Can someone please point me to the origin of the system of coupled differential equations (1) in Section 2 of Shahshahani's book$^\star$?

$$ \dot{x}_i = x_i \sum_{j=1}^n m_{ij} \frac{x_j}{|x|} $$

Shahshahani claims this equation is well known as a simple consequence of random mating and cites Crow, J. F., Kimura, M., et al. (1970). An introduction to population genetics theory, but I have not been able to find the equation there.

This is a dynamical system with $x \in \mathbb{R}^n_{>0}$, and $m$ is a symmetric matrix with constant coefficients. The equation models the evolution of a genetic system subject to selection in population genetics. The variables represent

  • $n \in \mathbb{N}$ number of types of gametes in a diploid population
  • $i \in \{1, \dots, n\}$ labels the types of gametes
  • $x_i \in \mathbb{R}_{>0}$ number of gametes of type $i$
  • the unordered pair $\{i,j\}$ represents the genotype determined by the gametes $i$ and $j$
  • $m$ is the selection matrix and $m_{ij}=m_{ji}$ is the selective advantage of the genotype $\{i,j\}$
  • $|x| := x_1 + \dots + x_n$

$\star$ S. Shahshahani, A new mathematical framework for the study of linkage and selection, Memoirs of the American Mathematical Society, Volume 17, Number 211, 1979.

  • 5
    $\begingroup$ Can you please provide enough information in your question so that it can be understood without reading external references? What is the meaning of the variables in the equation? $\endgroup$
    – Bryan Krause
    Oct 18 '21 at 22:08
  • 1
    $\begingroup$ $m_{ij}$ is the mean number of descendants at the next generation from a single individual carrying $i,j$, so $m_{ij}$ has to be multiplied by the number of individuals carrying $i,j$ at the current generation. $ x_i \frac{x_j}{|x|}$ is supposed to be that number of individuals. This is a very simplified model assuming perfect mixing between individuals carrying $x_i$ and those carrying $x_j$ and I'm not sure the "unordered" is taken in account correctly. But anyway it gives you a rough idea of what this equation says and how useless/simplified it is. $\endgroup$
    – reuns
    Oct 22 '21 at 13:59

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