# Genetic evolution without crossover

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.

• 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? Oct 18, 2021 at 22:08
• $m_{ij}$ 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. Oct 22, 2021 at 13:59