An nth-cousin mating model and the n-anacci numbers

TL;DR

This paper explores a mathematical model of inbred pedigrees where mating occurs between nth cousins, revealing a connection to the n-anacci sequence and analyzing the growth rate of pedigree size over generations.

Contribution

It establishes a link between nth-cousin mating models and n-anacci sequences, deriving generating functions and asymptotic growth rates for pedigree sizes.

Findings

Pedigree growth asymptotically follows the n-anacci sequence.

Growth rate approaches the golden ratio for second cousins.

Growth approaches 2 as the cousin degree increases.

Abstract

In seeking to understand the size of inbred pedigrees, J. Lachance (J. Theor. Biol. 261, 238-247, 2009) studied a population model in which, for a fixed value of $n$, each mating occurs between $n$th cousins. We explain a connection between the second-cousin case of the model ($n=2$) and the Fibonacci sequence, and more generally, between the $n$th-cousin case and the $n$-anacci sequence $(n \geq 2)$. For a model with $n$th-cousin mating $(n \geq 1)$, we obtain the generating function describing the size of the pedigree $t$ generations back from the present, and we use it to evaluate the asymptotic growth of the pedigree size. In particular, we show that the growth of the pedigree asymptotically follows the growth rate of the $n$-anacci sequence -- the golden ratio $\phi = (1 + \sqrt{5})/2 \approx 1.6180$ in the second-cousin case $n=2$ -- and approaches 2 as $n$ increases. The computations explain the appearance of familiar numerical sequences and constants in a pedigree model. They also recall similar appearances of such sequences and constants in studies of population biology more generally.