Markov-chain approach to the distribution of ancestors in species of biparental reproduction

We studied how to obtain a distribution for the number of ancestors in species of sexual reproduction. Present models concentrate on the estimation of distributions repetitions of ancestors in genealogical trees. It has been shown that it is not possible to reconstruct the genealogical history of each species along all its generations by means of a geometric progression. This analysis demonstrates that it is possible to rebuild the tree of progenitors by modeling the problem with a Markov chain. For each generation, the maximum number of possible ancestors is different. This presents huge problems for the resolution. We found a solution through a dilation of the sample space, although the distribution defined there takes smaller values with respect to the initial problem. In order to correct the distribution for each generation, we introduced the invariance under a gauge (local) group of dilations. These ideas can be used to study the interaction of several processes and provide a new approach on the problem of the common ancestor. In the same direction, this model also provides some elements that can be used to improve models of animal reproduction.

Figure 1

Examples of three kinds of genealogical trees (only the first few $g$ generations). (a) No restrictions by blood relationship. (b, c) Two kinds of restriction in third generation: ancestors sharing one parent (b) or ancestors sharing two parents (c). The restriction by blood relationship increases according to the degree of endogamy.

Figure 2

The four values of $\{\mathfrak{a},\mathfrak{b}\}$ are obtained from (19) and parametrized from $(t_1,{\alpha }_1,t_2,{\alpha }_2)=(t_1,\xi 2^{t_1+1},T,2)$, where $0\le \xi \le 1$. In $t_1$ the curve reaches the fraction $\xi$ of the total number of possible ancestors for that generation, while $t_2$ defines the maximum generation range denoted by $T$. The geometric progression $2^{t+1}$ is in dash black line. The $\{a,b,c,d\}$ ({blue, green, orange, red}) lines can be obtained respectively from $(t_1,{\alpha }_1,t_2,{\alpha }_2)\in \{(3,0.4\times 2^4,12,2),(3,0.7\times 2^4,10,2),(3,0.9\times 2^4,10,2),(3,0.7\times 2^4,12,2)\}$.

Figure 3

A band of curves, defined by the set $\mathcal{B}=\{\widehat{\alpha }\left(t\right):\widehat{\alpha }\left(t\right)\in [\alpha \left(t\right)-\sigma \left(t\right),\alpha \left(t\right)+\sigma \left(t\right)]\}$, contains the expectation value $\alpha \left(t\right)$ given by the equation (18) and its statistical fluctuation at 1-$\sigma$. The $\{a,b,c\}$ ({blue, green, red}) lines corresponds to the realization of $\left\{\alpha \right(t)-\sigma (t),\alpha (t),\alpha (t)+\sigma (t\left)\right\}$, with the values of $(t_1,{\alpha }_1,t_2,{\alpha }_2)=(6,0.9\times 2^7,10,2)$, respectively.