Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometry and stability of species complexes: larger species speciate less often

Amaury Lambert, Emmanuel Schertzer, Yannic Wenzel

Abstract

Species complexes are groups of closely related populations exchanging genes through dispersal. We study the dynamics of the structure of species complexes in a class of metapopulation models where demes can exchange genetic material through migration and diverge through the accumulation of new mutations. Importantly, we model the ecological feedback of differentiation on gene flow by assuming that the success of migrations decreases with genetic distance, through a specific function $h$. We investigate the effects of metapopulation size on the coherence of species structures, depending on some mathematical characteristics of the feedback function $h$. Our results suggest that with larger metapopulation sizes, species form increasingly coherent, transitive, and uniform entities. We conclude that the initiation of speciation events in large species requires the existence of idiosyncratic geographic or selective restrictions on gene flow.

Geometry and stability of species complexes: larger species speciate less often

Abstract

Species complexes are groups of closely related populations exchanging genes through dispersal. We study the dynamics of the structure of species complexes in a class of metapopulation models where demes can exchange genetic material through migration and diverge through the accumulation of new mutations. Importantly, we model the ecological feedback of differentiation on gene flow by assuming that the success of migrations decreases with genetic distance, through a specific function . We investigate the effects of metapopulation size on the coherence of species structures, depending on some mathematical characteristics of the feedback function . Our results suggest that with larger metapopulation sizes, species form increasingly coherent, transitive, and uniform entities. We conclude that the initiation of speciation events in large species requires the existence of idiosyncratic geographic or selective restrictions on gene flow.

Paper Structure

This paper contains 27 sections, 12 theorems, 80 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Model description
  3. Evolutionary divergence feedback.
  4. Technical assumptions.
  5. The model.
  6. Mathematical analysis
  7. ODE approximation.
  8. Duality approach.
  9. A special case: two populations
  10. Intransitive species
  11. Friendship graph.
  12. Ring species.
  13. Subspecies clustering
  14. Large metapopulations
  15. Fluctuating migration
  16. ...and 12 more sections

Key Result

Theorem A.1

The process of stochastic genetic proximities $(P^L_{i j}(t))_{i,j\in[N]}; t\geq 0)$ converges in distribution as $L \rightarrow \infty$ to $(P_{i j}(t))_{i,j\in[N]}; t\geq 0)$, solution to the system of ordinary differential equations given by

Figures (14)

  • Figure 1: Toy realisation of the model and a migration event. Here, $N = 2, L = 3$, and the migration event occurs from population 1 to 2, affecting locus 3. The genetic proximity between 1 and 2 changes from $P_{12}=1/3$ to $P_{12} = 2/3$.
  • Figure 2: Convergence of the stochastic genetic proximities to the solution of the ODE for 3 populations as the number $L$ of loci gets large. The strong, solid lines are numerically simulated solutions to the ODE (\ref{['eq:ODE']}). The transparent lines are simulations of the stochastic model for different numbers of loci, namely $L = 50, 500, 1000$ from left to right. Additionally, we varied mutation rates, namely $\mu = 0.1, 0.08, 0.05$ from left to right, while keeping the migration matrix constant: $M=((0, 0.1, 0.8), (0.1, 0, 0.5), (0.8, 0.5, 0))$.
  • Figure 3: Realisation of the genetic partitions induced by the single-locus Moran model, and its dual for $N = 5$. On the left, colours represent genetic types, whereas on the right, colours represent ancestral lineages.
  • Figure 4: Migration and interbreeding graphs corresponding to intransitive equilibria. Intransitive interbreeding graphs ((a2): the Friendship graph) can emerge in complete and symmetric migration (a1). A ring of populations connected through migration (b1) can give rise to the ring species interbreeding graph (b2): the terminal forms of the ring species complex are reproductively isolated, despite ongoing gene flow through the chain of intermediary populations.
  • Figure 5: Existence of stable ring species equilibria depending on the mutation / migration ratio and the threshold value. We performed a systematic root search (as described in Fig. \ref{['fig:asym_N']}), but for ring species equilibria. Here, we set $h(x) = \mathbf{1}_{\{ x \geq c\}} \frac{x - c}{1 - c}, N = 6.$
  • ...and 9 more figures

Theorems & Definitions (17)

  • Theorem A.1
  • Theorem A.2
  • Lemma A.3
  • Lemma A.4
  • Lemma A.5
  • Proposition A.6
  • Corollary A.7
  • Definition B.1: Dual effective migration graph
  • Theorem B.2: Fixed point problem
  • Remark B.3
  • ...and 7 more