On the path to extinction

The Time to Extinction

Now consider processes with x ancestors, Z_t^x.

By 2 the time to extinction, satisfies for some c_t tending to c, as t → ∞. As a consequence, where γ_x → Euler's γ.

To verify 6 note that for any fixed number v, 0 < v < ∞ On the other hand, where sup_t≥v |c_t − c| < ε for any ε > 0, given that v is sufficiently large. Thus it remains to observe that as x → ∞ where o(1), denotes a remainder term, vanishing as x → ∞.

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 η_x by Then, as x → ∞, This extends Pakes's (9) result for Markov branching processes.

In Between Dawn and Demise

Let Z_uTx^x denote the population size at some time uT_x, 0 ≤ u ≤ 1, between its inception at size Z ₀ ^x = x and extinction. We consider nonlattice processes, and to ease exposition, write t_x = ln x, so that T_x ∼ t_x/r, by 7 and 8. Since 𝔼[Z_t^x] ∼ xCe^−rt, as t → ∞, also x^u−1 𝔼[Z _u(tx+t)/r ^x] ∼ Ce^−ut, x → ∞, and it is natural to guess that x^1−u provides the right norming and that in distribution, as the initial population size x → ∞, 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/(2p ₀ − 1), which is 1.5 in the present case.

Fig. 1 displays the simulation results. The two upper panels show that the process Z_uTx^x, 0 < u < 1 displays much more of variation than does Z_t^x. 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 in terms of the classical Gamma function. Thus, for b/C close to one (it cannot be smaller), the Gamma function exerts a strong influence on the behavior of expected size, and since Γ(1) = Γ(2) = 1 the overall picture is that of a fairly constant mean. For large b/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.

The Path-to-Extinction Theorem and Its Proof

Our proof of the convergence follows a three-step programme, which requires acquaintance with weak convergence theory (10). The reader interested in the basic pattern of extinction, rather than mathematical technique is thus advised to jump to Theorem 1.

1. For fixed 0 < u < 1, write ζ_x(t) = x^u−1 Z _u(tx+t)/r ^x and prove that as x → ∞, the arrow indicating weak convergence in the Lindvall–Skorohod topology extended to the whole real line.

2. Consider ζ_x at the random argument τ_x : = ln c + η_x → τ : = ln (C/b) + η, where η is Gumbel by 7 and 8, and check conditions for ζ_x(τ_x) → Ce^−uτ = C^1−u b^u e^−uη in distribution for fixed u.

3. Study the asymptotics of ζ_x(τ_x)(u) = x^u−1 Z_uTx^x as a function of u varying inside the unit interval.

Step 1.

We already noted that Similarly, We can conclude that for any fixed t, ζ_x(t) → Ce^−ut in mean square. The finite dimensional convergence is obvious.

Turning to tightness, we note that for any v ∈ ℝ and a > 0 But the total progeny Y of a population from one newborn ancestor clearly dominates the maximum of the numbers ever simultaneously present, and the sum of the maxima majorises the supremum of the sum. Since 𝔼[Y] = 1/(1 − m), it follows that We can conclude that for some constant K.

Now consider the population counted with a characteristic χ (ref. 3, p. 167) that for any h > 0 gives to each individual present the value of

1. an indicator of dying the next h/r time units plus

2. the total number of progeny of the individual born within the same next time span of length h/r.

Denote the process thus counted by D_h^x(t). Clearly, for any t ∈ ℝ, Write δ = h/r, and denote the total progeny within t time units stemming from one newborn individual by Y_t. Then, For nonlattice processes it follows that, as t → ∞, If we assume that reproduction is L ²-Lipschitz in the sense that the second moment of the number of births between ages t and t + h, v_h(t), satisfies then a similar, but prohibitive, analysis of the convolution equation for second moments, shows that also 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(μ(t + h) − μ(t), so that such qualities of the reproduction function μ transfer.

By Chebyshev's inequality, therefore, for a suitable K, for h little enough and K′ some constant.

Tightness follows along the lines of the Corollaries in ref. 10 (p. 83 and p. 142). The process ζ_x(t) thus has the weak nonrandom continuous limit Ce^−ut, t ∈ ℝ, as x → ∞. 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 3.

The preceding two steps can now be repeated for any linear combination yielding the finite dimensional distributional convergence where η is standard Gumbel.

We summarize the situation for nonlattice populations:

Theorem 1.

Consider subcritical Malthusian general, single-type, nonlattice branching processes with finite reproduction variances. Assume conditions 1 and 9. Then the finite dimensional distributional convergence 10 holds.

It remains to note the discontinuity at the endpoints of the unit interval in that x^u−1 Z_uTx^x equals 1 at u = 0 and 0 at u = 1. The former disappears in the Markov case, where age distribution does not matter and C = 1. The latter mirrors the drastic fall of population size from u close to but < 1 to u = 1, at which we proceed to have a closer look.

Markov Branching Processes on the Eve of Extinction

Markov branching processes, i.e., splitting processes (children are only born at their mother's death) where individuals have exponentially distributed life spans, have been much used in mathematical biology, even though they are of limited biological relevance, since exponential life spans imply the absence of aging. Being nonlattice, Markov branching processes are covered by our general results on extinction time and path to extinction.

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 1 under slightly relaxed conditions, and adds new knowledge because it renders it possible to study the process immediately before extinction, viewing the discontinuity from the left, as u → 1, through a magnifying glass, as it were (Theorem 2).

Now, let Z_t^x be a Markov branching process in which particles have exponentially distributed lifetimes with parameter a. Then exactly. Hence, as pointed out, C = 1; also r = a(1 − m), as any textbook would tell you.

Further, consider a function v(t) < t. A total probability argument yields In words, the last equation holds because, conditionally on Z _v(t) ^x = y, the probability of the original population, initiated by x 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 valid for any positive integer y and helpful in calculating an integral representation of the probability generating function Analysis of the latter for the case v(t) = ut, 0 < u < 1 yields Theorem 1 for the Markov branching case, as mentioned under somewhat less restrictive conditions: the finite reproduction moment can be replaced by the x log x condition where p _k is the probability of splitting into k children.

In the present context, it is more interesting that the case v(t) = t − u, 0 < u < t, yields distributional convergence of Z_Tx^x−u, as x → ∞, the limiting process Y_u, thus displaying the properties of extinct populations on the eve of their disappearance.

Indeed, write π_y, y = 1, 2, … for the unique solution (up to multiplicative factors) of the system the stationary measure of the process (ref. 12, p. 24). Then, the following can be proved to hold: