When a bisexual process starts with \(j\) mating units, we will denote its
probability of extinction by \(q_j\).
Recall from Lecture 14 the notation \[
\mu_j=\frac{1}{j}E(Z_n\ |\ Z_{n-1}=j), \qquad j\ge 1,
\]
for the average unit reproduction means. Also,
according to Theorem 15.1, \(q_j=1\)
for all \(j=1,2,\ldots\) if and only if
\[
\mu_\infty= \lim_{j\to \infty} \mu_j = \sup_{j>0} \mu_i=1.
\] The value \(\mu_1\) tells us
all we need to know about certain extinction in the (asexual)
Galton-Watson process. But the sequence \(\{\mu_j\}_{j=1}^\infty\) must be considered
to insure an extinction probability of one in a process with mating.
Since \(\{ Z_n\}_{n=0}^\infty\) is a
Markov chain with a single absorbing state (zero) and all other states
(positive integers) transient, either the process is absorbed in state
zero (extinction) or \(\lim_{n\to
\infty}Z_n=\infty\) (survival). Since \(\mu_\infty\) is both \(\lim\) and \(\sup\) of \(\{\mu_j\}\), when \(\mu_\infty<1\), each \(\mu_j<1\) and Theorem 15.1 implies that
extinction is certain (eventually) no matter how many mating units start
the process. On the other hand, if \(\mu_\infty >1\), there exists a positive
integer \(k\) such that if \(j>k\), then \(\mu_j>1\). It then follows that if \(Z_0\) is large enough, there will be a
positive probability of survival.
The branching process with mating has four parameters as follows:
\(Z_O\), the number of mating units in the initial (zeroth) generation;
the mating function \(\zeta\) tells us the number of mating units that will be formed in a generation having \(x\) females and \(y\) males. This is a nonnegative integer-valued function defined on all ordered pairs of nonnegative integers subject to the conditions
\(0<\zeta(x, y)<xy\) (which implies that \(\zeta(t,0)= \zeta(0,t)=0\) for any non-negative integer \(t\)), and
\(\zeta\) is nondecreasing in \(x\) and in \(y\).
the offspring probability distribution \(\{ p_k\}_{k\ge 0}\), where \(p_k\) is the probability that a mating unit will produce \(k\) offspring; and
\(\alpha\), the probability that an individual offspring will be female.
Theorem 15.1 motivates the question, if \(\mu_\infty>1\) when are there enough mating units in the initial generation to have an extinction probability less than one? In contrast to the Galton-Watson process it is possible in a process with mating to lower an extinction probability of one to a value less than one by increasing \(Z_0\) while fixing the other three parameters.
Example 1. Consider a process with the perfect fidelity mating function \(\zeta(x,y)=\min(x,y)\), \(\alpha = 1/2\), and \(p_3=1\). If \(Z_0=1\), then \(q_1\) will be one. (Indeed, the process will have no more than 1 mating unit in each generation.) Daley showed that extinction is certain if and only if, the minimum of the mean number of females produced per mating unit \((\alpha \mu)\) and the mean number of males produced per mating unit \(((l - \alpha)\mu)\) is less than or equal to one. If we increase \(Z_0\) to two, Daley’s criterion applies. Since \[ \min\{\alpha \mu, (1 - \alpha) \mu\} = \min\left\{\frac{1}{2}\cdot 3, \left(1-\frac{1}{2}\right)\cdot 3\right\}=\frac{3}{2}>1, \] it follows that \(q_2<1\). (It is known that \(\{q_j\}\) is a non-increasing sequence.)
It is now our intention to state a simple criterion that will indicate which values of \(j\) will give \(q_j<1\) when \(\mu_\infty >1\). To establish this criterion, we need to state two definitions. But first, it looks like the mating function definition along with the added condition of superadditivity is still too broad. It would appear that relevant mating functions should also satisfy the following additional two conditions. (It is not by accident that these two conditions are needed to have our criterion for determining when \(q_j<1\).)
Condition 1 \(\zeta (1,1)=1\). This is certainly a reasonable expectation. If a generation consists of one male and one female (forgetting what differences they may have) they will mate.
Condition 2 \(\zeta (x,y)<\min (xy, x+y)\). (Note that \(xy<x+y\) only when \(x=1\) or \(y=1\).) Many superadditive mating functions are bounded by the number of individuals of one or both sexes in any generation.
Definition 1. If \(\zeta\) is a superadditive mating function that satisfies Conditions 1 and 2 we say that \(\zeta\) is a superadditive population bounded (SPB) mating function.
Note that conditions 1 and 2 along with superadditivity imply that when \(\zeta\) is SPB the following inequalities hold: \(\min(x,y)< \zeta (x,y)<x+y\). This is a reasonable range for the number of mating units in any generation of a human or animal population.
Definition 2. Since it is known that \(\{ q_j\}_{j=0}^\infty\) is a non-increasing sequence, when \(\mu_\infty>1\), there will be a largest nonnegative integer, say \(N_\zeta\), such that \(q_{N_{\zeta}}= 1\).
In Example 1 we see that \(q_1=1\) and \(q_2<1\) implies \(N_\zeta =1\).
We can now state the main result of this lecture.
Theorem 16.1. Assume \(\zeta\) is a SPB mating function in a process with \(\mu_\infty>1\), then \(q_j<1\) if and only if, \(P(Z_{n+1}>j\ |\ Z_n=j)>0\).
Theorem 16.1 says that under the given assumptions if we start with \(j\) mating units and there is a positive probability of having more than \(j\) mating units in the next generation, there is a positive probability of survival.
The next two examples show that Conditions 1 and 2 in the definition of a SPB mating function are needed in order for \(P(Z_{n+1}>j\ |\ Z_n=j)>0\) to imply \(q_j<1\).
Example 2. \(\zeta(x,y)=0\) when \(x\le 2\), \(y\le 3\) and when \(x\le 3\), \(y\le 2\); \(\zeta(4,4)=\zeta(5,4)=\zeta(4,5)=3\), \(\zeta(x,y)=\min(x,y)\) otherwise, \(p_3\equiv 1\), \(\alpha=1/2\) and \(Z_0=2\). Note that this mating function is superadditive; Condition 2 holds but Condition 1 does not. Note that \(P(Z_{n+1}=3\ |\ Z_n=2)>0\). But, since \(\zeta(i,9-i)\le 3\) for \(i=0,1,2,\ldots,9\), it follows that \(q_2=q_3=1\).
Example 3. \(\zeta(2,4)=\zeta(3,3)=\zeta(4,2)=6\), \(\zeta(x,y)=xy\) otherwise, \(p_1\equiv 1\), \(\alpha=1/2\) and \(Z_0=5\). Here the mating function is superadditive, Condition 1 holds but Condition 2 does not. It is obvious that \(P(Z_{n+1}=6\ |\ Z_n=5)>0\). However, \(\zeta(i,6-i)<6\) for \(i=0,1,\ldots ,6\), implies that \(q_5=q_6=1\).