##Branching Processes with Mating, Part II
To examine the effect of sexual reproduction on extinction probabilities, we turn to the Galton–Watson process with mating, which is introduced in Lecture~14. Recall that in this model the nth generation consists of \(F_n\) females and \(M_n\) males, who form \(Z_n=\zeta (F_n, M_n)\) couples where \(F_n\) and \(M_n\) are random variables and \(\zeta\) is a deterministic function, called a mating function. Each couple produces offspring independently of all other couples and according to the same distribution. Thus, let for each couple \(p_{j,k}\) denote the probability of producing \(j\) female and \(k\) male children. With \(X_k\) and \(Y_k\) denoting, respectively, the number of female and male offspring of the \(k\)th couple of the \(n\)th generation (labeled in arbitrary fashion), we arrive at \[ F_{n+1}=\sum_{k=1}^{Z_n} X_k\qquad \mbox{and}\qquad M_{n=1}=\sum_{k=1}^{Z_n} Y_k \] for \(n\ge 0\), where the \((X_k,Y_k)\) are independent and identically distributed. This is the familiar structure for the Galton–Watson branching process with the one, but important, difference that here the summation ranges over the number of couples of the preceding generation. Choosing the asexual mating function \(\zeta (x,y)=x\), we see that \(Z_n\) just equals the number of females in the \(n\)th generation (\(Z_n = F_n\) for all \(n\ge 0\)) and is, indeed, a classic branching process.
From a mathematical viewpoint it is desirable to restrict the class of offspring distributions to facilitate explicit computations. Daley (1968) gave two alternative possibilities:
Assumption 1 Conditionally on the total number of offspring, their sex is determined at random (analogously to flipping a, possibly biased, coin). More formally, if \(\hat{p}_{j+k}\) denotes the probability that a couple produces \(j+k\) children, and if the probability that a child is female equals \(\theta\), then for all \(j,k\ge 0\) \[ p_{j,k}={ j+k \choose j} \theta^j (1-\theta)^k \hat{p}_{j+k}.\qquad (1) \] Assumption 2 Another possibility is that the numbers of male and female offspring are independent with a possibly different distribution. In this case for all \(j,k \ge 0\) \[ p_{j,k}=p_j^F p_k^M, \] where \(p_j^F\) and \(p_k^M\) denote the probabilities that a couple has respectively, \(j\) female and \(k\) male children, respectively.
Note that the mechanism that corresponds to Assumption~1 is most common in mammals, and occurs in humans with \(\theta =0.5\). Assumption~2 may be reasonable in situations with environmental sex determination, such as temperature dependence in many reptiles.
Let us stipulate without further discussion that hereafter \((p_{j,k})_{j,k\ge 0}\) always satisfies Assumption~1 or 2 and, further, that there is a positive probability of producing offspring that are of one sex only, that is, \[ \max (p_{0,\ast}, p_{\ast, 0})>0, \] where \(p_{0,\ast}=\sum_{k\ge 0} p_{0,k}\) and \(p_{\ast, 0}=\sum_{j\ge 0} p_{j,0}\) denote the respective probabilities that a couple has only male or only female offspring. This condition holds automatically under Assumption~1 because \(0<\theta<1\), and is equivalent to \(\max(p_0^F , p_0^M)>0\) under Assumption~2. As before, we assume that the mating function \(\zeta\) is common sense and superadditive (see Lecture~14).
We now turn to the fundamental question of finding conditions that guarantee certain ultimate extinction of a Galton–Watson process with mating \((Z_n)_{n\ge 0}\). To be more precise, let \[ R_j:=P(Z_n=0\ \mbox{eventually}\ |\ Z_0=j) \] denote the extinction probability given \(j\ge 1\) ancestor couples. Then the question in its most ambitious form may be restated as: Is there an intuitive condition for \(R_1 = R_2 =\ldots =1\), as for the simple Galton–Watson process, where we know that certain extinction occurs if, and only if, each individual produces at most one child on average and has a positive chance of having no children?
The following example, from Hull (1982), shows that one cannot expect an equally simple answer for processes with sexual reproduction. Consider the mating function \(\zeta (x, )=0\) if \(x=0\) or \(y=0\), and \(\zeta (x,y)=x+y−1\) otherwise. Let \(p_{j,k}\) be of the form of (1) for some \(0<\theta <1\) and with \((\hat{p}_j)_{j\ge 0}\) defined through \(\hat{p}_3=1\), and hence \(\hat{p}_j=0\) otherwise. Then, every couple has exactly three children. Nonetheless, extinction occurs if, for some \(n\ge 0\), all couples of the \(n\)th generation produce only female or only male offspring.
The following general result is due to Dalley et al. (1986).
Theorem 15.1 For a Galton–Watson process \((Z_n)_{n\ge 0}\) with a common sense, s uperadditive mating function \(\zeta\), the average reproduction means \(\mu_j\) are convergent to the limit \(\mu_\infty= \sup_{k\ge 1} \mu_k\) Furthermore, \(\mu_\infty\le 1\) implies certain extinction for any initial population size (i.e., \(R_1=R_2=\ldots =1\)), while in the case \(\mu_\infty >1\) (ultimate supercriticality) the population survives with positive probability for a sufficiently large initial population size, in fact \(1>R_{i_0}\ge R_{i_0+1}\ge \ldots\) for some positive integer \(i_0\).
Proof In the following we present the main arguments of the proof, without technicalities. The first observation to make is that for all \(i, k \ge 0\) \[ P(Z_{n+1}=j\ |\ Z_n=i)=P\left(\zeta\left(\sum_{k=1}^i X_{n+1,k}, \ \sum_{k=1}^i Y_{n+1,k}\right)=j\right) \] and, since the mating function ζ is monotonic in each argument, this implies \[ P(Z_1>k\ |\ Z_0=i)=P\left(\zeta\left(\sum_{j=1}^i X_{j}, \ \sum_{j=1}^i Y_{j}\right)>k\right)\le P\left(\zeta\left(\sum_{j=1}^{i+1} X_{j}, \ \sum_{j=1}^{i=1} Y_{j}\right)>k\right) \] \[ =P(Z_1>k\ |\ Z_0=i+1). \] So the probability of exceeding a size \(k\) in the next generation forms an increasing function of the current population size. A Markov chain with this property is called stochastically monotone. By an easy inductive argument, one can prove that the above inequality generalizes for all \(i,k =0,1,\ldots\) and \(n=1,2,\ldots\) to \[ P(Z_n>k\ |\ Z_0=i)\le P(Z_n>k\ |\ Z_0=i+1), \] which in turn yields the important fact that the extinction probability \(R_i\) is a decreasing function of the initial population size \(i\). Namely, by letting \(n\) tend to infinity for all \(i\ge 0\) \[ 1-R_i=\lim_{n\to \infty}P(Z_n>0\ |\ Z_0=i)\le \lim_{n\to \infty}P(Z_n>0\ |\ Z_0=i+1)=1-R_{i+1}. \] For a more intuitive comparison argument, suppose the population starts with \(i+1\) ancestor couples (\(Z_0=i+1\)). Choose an arbitrary subset of \(i\) couples and denote by (\(Z_n)_{n\ge 0}\) the process based on this subset, hence \(Z_0=i\). Then the \(Z_1\) couples that form the first generation of the original population are those formed by the offspring of the \(i\) ancestor couples of the subpopulation plus, generally, some more because of the one additional ancestor couple in the original population and the monotonicity of the mating function. This shows \(Z'_1\le Z_1\) and finally leads to the conclusion that \(Z'_n\le Z_n\) for all \(n\ge 0\) when repeating the argument for the subsequent generations. Since the extinction probabilities of \((Z_n)_{n\ge 0}\) and \((Z'_n)_{n\ge 0}\) are \(R_{i+1}\) and \(Q_i\), respectively, the inequality \(Q_i\ge Q_{i+1}\) follows as a consequence.
We now show that \(\mu_j\) converges to \(\mu_\infty = \sup_{k\ge 1} \mu_k\). Indeed, by the definition of \(\mu_j\) \[ (j+k)\mu_{j+k}=E(Z_1\ |\ Z_0=j+k)=E\left(\zeta\left(\sum_{l=1}^{j+k} X_l, \ \sum_{l=1}^{j+k} Y_l\right)\right)\qquad (2) \] and from the superadditivity of \(\zeta\), the result is larger than or equal to \[ E\left(\zeta\left(\sum_{l=1}^{j} X_l, \ \sum_{l=1}^{j} Y_l\right)\right)+E\left(\zeta\left(\sum_{l=j+1}^{j+k} X_l, \ \sum_{l=j+1}^{j+k} Y_l\right)\right). \] From the independence and identical distribution of the offspring variables \((X_l, Y_l)\), it follows that this equals for all \(j,k\ge 1\) \[ E\left(\zeta\left(\sum_{l=1}^{j} X_l, \ \sum_{l=1}^{j} Y_l\right)\right)+E\left(\zeta\left(\sum_{l=1}^{k} X_l, \ \sum_{l=1}^{k} Y_l\right)\right)=E(Z_1\ |\ Z_0=j)+E(Z_1\ |\ Z_0=k) \] \[ =j\mu_j+k\mu_k\qquad (3) \] Combining Equation (2) with Equation (3), we find that \((j+k)\mu_{j+k}\ge j\mu_j + k\mu_k\) for all \(j,k\ge 1\), which implies that \(j\mu_j\) is superadditive. Applying standard results on superadditive functions to \((j\mu_j)_{j\ge 1\) then yields the asserted convergence of the \(\mu_j\) to \(\mu_\infty = \sup_{k\ge 1} \mu_k\).
Suppose now that \(\mu_\infty \le 1\) and thus \(\mu_j\le 1\) for all \(j\ge 1\). Then for all \(i, n\ge 0\) \[ E(Z_{n+1}\ |\ Z_n=i)=i\mu_i\le i \] holds. A stochastic sequence with this property is called a supermartingale. A fundamental result from the theory of stochastic processes says that every non-negative supermartingale converges to a finite random variable, hence \(Z_n \to Z_\infty\) (for any given initial population size). However, \(Z_\infty\) must then be identical to 0 by the extinction–explosion dichotomy, and so \(R_1=R_2=\ldots =1\), as asserted.
To see that \(R_i<1\) for all sufficiently large \(i\) in the case \(\mu_\infty>1\) is more difficult and too technical to be presented here. However, a rather simple argument from Hull (1982) exists under the stronger condition \(\mu_1>1\), and is again based on a comparison of \((Z_n)_{n\ge 0}\) with another process, a supercritical Galton–Watson process. Define \(Z'_0=Z_0\) and the, recursively, for \(n\ge 2\) \[ Z'_n=\sum_{j=1}^{Z'_{n-1}} \zeta (X_j, Y_j). \] One may think of \((Z_n)_{n\ge 0}\) as describing an inbreeding population in which couples are formed according to the same mating function, but only by children of the same parents. The superadditivity of \(\zeta\) implies \[ Z'_1=\sum_{j=1}^{Z_0} \zeta(X_j, Y_j)\le \zeta\left( \sum_{j=1}^{Z_0} X_j, \sum_{j=1}^{Z_0} Y_i\right)=Z_1, \] and then, inductively, \(Z'_n\le Z_n\) for all \(n\ge 0\). Since all \(\zeta (X_j, Y_j)\) are independent with the same distribution \((p_k)_{k\ge 0}\), say, \((Z_n)_{n\ge 0}\) is distributed as a simple Galton– Watson process with offspring distribution \((p_k)_{k\ge 0}\). It is further supercritical because \(E(\zeta (X_1, Y_1))=E(Z_1\ |\ Z_0=1)=\mu_1>1\). Consequently, \((Z_n)_{n\ge 0}\) survives with positive probability for any initial population size and so \((Z_n)_{n\ge 0}\) also does (i.e., \(1>R_1\ge R_2\ge \ldots )\).