# Diversity patterns and speciation processes in
a two-island system with continuous migration

Débora Princepe  Instituto de Física ‘Gleb Wataghin’, Universidade Estadual de
Campinas, Campinas, Brazil  Simone Czarnobai  Programa de Pós Graduação em
Ecologia e Conservação, Universidade Federal do Paraná, Curitiba, Brazil
Thiago M. Pradella  Instituto de Física ‘Gleb Wataghin’, Universidade Estadual
de Campinas, Campinas, Brazil  Rodrigo A. Caetano  Departamento de Física,
Universidade Federal do Paraná, Curitiba, Brazil  Flavia M. D. Marquitti
Instituto de Física ‘Gleb Wataghin’, Universidade Estadual de Campinas,
Campinas, Brazil  Instituto de Biologia, Universidade Estadual de Campinas,
Campinas, Brazil  Marcus A. M. de Aguiar  Instituto de Física ‘Gleb Wataghin’,
Universidade Estadual de Campinas, Campinas, Brazil  Sabrina B. L. Araujo
Departamento de Física, Universidade Federal do Paraná, Curitiba, Brazil

Geographic isolation is a central mechanism of speciation, but perfect
isolation of populations is rare. Although speciation can be hindered if gene
flow is large, intermediate levels of migration can enhance speciation by
introducing genetic novelty in the semi-isolated populations or founding small
communities of migrants. Here we consider a two-island neutral model of
speciation with continuous migration and study diversity patterns as a
function of the migration probability, population size, and number of genes
involved in reproductive isolation (dubbed as genome size). For small genomes,
low levels of migration induce speciation on the islands that otherwise would
not occur. Diversity, however, drops sharply to a single species inhabiting
both islands as the migration probability increases. For large genomes,
sympatric speciation occurs even when the islands are strictly isolated. Then
species richness per island increases with the probability of migration, but
the total number of species decreases as they become cosmopolitan. For each
genome size, there is an optimal migration intensity for each population size
that maximizes the number of species. We discuss the observed modes of
speciation induced by migration and how they increase species richness in the
insular system while promoting asymmetry between the islands and hindering
endemism.

Keywords: Continuous migration, speciation with gene flow, neutral model,
island biogeography.

##  1 Introduction

Geography plays a central role in speciation. The isolation of populations
imposed by geographic barriers, allopatry, is indeed the most straightforward
process of diversification [1, 2]: when isolation is complete, gene flow is
interrupted, and mutations accumulated in individuals of one group are not
shared with individuals of the other group, increasing the genetic
discrepancies between the populations and eventually leading to reproductive
isolation. Speciation in the presence of gene flow, however, is also very
frequent, occurring when isolation is partial or even when geographic barriers
are completely absent [3, 4, 5]. For instance, speciation with restricted gene
flow in spatially structured populations, called parapatric, has been studied
in different contexts [6, 7, 8, 9, 10]. Likewise, populations inhabiting a
single geographic area, with no restriction to gene flow, can also split into
different species, a process termed sympatric speciation. The possibility of
speciation in sympatry has been theoretically demonstrated for populations
under disruptive selection [11, 12, 13], strong competition [14], mating
preference [15] and even in neutral scenarios [16], but its occurrence in
nature is rare and still controversial [2, 17].

Migration between groups is a common behavior that prevents complete isolation
of populations [18, 19, 20, 21]. For instance, estuarine-river environments
and tide pools are systems under pulse-driven biotic events [22]. These
environments are cyclically isolated and expanded as a consequence of the
water levels [23]. Thus, the system behaves like islands subject to periodic
exchanges of migrants from neighboring sites. Similarly, species that have
seasonal migration behavior are constantly mixing in a common area nesting or
breeding place and irradiating after to other sites [24, 25, 26, 27]. In this
case, the migration itself (associated with dispersal abilities and spatial
structure of the landscape) plays the role of an intermittent geographic
barrier [4, 28, 29, 30, 31, 32]. Such intermediate geographic situations can
affect the process of speciation in complex ways since diversification in the
presence of migration depends on a balance between colonization, local
selection, and the homogenizing effects of gene flow [33]. On the one hand,
high levels of migration can constrain diversification via maintenance of gene
flow [34, 35] and hinder speciation [2, 28]. It has been suggested that even
one or a few migrants should be sufficient to avoid genetic diversification
altogether [36]. On the other hand, migration between partially isolated
populations can foster speciation by increasing genetic variation [5, 37] and
by promoting founder events [38], which is the establishment of a small number
of migrants in a new area, favoring rapid genetic changes and eventual
diversification into a new species [39, 40, 41, 42]. Understanding how
migration increases diversity in semi-isolated populations has become
increasingly important in a world of fragmented environments [43, 44].

Investigation of the role of migration on speciation has mainly focused on the
time to speciation and the influence of population structure on the rate of
species creation, with evolution essentially driven by mutation and genetic
drift [45, 46]. Notably, migration increases the time for speciation [7], but
species divergence can be favored by subdivision of the population, even in
the absence of local adaptation [6, 8]. When considering isolated populations
under rare but recurrent migration, optimal intermediate migration intensities
maximize the rate of species creation in two-islands systems [46, 47, 48].
Still, some patterns are yet to be addressed and discussed in more detail,
such as the distribution of species richness, endemism, and species
persistence. In this aspect, models can help elucidate and differentiate
processes from patterns. Furthermore, current advances in obtaining genome-
wide datasets provide robust estimates of gene flow and demographic history
[49, 50], evincing that speciation with gene flow in insular systems is common
and may rise even in the absence of ecological divergence [50]. Therefore,
understanding how migration and diversification are related can provide a
better interpretation of speciation events in these intermediary geographic
configurations that are not fully sympatric or allopatric.

In the present study, we focus on how migration intensity affects speciation
and species richness in insular communities. We consider a neutral model for
two islands subject to frequent exchange of individuals, where mating occurs
locally and is only restricted by genetic similarity. The model is based on
the work of Manzo & Peliti [45], who considered a similar dynamic process of
local reproduction and migration between two islands in the limit of
infinitely large genomes, and investigated the possibility of allopatric
speciation in the presence of gene flow. Here we show that population size and
genetic features, particularly the number of loci involved in reproductive
isolation, are key to determine not only the total number of species in the
system but also the fraction of species that are endemic or cosmopolitan. Our
simulations show that migration promotes the appearance of new species through
two distinct processes: founding populations and induced sympatric speciation.
In the first case, the small but continuous flow of migrants creates a sub-
population that differentiates from the residents. In the second case, the
migrants mix with the residents, creating a genetically diverse population
that eventually splits into species.

##  2 Methods

Our model for insular populations is based on the theory proposed by Manzo &
Peliti [45], which is itself based on the sympatric speciation model by Higgs
& Derrida [16]. In these models, each individual is characterized by a string
of infinite biallelic genes that represents the loci involved in reproductive
isolation, dubbed as the individual’s genome. Using infinite genes allows the
derivation of several analytic results by mean-field calculations. However,
important features of the infinite genes model disappear if the genome size is
small, including the possibility of sympatric speciation [51]. Here we review
the model of speciation in islands as proposed by Manzo & Peliti and extend it
to finite genomes [9, 52]. We shall see that genome length plays a key role in
the diversification process and that the limit of infinite genes is a very
particular case.

###  2.1 Model overview

We consider two identical spatial regions (hereafter called islands) occupied
by populations that can migrate and evolve through mutations and
recombination. Each individual $\alpha$ is represented by a binary string of
independent $B$ genes,
$\\{S_{1}^{\alpha},S_{2}^{\alpha},\dots,S_{B}^{\alpha}\\}$, where each locus
$S_{i}^{\alpha}$ can assume the alleles $\pm 1$. Individuals are
hermaphroditic and reproduction is sexual. Population size is kept constant at
the carrying capacity $M$ in each island, except during the migration period
when small fluctuations are allowed. The dynamics starts with $2M$ clonal
individuals ($M$ on each island) and goes through cyclical events of
migration, reproduction, and species identification (see Fig. 1), as detailed
below:

Figure 1: Schematic of the model dynamics. We consider two insular populations
of initially identical individuals, represented by dots. At each time step,
the islands exchange individuals with a probability $\epsilon$. Generations
are non-overlapping and sexual reproduction occurs between individuals in the
same island having a minimal genetic similarity. Species are identified at the
end of each cycle based on the possibility of gene flow, illustrated by links
and enclosed in ellipses. The ellipses A, B, C and D (full lines) correspond
to species classification within each island. Species can be exclusive to an
island (B and D), or can be common to both (A and C, dashed lines).

Migration: at each generation, each individual has a probability $\epsilon$ of
migrating to the other island. Thus the population size in each island can
vary right after migration, but the total remains $2M$ individuals. We refer
to this process as continuous migration, as opposed to rare migrations of
larger groups [46].

Reproduction: reproduction occurs after migration only between individuals on
the same island. On each island, $M$ offspring are born and replace the
previous population, compensating for fluctuations in populations size that
may have occurred after migration. Generations do not overlap, and offspring
are generated as follows: a first parent $P_{1}$ (focal parent) is chosen at
random. A second parent $P_{2}$ is selected, also at random, among the
remaining individuals. Reproduction between these two individuals will only
occur if they are genetically compatible, i.e., if they have a minimal genetic
similarity, as defined below. If the selected $P_{2}$ is not compatible with
$P_{1}$, another second parent is randomly selected until this condition is
met. If after $M$ trials no such individual is found, $P_{1}$ is discarded and
a new first parent is selected. The offspring inherits, gene by gene, the
allele of either parent with equal probability. The offspring’s genome is also
subjected to a mutation rate $\mu$ per loci, which can be reversed during the
dynamics.

Reproduction is restricted by the genetic compatibility of the mating pair.
Two individuals $\alpha$ and $\beta$ can reproduce only if their genetic
similarity, defined by

$q^{\alpha\beta}=\frac{1}{B}\sum_{i=1}^{B}S_{i}^{\alpha}S_{i}^{\beta},$ (1)

satisfies the condition $q^{\alpha\beta}\geq q_{min}$, where the minimum
similarity $q_{min}$ is a model parameter. If the genomes are identical then
$q^{\alpha\beta}=1$, while two genomes with random entries have
$q^{\alpha\beta}$ close to zero. The model is neutral in the sense that there
is no fitness assigned to the individuals, and the choice of mating partner is
random, limited only by a minimum genetic similarity, i.e., a weak form of
assortative mating.

Species identification: speciation occurs when the gene flow between groups of
individuals is disrupted. The population maps to a network where the
individuals (nodes) are connected if they are genetically compatible. Species
correspond to the components of the network. Not all individuals of a species
need to be compatible since the genetic flow can be established through
intermediary individuals. In network terms, there is a path connecting them
within the component, as illustrated in Figure 1. We identify species
considering local and global classifications. In the first case, only
individuals from the same island are considered (components circled by
continuous lines in Fig. 1), while in the second, the genetic compatibility
network is built regardless of which island individuals belong (components
circled by dashed lines in Fig. 1).

###  2.2 Analytical results

In the limit where genome size becomes infinite, the entire dynamics can be
obtained by simply updating the similarity matrix. A number of simplifications
occur and analytical results can be obtained [16], as we summarize below.

####  2.2.1 One island model

If there is no restriction on mating, $q_{min}=0$, the overlaps
$q^{\alpha\beta}$ are all very similar. The distribution of overlaps is a
peaked function whose average value satisfies the evolution equation [16] (see
section LABEL:app1 of the Supporting Material for a derivation)

$q_{t+1}=e^{-4\mu}\left[\left(1-\frac{1}{M}\right)q_{t}+\frac{1}{M}\right]\equiv
e^{-4\mu}Q(q_{t},M)$ (2)

where $t$ is measured in generations. Setting $q_{t+1}=q_{t}=\bar{q}$, we find
the equilibrium $\bar{q}\equiv q_{0}\approx(1+4\mu M)^{-1}$. If
$q_{min}>q_{0}$, the distribution of similarities cannot reach the equilibrium
and the population splits into species, formed by groups of individuals whose
average similarity is larger than $q_{min}$. Inter-species similarities are
smaller than $q_{min}$, tending to zero with time [16]. Average species
abundance $m$ can be estimated by setting $q_{0}\rightarrow q_{min}$ and
$M\rightarrow m$, as this gives the population size that would naturally
equilibrate at $q_{min}$. We obtain $m=({q^{-1}_{min}}-1)/(4\mu)$ and the
average number of species is $N_{(one)}=M/m=4M\mu/(q^{-1}_{min}-1)$. We find
numerically that the distribution of abundances has an average slightly
smaller than $m$, which implies average number of species slightly larger than
$N_{(one)}$.

We note that this dynamic behavior can change drastically for finite genomes.
If the number of loci $B$ is small enough, the distribution of similarities
becomes wider and, instead of splitting into separate peaks as it crosses
$q_{min}$, it equilibrates as a single peak around $q_{min}$, preventing
speciation. Only for $B$ above a critical value, which depends on population
size, mutation rate and $q_{min}$, the above description (and therefore
sympatric speciation) is recovered [51].

####  2.2.2 Two-islands model

In the case of two islands, we need to distinguish the similarities between
individuals inhabiting the same island, $q$, or different islands, $p$. If
individuals $\alpha$ and $\beta$ were born on the same island, then their
average $q$ similarity in the next generation, before migration, is given by
Eq. 2. If, on the other hand, they were born in different islands, then their
$p$ similarity before migration is $p_{t+1}=e^{-4\mu}p_{t}$ as they cannot
have a common parent [45] (see Eq. 3 of Supplementary Material). The dynamics
of $q$ and $p$ after migration is, therefore, given by the following
equations:

$\displaystyle q_{t+1}$ $\displaystyle=$ $\displaystyle
a(\epsilon)e^{-4\mu}Q(q_{t},M)+b(\epsilon)e^{-4\mu}p_{t}$ $\displaystyle
p_{t+1}$ $\displaystyle=$ $\displaystyle
b(\epsilon)e^{-4\mu}Q(q_{t},M)+a(\epsilon)e^{-4\mu}p_{t}$ (3)

where $a(\epsilon)=(1-\epsilon)^{2}+\epsilon^{2}$ is the probability that both
$\alpha$ and $\beta$ did not migrate or that they both migrated, therefore
keeping their original geographic relation, and
$b(\epsilon)=2\epsilon(1-\epsilon)$ is the probability that one of them two
exchanged places, altering the geographic relative position of the pair. The
equilibrium solutions are obtained by setting $q_{t+1}=q_{t}$ and
$p_{t+t}=p_{t}$. For $\epsilon$, $\mu$ and $1/M$ all much smaller than 1, we
obtain [45]

$\displaystyle q_{0}(\nu,\sigma)$ $\displaystyle=$
$\displaystyle\frac{\nu+2\sigma}{2\sigma(2\nu+1)+\nu(\nu+1)}$ $\displaystyle
p_{0}(\nu,\sigma)$ $\displaystyle=$
$\displaystyle\frac{2\sigma}{2\sigma(2\nu+1)+\nu(\nu+1)}$ (4)

where $\nu=4\mu M$ and $\sigma=M\epsilon$ is the average number of exchanged
migrants at each generation. Notice that for $\sigma\gg\nu$, we have
$p_{0}(\nu,\sigma)=q_{0}(2\nu,0)$, indicating that for a large migration
intensity the two islands behave as a single island with twice the population.

Two new important results are derived here: first, the expected number of
species in each island, $N$, can be estimated by replacing $q_{0}(\nu,\mu)$ by
$q_{min}$ in Eq. 4, as similarly done before, and solving for $M\rightarrow
m$. The number of species is $N=M/m$:

$N=\frac{\nu(4\sigma+\nu)}{(2\sigma+\nu)(q_{min}^{-1}-1)}.$ (5)

Second, the total number of species in the islands, $N_{T}$, can also be
estimated as

$N_{T}=N\left(2-\frac{p_{0}(\nu,\sigma)}{q_{0}(2\nu,0)}\right).$ (6)

When $\sigma=0$ the islands are independent and $N_{T}=2N$. When migration
dominates over mutations, $\sigma\gg\mu$, $N_{T}=N$.

###  2.3 Numerical simulations and data analysis

We performed simulations with both finite and infinite genomes to evaluate how
migration probability ($\epsilon$), population size per island ($M$) and
amount of loci involved in reproductive isolation ($B$) affects patterns of
diversity in a two island system. The mutation rate per locus and the genetic
threshold for reproduction were fixed for all simulations, $\mu=0.001$ and
$q_{min}=0.9$, respectively. The parameters of interest were varied with the
following values: $B=1000,2000,3000,10000,$ and the infinite model; $M$ from
25 to 500 individuals; and $\epsilon$ from 0 to 0.5. Populations were evolved
during 2000 generations, with observation of species richness at $T=500$,
$1000$, and $2000$ generations. We ran 50 simulations for each set of
parameters, unless stated otherwise (see Supplementary Material, Fig.
LABEL:fig:Samples).

When probing the species richness, we considered both local and global
classifications (see Species identification in 2.1). In the local
classification, the species richness per island were designated by $N_{L1}$
and $N_{L2}$. Under the global classification, the species richness in each
island was $N_{1}$ and $N_{2}$, and the total number of species was $N_{T}$.
Local and global classifications differed when individuals in one island, say
island 1, could establish gene flow with two species of island 2 that would
otherwise be reproductively isolated, a configuration that resembles the ring
species assemblage [53]. In this case these two species that contributed to
$N_{L2}$ would count as one for $N_{2}$. We quantified these events, named
here “ring-like species”, by calculating
$N_{ring}=(N_{L1}-N_{1})+(N_{L2}-N_{2})$ (the local classification was
exclusively employed for this analysis).

We also evaluated the asymmetry $\Delta N$ between the islands as the
proportion of unbalance in species richness (under the global classification)
in respect to the average,

$\Delta N=\frac{\sqrt{(N_{1}-N_{2})^{2}}}{\bar{N}},$ (7)

where $\bar{N}=(N_{1}+N_{2})/2$. The diversity between the islands was
measured based on the Jaccard distance index, whereas we calculate the number
of exclusive species given by $(N_{T}-N_{1})+(N_{T}-N_{2})$. We call here the
beta-diversity index $\beta_{I}$ the normalization of this quantity, defined
as

$\beta_{I}=\frac{2N_{T}-N_{1}-N_{2}}{N_{T}}.$ (8)

If species are exclusive to each island, $N_{T}=N_{1}+N_{2}$ and
$\beta_{I}=1$, indicating maximum endemism. In the other hand, if species are
all common to both islands, we have $N_{T}=N_{1}=N_{2}$ and $\beta_{I}=0$,
i.e., all species are cosmopolitan.

##  3 Results

Figure 2: Diversity patterns in the islands as function of population size
($M$) and migration probability ($\epsilon$) for genome lengths $B=1000$,
$2000$ and $3000$ at $T=2000$: average species richness per island,
$\bar{N}=(N_{1}+N_{2})/2$ (first row); the proportion of species that is
exclusive of an island, measured by the beta-diversity index $\beta_{I}$
(second row); and the number of ring-like species, $N_{ring}$ (third row).
Migration increases the species richness in each island but reduces the beta-
diversity, followed by the formation of ring-like species. There are optimal
combinations of population size and migration probability that favor
speciation for each genome length $B$. The patterns are similar through the
values of $B$, with larger $B$ sustaining a higher number of species.

We first explored how migration probability, genome size, and population size
affected diversity at the end of the simulations ($T=2000$) for genome lengths
$B=1000$, $2000$, and $3000$ with varying migration probability,
$0\leq\epsilon\leq 0.08$. When the islands were completely isolated from each
other, $\epsilon=0$, speciation did not occur within the islands for the
smallest genome size, $B=1000$ (Fig. 2, upper left). In this case, the
populations differentiated only through the accumulation of mutations in each
island (allopatry), and each island had a unique exclusive species (indicated
by $\beta_{I}=1$ in Fig. 2, bottom left). This inter-island divergence was
limited by the population size ($M\lesssim 350$ individuals), as larger $M$
required longer times for speciation (see Fig. LABEL:fig:B1000). For $B=2000$,
sympatric speciation in isolated islands was rare, occurring scarcely only for
small values of $M$ (Fig. 2, upper center). On the other hand, for $B=3000$,
isolated islands presented sympatric speciation for all $M\gtrsim 50$ (Fig. 2,
upper right). Increasing $B$ also led to an increase in the maximum population
size that supported $\beta_{I}=1$, the maximum endemism (Fig. 2, middle).
Strict allopatry ($\epsilon=0$) represented the peak of endemism for all
scenarios (see Fig. 2, middle).

Migration even at low levels was sufficient to increase the number of species
on each island. However, the beta-diversity decreased with migration, as
expected, with a slower decay the larger the genome size. Therefore, although
species in each island were more numerous, they were more likely to be shared.
Regardless of the genome size, a curve depending on population size and
migration probability maximized the per island and total species richness,
with peak values increasing with $B$ (Fig. 2, top; for the values of the total
species richness, $N_{T}$, see Fig. LABEL:fig:B1000, LABEL:fig:B2000 and
LABEL:fig:B3000). The increase of the average number of species per island,
$\bar{N}$, was followed by the formation of ring-like species (Fig. 2,
middle), that is, groups of individuals with disrupted gene flow in the same
island but with the possibility of gene flow through individuals from the
other island. The increase in $N_{ring}$ was simultaneous to the loss of
endemism, indicating the genetic homogenization due to more intense migration.
Those patterns were consistent over time, with a direct dependence on the time
for allopatry with $M$ (see Fig. LABEL:fig:B1000, LABEL:fig:B2000 and
LABEL:fig:B3000).

Figure 3: Species richness and asymmetry between the islands at $T=2000$ for
varying migration probability ($\epsilon$) and genome sizes $B$ (colors) with
fixed population size ($M=200$). Results are shown for 50 independent
simulations. The solid lines represent the average value of all realizations,
and the shadowed areas show a confidence interval of 90%. The full black lines
represent the expected values from Eqs. 5 and 6. (a) Average number of species
per island, $\bar{N}=(N_{1}+N_{2})/2$. (b) Total number of species, $N_{T}$.
(c) Species asymmetry $\Delta N$, calculated with Eq. 7. The higher asymmetry
occurs for values of $B$ for which migration is essential for speciation.

We also simulated larger genomes ($B=10000$ and infinite loci) to compare with
the predictions from the analytical solutions. The results were analyzed in
more detail by fixing the population size at $M=200$ and varying the migration
probability in the range $0\leq\epsilon\leq 0.5$. The theoretical prediction
for the species richness in each island $N$ (Eq. 5), based on a model of
infinite loci, was slightly lower than the average value from the simulations.
It occurs because the prediction considers a uniform distribution of species
abundances (constant $m$), and the simulated populations resulted, in fact, in
heterogeneous species abundances. For a better comparison, we adjusted the
result from Eq. 5 for isolated islands ($N=7.2$ for $\epsilon=\sigma=0$) to
the simulations by adding a correction ($\delta=1.8$), and shifted the whole
curve by the same constant (black line in Fig. 3a). This theoretical
expectation of the number of species in one island showed a monotonic increase
with $\epsilon$ and was also a reasonable estimate of the number of species
for $B=10000$. For large genomes ($B\geq 10000$), the population split into
several species even in strict isolation (sympatric speciation), and migration
increased species richness further. However, intense migration, ranging from
$\epsilon=0.1$ to $0.5$, led to the homogenization of the populations that
evolved effectively as contiguous $2M$ individuals. For small genomes,
$B=1000$ and $2000$, migration was essential to increment species richness, as
seen before. However, here we observe that this effect was limited to low
levels of migration: above a critical point, the number of species within the
island collapsed to a single one shared by the two islands ($\epsilon\geq
0.014$ for $B=1000$ and $\epsilon\geq 0.03$ for $B=2000$). $B=3000$ had an
intermediary behavior, presenting several species for $\epsilon=0$, but also
collapsed for high levels of migration ($\epsilon\geq 0.2$) with a continuous
transition. The increment in total species richness ($N_{T}$) beyond the
number of total species in strict allopatry ($\epsilon=0$) showed that
migration induced some form of speciation in the insular community, and the
effect was more intense for small genome lengths (Fig. 3b). The approximation
for $N_{T}$ from Eq. 6 was a reasonable estimate for $B\geq 10000$, predicting
the decay in $N_{T}$ with $\epsilon$ but in a steeper way. The qualitative
patterns of species richness with respect to the migration probability when
increasing $B$, keeping all other parameters fixed, are reproduced in a
similar way with decreasing $q_{min}$ (Fig. LABEL:fig:qmin a) and increasing
$\mu$ (Fig. LABEL:fig:qmin b) with all other parameters fixed – see Section
LABEL:SM-tradeoff of the Supplementary Material for a discussion of the trade-
off between the model parameters not explored here.

Although the migration probability was symmetric between islands, the number
of species in each one was asymmetric, especially for low and intermediary
migration intensities, $0\leq\epsilon\leq 0.03$ (Fig. 3c). This asymmetry
occurred most of the time but was ephemeral, as we observed an alternation of
the island with more species and fluctuation of the species richness on both
islands (videos included in the Supporting Material). Comparing to a null
model of random distribution of the species between the two islands with equal
probability (see the Supplementary Material, section LABEL:SM-asy), we
verified that some of the asymmetry results from inherent stochasticity of the
system, but the highest asymmetry, beyond the expected by chance, occurred for
values of $B$ and $\epsilon$ for which migration was the essential mechanism
for the speciation. We inferred that asymmetry occurred when speciation
induced by migration enhanced random imbalances between the number of species
on each island. For instance, when one island was composed of a single
species, migrants were likely to establish a new species on the arrival island
as they were genetically compatible. On the other hand, as the other island
then had more than one species, its migrants had a reduced chance of being
conspecific and were more likely to go extinct on the island of arrival for
lack of compatible mates, keeping the first island with a single species.

Figure 4:  Beta-diversity index of the insular community and ring-like species
at $T=2000$ for varying migration probability ($\epsilon$) and genome sizes
$B$ (colors), with fixed population size ($M=200$). (a) The beta-diversity
index $\beta_{I}$ decreases monotonically with migration from full allopatry
($\epsilon=0$ and $\beta=1$) to the mixed populations sharing most of the
species ($\epsilon\approx 0.5$ and $\beta\approx 0$). (b) As migration
increases, gene flow disrupted within one island can be re-established by
individuals from the other island, in a configuration similar to ring species.
(c) Example of ring-like species, for $B=2000$ and $\epsilon=0.02$ at $T=430$.
Locally there is one species at the left island and two species at the right
one, but, at the global classification, individuals of the first island
reconnect the two clusters from the second, and then the islands share a
single species.

Analysis of the beta-diversity index in the insular community shows that, as
expected, increasing the migration probability led to a decline in $\beta_{I}$
(Fig. 4a), from the complete endemism when islands were isolated ($\epsilon=0$
and $\beta_{I}=1$) to the mixed populations, having most of the species in
common ($\epsilon\approx 0.5$ and $\beta_{I}\approx 0$). Large genomes were
more resilient, with slow decay, while the small genomes presented a sharp
drop, following the collapse in the number of per island and total species to
a single one. Although the proportion of exclusive species in the system
always decreased, the absolute number of exclusive species increased with
migration for small genomes (Fig. LABEL:fig:exclusive). The increase in the
migration probability also caused the appearance of ring-like species (Fig.
4b). Figure 4c illustrates how it occurred during the simulations. In this
example, using the local referential, the island at the left had one species
and the other two species, $N_{L1}=1$ and $N_{L2}=2$. However, the two species
in the second island were, in fact, a single one in the global classification,
with gene flux between the two groups reconnected by individuals of island
one, then $N_{1}=N_{2}=1$. This assemblage is similar to ring species [53] on
a much smaller scale and longing only through a few generations, although
appearing throughout the dynamics and even at $T=2000$, as shown in Fig. 4b.
Such a configuration either preceded speciation (when the disrupted gene flow
later led to the formation of species) or disappeared with the subsequently
re-establishment of gene flow by the arrival of migrants or due to mutation.
Therefore we classified them as “ring-like species”. For small genomes, they
occurred in a small number, restricted to the range of migration intensity
that made $N_{i}>1$. For large genomes, they were recurrent and appeared in
significant numbers under intense migration.

Figure 5:  Evolution of the populations for $B=1000$ and $\epsilon=0.004$
(left) and $B=2000$ and $\epsilon=0.02$ (right) exemplifying the speciation
modes induced by migration. Bars on the bottom indicate the distribution of
species abundances in each island over time. Individuals of the same species
have the same color and are connected by links or dashed lines when they
belong to different islands. In the speciation through founder populations
(left panel), a portion of migrants differentiates into a new species without
mixing with the native population. In the second mode (right panel), migrants
can incorporate into the resident population, then differentiation between
islands is hindered, but migration promotes sympatric speciation.

Finally, we investigated the processes by which migration leads to speciation.
Using genetic compatibility networks, we observed two speciation modes induced
by migration, illustrated in Figure 5. Speciation by founding populations
occurred when migrants arriving on an island could not reproduce with the
resident population for being of a different species and, when they were able
to accumulate through several generations and establish a population, ended up
founding a new species (Fig. 5, left panel). We hypothesize that this was the
dominant mode for low migration probabilities based on the examples observed
with videos. The second mode, recognized as a sympatric speciation induced by
migration (shown in the right panel), seems to occur more frequently under
slightly larger migration probabilities. Here the islands presented common
species, then migrants could reproduce with the resident population, and their
incorporated genetic novelties promoted speciation. The described effects can
be observed in detail in the videos included with the Supporting Material.

##  4 Discussion

The present model was based on the work of Manzo & Peliti [45], who considered
only genomes with infinitely many loci. They showed that when mating was not
restricted by a minimum genetic similarity, i.e., $q_{min}=0$, the average
values of intra and inter-island similarities ($q$ and $p$) converged to the
constant values $q_{0}$ and $p_{0}$ given by Eq. 4. They proposed that setting
$p_{0}<q_{min}<q_{0}$ would drive the distribution of intra-island
similarities to $q_{0}$, avoiding sympatric speciation and at the same time
creating inter-island reproductive isolation. For the parameter range studied
in [45], this scenario was only achieved by setting $q_{min}\approx 0.2$. For
finite and sufficiently small genome sizes as explored here, on the other
hand, sympatric speciation can be avoided even if $q_{min}>q_{0}$, as the
distribution of intra-island similarities becomes stationary around $q_{min}$
[51]. The dynamics with finite genome sizes also brought up a richness of
processes and outcomes that do not occur in the extreme case of infinite
genomes. The time for allopatry, for example, does not depend on population
size for infinite loci (Eq. LABEL:eq:tallop SM), but strongly depends on $M$
for finite genomes (Fig. LABEL:fig:B1000, LABEL:fig:B2000 and
LABEL:fig:B3000). Furthermore, we extended the Manzo & Peliti model, obtaining
expressions for the number of species and estimating the percentage of
exclusive species on each island. Importantly, we studied the mechanisms of
speciation induced by migration for infinite and finite genomes, showing how
migration increases species richness while hindering endemism and promoting
asymmetry between the islands and transient ring species.

Strict allopatry with one species per island was only maintained for small
genomes ($B=1000$ and $2000$) in the absence of migration with $M=200$. Even a
few migrants were sufficient to promote speciation. For instance, with
$\epsilon=0.004$ (equivalent to an average number of $\sigma=0.8$ migrants per
generation), speciation by founding populations already occurred (see Fig. 5
left panel and videos in the Supplementary Material). This mode was mainly
related to low migration intensity and benefited from the slow dynamics of
those genomes sizes (Fig. LABEL:fig:Temporal). When the two islands still had
the same species, the migrants were genetically similar and, in a small
number, did not bring sufficient novelties to cause a break in gene flow.
After the islands differentiated from each other (around $T>400$ generations),
recently arriving migrants in each island could not reproduce with the
resident population and either accumulated through several generations or went
extinct. When a population was established in the arrival island, they
differentiated into a new species (Fig. 5, left panel). No sympatry was
observed in those cases, in the sense that the island’s resident species did
not branch. Therefore, the ancestry of the new species was always fully
connected to migrants, i.e., they did not share common ancestors with the
former (native) species.

With a slightly higher level of migration, the islands had more than one
species and shared some of them. Although there was a loss of endemism,
sympatry induced by migration was favored: migrants could reproduce with
residents, and their mixing led to speciation. Remarkably, despite the higher
proportion of shared species, the absolute number of exclusive species
increased with migration for small genomes (Fig. LABEL:fig:exclusive). This
connection between high species richness with low endemism or low species
richness and high endemism in insular systems has been observed in Darwin
finches, for instance [54], and explained by the spatial structure of the
islands [55]. Low levels of migration can be understood as a great physical
distance between the islands, while a moderate level indicates islands closer
to each other, establishing an analogy between our results and previous
findings. Also, an intermediary level of migration seemed to optimize the
evolution of diversity, which has been suggested in different contexts [46,
33], for instance, in the intermediate dispersal model [29, 32]. Based on the
theory of island biogeography and associating migration with dispersal
abilities, it proposes that colonization promotes speciation at the same time
that may increase the probability of extinction; therefore, moderate levels of
migration optimize species richness in islands [29, 32]. Our model reproduces
this effect while also providing a mechanistic view of the processes by which
migration promotes speciation.

Small genome sizes were positively correlated to the asymmetry in species
richness between the islands beyond the expected by chance (i.e. if the given
total number of species was randomly distributed between the islands with
equal probability). The effect was more significant for low levels of
migration (Fig. 3c, also LABEL:fig:sup_asy and LABEL:fig:Histograms), and we
hypothesized it was related to speciation by founding populations that, as
mentioned before, enhanced random imbalances of species richness. When
sympatric speciation took place, increasing the number of species, asymmetry
was less observed. Asymmetry in the geographic range of recently branched
sister species is expected under peripatric speciation and is more likely to
occur in small populations [56]. Here asymmetry in species richness resulted
from gene flow in a symmetric setup (same number of individuals and migration
probability in both islands) with a small number of species and not from
landscape heterogeneity or different species ranges. Further investigation
might address how such imbalances affect species abundances and the structure
of the phylogenies.

Diversity under high migration could only be sustained with large genomes,
which can be explained by the trade-off between population size and genome
length (see also Fig. LABEL:fig:fig2rev). In this model, when considering a
single island, sympatric speciation does not occur if the genome is too small
due to the low genetic variability and the slow accumulation of mutations
[51]. The required variability is readily provided by low and moderate
migration. However, under higher exchange of migrants, the populations evolve
as contiguous, and then the time to speciation increases for finite genomes
(see [51] and Fig. LABEL:fig:B1000, LABEL:fig:B2000 and LABEL:fig:B3000),
hindering speciation. Large genomes, however, can sustain higher variability
and present fast dynamics. Nevertheless, the robustness of large genomes made
their response to migration less compelling.

Along the whole dynamics, we observed the formation of ring-like species, but
they were more important for large genomes under high migration intensity,
indicating how gene flow was actively connecting the populations. Previous
models for islands with migration predicted the occurrence of ring species but
in larger chains of patches [6]; here we find a similar effect with only two
islands. However, they do not resemble the observed ring species [53], for
having a much smaller extension and low durability, but can be later used as
an indicative or step of the speciation process.

Finally, we note that the results presented here are conditioned to the regime
of continuous migration and relatively large mutation probability ($\mu
M\approx 0.1$). Rare migrations of larger groups can lead to different
outcomes if the time between migrations is larger than $M$ and $\mu\ll 1/M$,
allowing fixation of migrants alleles between migrations [48, 57]. Hence,
diversity in island systems depends on a large number of factors that include
periodicity of migration, number of individuals, mutation probability and
number of loci and alleles involved in reproductive isolation. Therefore,
comparing models with data requires careful analysis of the situation at hand.

### Code availability

All Fortran codes used in this study are available in the GitHub repository at
https://github.com/deborapr/two-islands.

### Conflict of interest

The authors declare that they have no conflict of interests.

### Acknowledgment

This work was partly supported by the São Paulo Research Foundation (FAPESP),
grants #2018/11187-8 (DP), #2019/24449-3 (DP), #2019/20271-5 (MAMA) and
#2016/01343-7 (ICTP-SAIFR). FMDM was supported by Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior – Brazil (Finance Code 001). MAMA
and SBLA were supported by Conselho Nacional de Pesquisas Científicas (CNPq),
grants #301082/2019-7 and #311284/2021-3 respectively. SBLA acknowledge the
computational support from Professor Carlos M. de Carvalho at LFTC-DFis-UFPR.

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