The transition to speciation in the finite genome Derrida–Higgs model: a heuristic solution

Abstract The process of speciation, where an ancestral species divides in two or more new species, involves several geographic, environmental and genetic components that interact in a complex way. Understanding all these elements at once is challenging and simple models can help in unveiling the role of each factor separately. The Derrida–Higgs model describes the evolution of a sexually reproducing population subjected to mutations in a well mixed population. Individuals are characterized by a string with entries ±1 representing a haploid genome with biallelic genes. If mating is restricted by genetic similarity, so that only individuals that are sufficiently similar can mate, sympatric speciation, i.e. the emergence of species without geographic isolation, can occur. Only four parameters rule the dynamics: population size N , mutation rate µ , minimum similarity for mating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>q</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>min</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and genome size B . In the limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> , speciation occurs if the simple condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>q</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>min</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mstyle scriptlevel="0"/> <mml:mo>+</mml:mo> <mml:mstyle scriptlevel="0"/> <mml:mn>4</mml:mn> <mml:mstyle scriptlevel="0"/> <mml:mstyle scriptlevel="0"/> <mml:mi>μ</mml:mi> <mml:mi>N</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> is satisfied. However, this condition fails for finite genomes, and speciation does not occur if the genome size is too small. This indicates the existence of a critical genome size for speciation. In this work, we develop an analytical theory of the distribution of similarities between individuals, a quantity that defines how tight or spread out is the genetic content of the population. This theory is carried out in the absence of mating restrictions, where evolution equations for the mean and variance of the similarity distribution can be derived. We then propose a heuristic description of the speciation transition which allows us to numerically calculate the critical genome size for speciation as a function of the other model parameters. The result is in good agreement with the simulations of the model and may guide further investigations on theoretical conditions for species formation.