬍ for any ⬎ 0, given that v is sufficiently large. Thus it remains to observe that as x 3 ⬁
共1 ⫺ ct e⫺rt兲kdt,
k⫽0
6108 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0610816104
[6]
k⫽0
⬁
ce⫺rt共1 ⫺ ce⫺rt兲kdt ⫽
v
⫽
1 r
冘
x⫺1
k⫽0
1 共1 ⫺ 共1 ⫺ ce⫺rv兲k⫹1兲 k⫹1
1 共ln x ⫹ ␥ ⫹ ln c兲 ⫺ v ⫹ o共1兲, r
where o(1), denotes a remainder term, vanishing as x 3 ⬁. Extreme 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 exponential tail that implies a classical Gumbel limit (ref. 8, p. 17). Indeed, for each initial number x, define x by Tx ⫽ 共ln x ⫹ ln c ⫹ x兲兾r.
[7]
⺠共x ⱕ y兲 3 exp共⫺e ⫺y兲, ⫺⬁ ⬍ y ⬍ ⬁.
[8]
Then, as x 3 ⬁,
This extends Pakes’s (9) result for Markov branching processes. In Between Dawn and Demise x Let Z uTx denote the population size at some time uT x, 0 ⱕ u ⱕ 1, between its inception at size Z x0 ⫽ x and extinction. We consider nonlattice processes, and to ease exposition, write t x ⫽ ln x, so that T x ⬃ t x兾r, by 7 and 8. Since ⺕[Z xt ] ⬃ xCe ⫺rt, as t 3 x ⬁, also x u⫺1 ⺕[Z u(tx⫹t)/r] ⬃ Ce ⫺ut, x 3 ⬁, and it is natural to guess 1⫺u that x provides the right norming and that x 3 C1⫺ubue⫺u xu⫺1ZuT x
in distribution, as the initial population size x 3 ⬁, at least for fixed 0 ⬍ u ⬍ 1. 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 0 ⫽ 0.75 for zero and p 2 ⫽ 0.25 for two offspring, and expected life span ⫽ 1. For birth-and-death, as pointed out, C ⫽ 1 and further b can be shown to equal p 0兾(2p 0 ⫺ 1), which is 1.5 in the present case. Fig. 1 displays the simulation results. The two upper panels x show that the process Z uTx, 0 ⬍ u ⬍ 1 displays much more of x 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 expectation and variance of the proposed limiting population size are ⺕关C1⫺ubue⫺u兴 ⫽ C1⫺ubu⌫共u ⫹ 1兲, Var关C 1⫺ub ue ⫺u兴 ⫽ 共C 1⫺ub u兲 2共⌫共2u ⫹ 1兲 ⫺ ⌫ 2共u ⫹ 1兲兲, 0 ⬍ u ⬍ 1, in terms of the classical Gamma function. Thus, for b兾C close to one (it cannot be smaller), the Gamma function exerts a strong influence on the behavior of expected size, and since ⌫(1) ⫽ ⌫(2) ⫽ 1 the overall picture is that of a fairly constant mean. For large b兾C it increases practically exponentially. This may seem intriguing, but is explained by the norming through x 1⫺u. The limiting variance always increases with u, from 0 to b 2. Fig. 2 shows these expectations and standard deviations for different values of b assuming C ⫽ 1. The situation for b ⫽ 1.5 thus seems to fit simulations. Jagers et al.
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