Download Persistence of the sexes in metapopulations under intense

Journal of
Ecology 2007
95, 937–950
Persistence of the sexes in metapopulations under intense
asymmetric competition
Blackwell Publishing Ltd
GISELA GARCÍA-RAMOS, CHRISTOPHER STIEHA,
D. NICHOLAS McLETCHIE and PHILIP H. CROWLEY
Department of Biology and Center for Ecology, Evolution & Behaviour, 101 Morgan Building, University of Kentucky,
Lexington, KY 40506, USA
Summary
1. When the two sexes compete so intensively that one of them may be consistently
excluded from patches of habitat, how can they coexist in the population as a whole?
2. To address this question, we constructed a population model capable of simulating
the dynamics of sex-specific life-history stages within frequently disturbed patches
and across a multipatch system strongly influenced by extinction and colonization
(metapopulation).
3. We parameterized the model based on the dioecious bryophyte Marchantia inflexa,
attempting to capture sufficient biological realism for the results to be quantitatively
comparable with natural patterns.
4. In nature, M. inflexa spreads by tissue extension and dispersal of asexual propagules
within patches or sexual spores between patches. Females have faster tissue expansion
and males greater production of asexual propagules. Some patches and even entire
populations lose males or, more rarely, females.
5. In the model, males were often eliminated by competition from individual patches,
but both sexes almost always persisted in the metapopulation (as in nature), with
females typically predominating. Male advantage during patch filling and spores
produced and dispersed where males (the fugitive sex) had not yet been eliminated
kept both sexes in the model system.
6. Similar mechanisms may maintain both sexes in other systems, particularly small,
highly disturbed populations where life-history traits and mortality differ between
the sexes.
Key-words: Asexual reproduction, clonality, loss of sex, Marchantia inflexa, overgrowth competition, sex-ratio dynamics, sexual dimorphism, sexual reproduction
Journal of Ecology (2007) 95, 937–950
doi: 10.1111/j.1365-2745.2007.01264.x
Introduction
© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society
The origin and maintenance of sex have long fascinated
ecologists and evolutionary biologists (Williams 1975;
Maynard Smith 1978; Bell 1982; Charnov 1982; Stearns
1988; Hurst & Peck 1996; Hardy 2002), yet relatively
little attention has been devoted in the literature to ecological mechanisms that may influence the coexistence
of the sexes in nature. In fact, for species having genetically fixed sex and the sexes in separate individuals
(dioecy), situations in which at least one sex is at risk of
local extinction are common, including those characterized by small population sizes, high disturbance
Correspondence: G. García-Ramos (e-mail [email protected]).
frequencies, or intense competition or predation that
may fall more heavily on one of the sexes. As a result,
some animal and plant species have extreme sex ratios
(Bowker et al. 2000; McGovern 2002) or are known
only from single-sex populations or species (Longton
& Schuster 1983; Philbrick & Les 1996); and some
fungi may have lost sexual reproduction through the
extinction of alternative mating types (Alexopoulus
et al. 1996). However, the complete loss of a sex from
species seems relatively rare (Longton & Schuster
1983), suggesting that other counteracting forces may
generally prevail.
One force affecting the proportion of sexes is the
co-occurrence of asexual reproduction. There are many
examples across kingdoms in which both sexes (or mating
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et al.
© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society,
Journal of Ecology
95, 937–950
types) are capable of asexual reproduction, including
protists (Raven & Johnson 2002), fungi (Alexopoulus
et al. 1996; Leslie & Klein 1996), animals (McGovern
2002) and plants (Mogie 1992; Newton & Mishler
1994). The many species that can reproduce both
sexually and asexually (Bell 1982) provide a ground for
understanding the adaptive significance of sex and
the mechanisms retaining the sexes in populations
(as Williams’s ‘balance’ argument emphasizes, the
stable maintenance of sexual and asexual reproduction
within a species implies that sex must have short-term
benefits; Hurst & Peck 1996). One relevant aspect in
species capable of both types of reproductions is the
ecological specialization of the propagules. In these
species, Williams (1975) pointed out that the asexually
produced offspring will develop near the parent, but
the sexually produced propagules will disperse more
widely. Also, Maynard Smith (1978) noted that sexually and asexually produced propagules in such species
are often adapted to different ecological situations
such that a mixed mode of reproduction could be maintained. These remarks imply the existence of ecological
opportunities for the recruitment of both types of
propagules. Therefore, another factor influencing the
proportion of sexes is a spatially structured population
with dynamics of extinction and colonization (metapopulation) that provides conditions for short and
large dispersers, and a range of population densities
for their recruitment. Furthermore, the sexes often
differ in many life-history traits, such as growth and
asexual propagule production, that may confer upon
them a differential performance with environmental
conditions. Sexual dimorphism in life-history traits has
been broadly documented (Grant & Mitton 1979;
Popp & Reinartz 1988; Geber et al. 1999; Eppley 2001)
and implicated to drive the proportion of sexes in
populations (Lloyd 1973; Allen & Antos 1993;
McGovern 2002).
In sexually produced offspring, an equal parent investment in males and females is predicted by frequencydependent selection (Fisher 1930). However, a biased
sex ratio can develop secondarily from this balanced
primary ratio as a result of differential life histories
of males and females that include clonal growth and
asexual reproduction in concert with various selection
pressures. This secondary sex ratio may vary extensively
over space (Longton & Schuster 1983; Willson 1983)
and time (McGovern 2002). The present study addresses
a focal species and follows the relative abundances of
males and females over time and space and the ecological
mechanisms that may facilitate retention of both sexes
within populations.
The liverwort Marchantia inflexa Nees et Mont is
usually capable of retaining both sexes in multipatch
assemblages, despite the frequent loss of one of either
sex in individual patches (McLetchie & Puterbaugh 2000).
This suggested to us that metapopulation structure and
dynamics involving extinction and recolonization in
the multipatch assemblage may play a crucial role in
maintaining the sexes and sexual reproduction (McLetchie
et al. 2002). M. inflexa is an ideal species for addressing
this issue because (i) each individual is only one of the
two sexes (sex is chromosomally determined); (ii) some
single-sex populations of this dioecious species are
known to exist (Schuster 1992), indicating that complete loss of a sex is at least possible; (iii) considerable
empirical information is available on sex-specific clonal
expansion by growth and asexual reproduction, both
from field and from glasshouse studies (McLetchie
& Puterbaugh 2000); and (iv) models of sex-specific
dynamics within individual patches have already been
developed and parameterized for this species (McLetchie
et al. 2002; Crowley et al. 2005a,b), predicting eventual
loss of one sex or the other from the patch. Therefore,
a multipatch study of M. inflexa based on these previous
investigations can determine whether metapopulation
characteristics may help retain both sexes, and thus
sexual reproduction in a patch assemblage, under conditions leading to loss of a sex in single patches. The
pattern of M. inflexa resembles that of many dioecious
bryophyte species. Single-sex populations and male
rarity commonly occur in liverworts and mosses (Bowker
et al. 2000) and hornworts (Renzaglia & McFarland
1999), although some dioecious bryophytes are singlesex species (Longton & Schuster 1983). We emphasize
that the basic issue is of fundamental importance,
yet details of the mechanism depend on biological and
ecological specifics; hence, we focus here on a species
for which these specifics can at least be estimated.
In the following section we summarize the life history and ecology of M. inflexa at a field site in Trinidad.
Next we describe our metapopulation model – an array
of patches, each linked to the others by spore dispersal,
subjected to extinction and following single-patch
dynamics as in McLetchie et al. (2002). We then present
our results, emphasizing the relationship between singlepatch and patch-assemblage dynamics and including
a thorough analysis of the key features. This study
demonstrates that metapopulation dynamics may sustain
coexistence of sexes under intense asymmetric competition where isolated patches result in the loss of males.
Finally, we address the implications of our findings for
the M. inflexa system and for the ecological maintenance
of the sexes in natural populations more generally.
MARCHANTIA INFLEXA
 
Marchantia inflexa is a thalloid dioecious liverwort that
ranges from northern Venezuela to the south-eastern
USA (Bischler 1984). At tropical latitudes the sexes
are found together within populations, but in the USA
some all-male and all-female populations have been
found (Schuster 1992; Fuselier & McLetchie 2004).
In sex-expressing populations, archegoniophores and
antheridiophores that produce eggs and sperm, respectively, are elevated above the horizontal plant body
(Fig. 1a). Sperm are splashed onto nearby archegoniophores by rainwater, fertilizing eggs and generating
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Fig. 1 (a) Stage dynamics of M. inflexa. There are seven stages defined by gender and reproductive conditions. Stages are
connected by transitions Til leaving stage l and entering stage i and by recruitment from asexual reproduction via gemmae Ai and
sexual reproduction via spores F1. Stage abundance may increase by growth Gi and decrease by overgrowth competition Oi. M7
represents a fertilization rate. Icons depict M. inflexa thallus units (0.5 cm wide) in different stages. At stages 3 and 6, the icons
show thalli with cups that produce gemmae (asexual propagules; centre picture). At stages 4 and 7, the icons represent sexexpressing thalli, with males showing antheridiophores and females archegoniophores (bottom picture). At stage 1, the icon
indicates a female thallus with an archegoniophore containing two sporophytes (black spots). Sporophytes produce spores
(sexual propagules). The picture at the top shows thalli in a non-reproductive stage. Modified from McLetchie et al. (2002).
(b) Phenology of the M. inflexa life cycle. Seasonal timing of non-zero stage transition rates are indicated in continuous bars.
Light bars represent male and dark bars female stage. Based on plant stages observed through a year at the Trinidad field station
(D.N. McLetchie, unpublished data).
© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society,
Journal of Ecology
95, 937–950
spores. In the Marchantiaceae group, chromosomes
determine sex (Bischler 1986), and the sex ratio at spore
formation is predicted to be 1 : 1.
Clonal expansion in M. inflexa occurs by extension
and bifurcation of thallus branches and by specialized
asexual reproductive propagules called gemmae,
dispersed by water from gemma cups on the thallus
surface (Fig. 1a). Each of these gemmae (~0.12 mm in
diameter) can produce a new individual genetically
identical to the parent plant.
Sex differences in clonal expansion rates via gemmae
and growth may favour one sex (usually females in
M. inflexa) sufficiently to threaten the other with local
elimination. New substrate is made available by disturbances (especially flooding) within patches and by wholepatch elimination. The asexual gemmae, dispersed by
water, establish primarily within their patch of origin;
the sexual spores are wind-dispersed and establish mainly
in patches that are completely or mostly unoccupied,
as suggested by the inhibition of spore germination
by fluids from established plants (K.R. Reniger et al.,
unpublished data).
At initial stages of patch colonization, Marchantia
males are expected to do as well or better than females
as a result of higher levels of asexual reproduction
(Voth & Hamner 1940; McLetchie & Puterbaugh 2000).
However, as the patch fills, the higher growth rates
of females (McLetchie & Puterbaugh 2000) become
increasingly important, reducing opportunities for
germination of gemmae.
There is a consistent seasonal pattern of life-history
stages in the field (as in Bischler 1984). At the field site
near Hollis Reservoir in north-central Trinidad, sex
expression begins in February and peaks in May, with
male sex structures usually emerging about 2 weeks
before females (D.N. McLetchie, unpublished data).
By November, most sex structures are gone. Gemma
cups are abundant in November but infrequent from
May until sex structures become rare again (Fig. 1b).
Growth occurs throughout the year.
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Table 1 Model formulation*
Stage
Stage dynamics
Eqn no.
Fertilized females
dN1 j
M7 N 4 j N7 j TRANSITION 1→5
N
=
−
− N1 j
T51N1 j
Gi ij
dt
K
K
i = 2,i≠ 4
OVERGROWN
6
FERTILIZATION
∑
eqn 1
GERMINATION −SPORES ASEX REPROD
n
dN 2 j 
=  F1
f jk N1k + A3
dt
 k=1
∑
n
∑a
jk
k =1

N3 k 


1 −

Nij 
7
∑ K  +
TRANSITIONS 3→2, 4→2
T23 N3 j + T24 N 4 j
i =1
OVERGROWN
TRANSITIONS 2→3, 2→4
Non-reproductive males
− (T32 + T42 ) N 2 j

N 
N
+ G2 N 2 j 1 − 2 j  − N 2 j
Gi ij
K 
K

i =3,5,6
GROWTH
∑
eqn 2
OVERGROWN
Asexually reproductive males
dN3 j TRANSITIONS 2→3, 3→2 GROWTH 
N 
N
= T32 N 2 j − T23 N3 j + G3 N3 j 1 − 3 j  − N3 j
Gi ij
dt
K 
K

i = 2,5,6
Sexually reproductive males
dN 4 j TRANSITIONS 2→4, 4→2
N
= T42 N 2 j − T24 N 4 j − N 4 j
Gi ij
dt
K
i = 2,i≠ 4
∑
eqn 3
OVERGROWN
6
∑
eqn 4
GERMINATION −SPORES ASEX REPROD
Non-reproductive females
n
dN5 j 
=  F1
f jk N1k + A6
dt
 k=1
∑
n
∑a
jk
k =1

N6 k  1 −

7
Nij 
∑ K  +
TRANSITIONS 1→5, 6→5
T51N1 j + T56 N6 j
i =1
eqn 5
OVERGROWN

N 
N
+ T57 N7 j − (T65 + T75 ) N5 j + G5N5 j 1 − 5 j  − N5 j
Gi ij
K 
K

i = 2,3,6
TRANSITIONS 7→5, 5→6, 5→7
GROWTH
∑
OVERGROWN
Asexually reproductive females
dN6 j TRANSITIONS 5→6, 6→5 GROWTH 
N 
N
= T65N5 j − T56 N6 j + G6 N6 j 1 − 6 j  − N6 j
Gi ij
dt
K 
K

i = 2,3,5
Sexually reproductive females
TRANSITIONS 5→7, 7→5
dN7 j
MN N
N
= − 7 4 j 7 j + T75N5 j − T57 N7 j − N7 j
Gi ij
dt
K
K
i = 2,i≠ 4
∑
FERTILIZATION
eqn 6
OVERGROWN
6
∑
eqn 7
*List of equations depicting the abundance dynamics for the seven life-history stages of M. inflexa in patch j, with stage numbers
and dynamics corresponding to Fig. 1(a). Nij is the abundance of stage i, Gi is growth rate, Ai is asexual reproduction rate, F1 is
sexual reproduction rate, M7 is fertilization rate, Til is transition rate from stage l to stage i, and K is carrying capacity. Equations
2 and 5 include germination of both local and dispersed spores and gemmae. In these equations, fjk and ajk represent, respectively,
the fractions of spores and gemmae produced by patch k that reach patch j. Spores may encounter restrictions for establishing
(Table 2). Seasonally, stage transitions are turned on and off according to the phenology in Fig. 1(b). Whole-patch extinction
occurs with probability pe, and within-patch disturbance removes a fraction of units occupying the patch with probability pd
(Table 2). Modified from McLetchie et al. (2002).
Model
- 
© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society,
Journal of Ecology
95, 937–950
Following the approach of McLetchie et al. (2002),
we depict the life cycle of M. inflexa within a patch as a
set of seven interacting stages – defined according to
gender and reproductive condition – that are directly
observable in the field and glasshouse (Fig. 1a).
Mathematically, this representation is implemented as
seven coupled ordinary differential equations similar
to the Volterra competition equations, with transitions
illustrated in Fig. 1(a) taken to be linearly determined
by the donor-stage (Table 1). The exceptions are (i) that
growth is logistic for each stage, with all space occupied
by other stages or unoccupied and considered available
to that stage; (ii) that the rate of sexual reproduction
reflects the multiplicative product of the abundance
of males and unfertilized females within the patch; and
(iii) that propagules are distributed within and among
patches using dispersal equations described below.
This formulation amounts to assuming that each stage
is subdivided into tiny segments randomly distributed
across the patch, with reproduction at random within
the patch. See McLetchie et al. (2002) for within-patch
details. Other work supports ignoring within-patch
spatial distributions of stages at least as a first approximation (Crowley et al. 2005a,b).
Fertilized archegoniophores (stage 1) produce spores
dispersed by wind both inside and outside of the patch
of origin. We assume that spores disperse within and
beyond the source patch according to an exponential
decay distribution (Nathan & Muller-Landau 2000;
Miller & McDaniel 2004). Germinating spores – sporelings – give rise to non-reproductive males (stage 2) or
females (stage 5) with equal frequency. Germination
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Persistence of sexes
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Table 2 Parameters of the model and default magnitudes. Stages 2, 3 and 4 represent non-reproductive, asexually and sexually
reproductive males, respectively. Stages 1, 5, 6 and 7 correspond to fertilized, non-reproductive, asexually and sexually
reproductive females, respectively. See derivation of values in Appendix S1
Symbol
Definition
Life-history
G 2, G 5
Growth rates in stages 2 and 5
G 3, G 6
Growth rates in stages 3 and 6
T42, T75
Stage transition rates from 2 to 4 and 5 to 7
T23, T24, T56, T57
Stage transition rates 3 – 2, 4 – 2, 6 – 5, 7 – 5
T32, T65
Stage transition rates from 2 to 3 and 5 to 6
T51
Stage transition rate from 1 to 5
M7
Fertilization rate
A 3, A 6
Asexual reproduction rates by stages 3 and 6
F1
Sexual reproduction rate per sex by stage 1
Dispersal and establishment
λ 1, λ 2
Spore dispersal parameters
p
First proportion for spore dispersal functions
z
Gemma dispersal parameter
q
Uncovered threshold for spore germination
r
Maximum spore establishment rate
Patch system
4h2
Patch size
u
Size of a unit of M. inflexa tissue
K
Patch carrying capacity, K = 4h2/u
n
Number of patches
x*
Distance between patches
Disturbance and extinction
pd
Probability of a patch disturbance in a month
pe
Probability of a patch extinction during a month
Magnitude
Units
0.585, 0.605†
0.220, 0.233†
0.714†
2.14†
1.11, 0.833†
0.638†
10.0‡
1.112, 0.519†
442†
month–1
month–1
month–1
month–1
month–1
month–1
month–1
month–1
month–1
0.7, 10‡
0.27‡
0.83‡
0.4‡
1000‡
m
–
m
–
unit month–1
1‡
5 × 10–5†
20 000‡
30‡
10‡
m2
m2 unit–1
units
–
m
0.2‡
0.002‡
–
–
†Based on field and glasshouse data.
‡Indirect estimation or best expected value.
© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society,
Journal of Ecology
95, 937–950
requires that the spores land on an unoccupied part of
the patch and are uninhibited by established neighbours,
represented in the model by the requirement that a
threshold proportion of the patch be empty for successful
germination (i.e. ≥ 0.4 empty under standard or ‘default’
conditions). An upper limit on the rate of sporeling
establishment (i.e. 1000 unit per month per patch in the
default) is also imposed to account for spatial limitations
(Nathan & Muller-Landau 2000). These restrictions
for successful germination were relaxed in the analysis,
and the impact of these and other parameter values is
addressed in the sensitivity analysis. Mature spores
germinate immediately, eliminating the possibility of a
spore bank (D. N. McLetchie, unpublished data).
Transitions between non-reproductive asexual, and
sexual stages are season-specific (Fig. 1b). For example,
stages 2 and 5 can develop into stages 3 and 6, respectively, during the season for asexual reproduction. The
asexual propagules thereby released within the patch
by these stages may germinate in unoccupied parts
of the patch, as may propagules arriving from other
patches. Dispersal distance for asexual propagules
follows an exponential decay distribution as for spores.
Asexual dispersal between patches is assumed to be low
but non-zero. Stage 7 females can be fertilized at a
rate proportional to the abundance of stage 4 males,
thereby shifting females from stage 7 to stage 1 at fertilization. This assumes sperm limitation is due to short
dispersal distances of sperm that fertilizes a limited
number of females (McLetchie 1996; Crum 2001). With
the release of spores, females shift from stage 1 to stage
5. Stages 2, 3, 5 and 6 can grow to occupy more of the
patch surface and may thereby overgrow or be overgrown by other stages. The sexual stages 1, 4 and 7 are
assumed to have negligible energy to expend on growth
or asexual reproduction.
Because of the complex geometry of the plant tissue
within an M. inflexa patch, the model tracks the population as a (continuous) number of small ‘units’ of
tissue representing thalli growing on top of the mat [cf.
the more overtly geometrical approaches of Rydgren &
Økland (2002) and Crowley et al. (2005b)]. The extent
of patch coverage or occupancy by these units and the
corresponding stages are simulated dynamically. The
within-patch dynamics of the model are mathematically similar to the patch model of McLetchie et al.
(2002).
  
The metapopulation dynamic model of M. inflexa
extends the analysis of McLetchie et al. (2002) to multiple patches connected by dispersal and experiencing
extinction. The primary variables of interest in the
model are the numbers of units of each stage i in patch
j at time t, Nij (t) (Table 1). We assume that the stages
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© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society,
Journal of Ecology
95, 937–950
expand, reproduce and suffer disturbance-related mortality within patches at rates consistent with glasshouse
and field data, as in McLetchie et al. (2002). In the
present study, the number of units established from
each type of propagule within a given patch j is the sum
over the patch index k of the multiplicative products of
four terms: the number Nik of units of stage i producing
propagules in patch k, the relevant reproductive rate
parameter (F1, A3 or A6 of Table 1), the fraction of the
propagules dispersed from patch k that land in patch
j ( fjk for spores and ajk for gemmae, defined below), and
the fraction of the patch that is unoccupied and thus
capable of permitting germination and establishment
(Table 1, term for germination in eqns 2 and 5). For
spores, establishment is limited to the maximum r and
is prevented all together unless at least some minimal
fraction of the patch area is unoccupied. With successful
spore establishment, equal numbers of stage 2 male units
and stage 5 female units are assumed to be produced.
The metapopulation consisted of n patches equal in
size (4h2 = 1 m2, h is half the length of a patch side) and
spacing (x* = 10 m) along a straight, idealized river
course. By assuming that dispersal followed a negative
exponential decay function, the fraction of spores
reaching a patch j from patch k is fjk(x) ≈ [2h2/(πλ1 |x|)]
exp(–|x |/λ1), where x is the distance from the source
patch and λ1 is a spore dispersal parameter. This supposes that spores disperse in two dimensions without
directional bias over the one-dimensional array of
patches. For gemmae, the fraction landing in patch
j from patch k is ajk(x) ≈ (h/z) exp(–|x | /z), where z is a
gemma dispersal parameter. This assumes dispersal
exclusively along the patch array with no bias to left or
right from the source patch. The fractions that land
in the source patch are f(0) = 1 – (1 + 1.12h/λ1)
exp(–1.12h/λ1), and a(0) = 1 – exp(–h/z). For spores, the
model also considers a weighted sum of two exponential functions to depict leptokurtic distributions with
‘fat tails’ often observed in dispersal data (Higgins &
Cain 2002; Levin et al. 2003). All propagules dispersed
to locations outside of all patches were ignored.
We considered two types of spatially independent,
random patch disturbances: intrapatch disturbances
affecting all stages equally (pd default averaging once
every 5 months and causing 20% removal), and wholepatch extinctions (pe default averaging once every
40 years). At the beginning of each time step in the
simulation, extinction and disturbances were implemented independently to each patch. The probability
of disturbance pd (or extinction pe) during a month was
recalculated for time step ∆t by using the function
p d′ = 1 – (1 – p d ) ∆t as indicated in MATLAB®. A
random-generated number was compared with pd′ for
applying disturbance to the patch. Data on patch
extinction and colonization are preliminary. Therefore,
the model was evaluated for reasonable parameter values
(see supplementary Appendix S1) and by using a sensitivity analysis that varied these values for determining
their effects on sex ratio.

Local and metapopulation sex ratios were the focal
response variables. We first explored the model’s
behaviour using the default parameter set with the
linear array of patches (Table 1). We then considered
effects of spore recruitment constraints, dispersal, disturbance, patch extinction frequency, vital rates and
metapopulation structure. Finally, we conducted a sensitivity analysis that evaluated changes in individual
parameter magnitudes relative to default values, based
on the linear array and single isolated immortal patches.
Equations were discretized using the finite difference
method (Hoffman 2001) and simulations run using
MATLAB® 7. At the beginning of each simulation,
five male and five female spores were added to each
patch. The time step was 3 days, simulations were
analysed over 300 years and replicated up to 40 runs.
Note that all of the stochasticity in the model is due to
disturbance and patch extinction.
Our focus is on sex ratio dynamics at the regional
level, where both sexes could be typically maintained
even when one sex (usually males) may tend to be
locally eliminated by competition, requiring patch
extinction before both sexes can again share the patch.
In this scenario, the regional processes of recolonization by the eliminated sex are central, and persistence
of the two sexes at the regional level hinges on asynchrony of dynamics among patches. These features are
consistent with the concept of a classic metapopulation
(Husband & Barrett 1996; Hanski & Gilpin 1997;
Harrison & Taylor 1997; Hanski 1999; Crowley &
McLetchie 2002; Freckleton & Watkinson 2002).
Results
A typical example of the sex ratio dynamics generated
by the model using the default parameter values is
illustrated in Fig. 2(a). For individual patches within
a metapopulation, there are distinct cycles of patch
filling, intense competition and sex elimination. These
single-patch cycles are extended by within-patch disturbances, interrupted by patch extinction and re-initiated
by spore establishment. Males predominate during the
interval of patch filling, with an increasing female bias
thereafter until males are eliminated by overgrowth or
both sexes are simultaneously eliminated by patch extinction (see McLetchie et al. 2002). At the metapopulation
level, however, both sexes persist, and the sex ratio
fluctuates with low amplitude around a mean male proportion of 0.128 ± 0.034 (temporal standard deviation).
Spatial variation in sex ratio along the linear array
of patches forming the metapopulation is shown in
Fig. 2(b). At any instant of time, even adjacent patches
may differ greatly in the proportion of males, leading
to a spatial standard deviation of 0.195 (standard deviation in male proportion between patches, averaged
over time). In the illustrated example, 13 of 30 patches
contained females only, 14 had more females but some
943
Persistence of sexes
in metapopulations
Fig. 2 Sex ratio dynamics. Parameters are at default values (Table 1). (a) A representative numerical simulation illustrates the sex
ratio dynamics for two patches within a linear metapopulation, and the dynamics over all patches taken together. (b) Sex ratio
spatial pattern along the linear metapopulation at a particular time (i.e. year 300; histogram and stars) and averaged over the last
200 years of the run (points joined by dotted lines).
males and three had more males but some females.
None of the patches went extinct. The time-averaged
sex ratio for each patch in the default case also varied
spatially, though far less than the instantaneous pattern,
with a consistent female bias.
In the remainder of this section, we address the effects
of key biological and physical factors on sex ratio and
persistence of the sexes. Specific ways that these factors
influence the results are characteristic of the M. inflexa
system, yet many of the basic mechanisms responsible
for these patterns link to other plant and animal systems,
as considered in the Discussion.
© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society,
Journal of Ecology
95, 937–950
     
 
Across a range of uncovered threshold values and
establishment rate maxima for spores, the proportions
of males for a linear metapopulation were female-biased,
ranging from approximately 0.10 to 0.43 (Fig. 3a).
Male proportions between 0.1 and 0.2 were obtained
over a broad range of establishment rate and threshold
combinations, including default conditions. Male
proportions above 0.2 were found for both the lowest
and the highest establishment rates. With establishment rate extremely limited, patch filling by growth and
asexual reproduction become comparable in importance with intense overgrowth competition during the
lifetimes of patches, reducing the female bias. At the
other extreme, with essentially unlimited spore establishment, areas reopened by disturbance were filled by
local and dispersed spores at a 1 : 1 sex ratio, rather than
by spreading of the dominant competitor. Neither of these
extremes seems likely to fit the focal system in nature.
When the spatial distribution of patches in the
metapopulation was changed from a linear array to an
island array, with all patches now taken to be equidistant from each other, the results were strikingly similar
(Fig. 3b). This and other comparisons of dynamics for
these two array geometries suggest that our qualitative
944
G. García-Ramos
et al.
Fig. 3 Spore limitation on isolated patch and metapopulation sex ratios. Effects of the uncovered threshold for spore germination
(i.e. the minimal fraction of the patch that must be unoccupied to permit spores to germinate) and the maximal spore establishment
rate on the sex ratio of M. inflexa: (a) linear metapopulation, with 30 equally spaced patches in a straight line; (b) island
metapopulation, with 30 equidistant patches; and (c) single isolated patch, invulnerable to extinction and under default
disturbances. Other parameters are at default values. Fraction of the patch uncovered for spore germination: ≥ 0.0 uncovered
(no restriction for germination), ≥ 0.2, –– ≥ 0.4 (default value), * ≥ 0.8 (high restriction, patch almost empty).
and even quantitative results may be relatively independent of the specific spatial arrangement of patches
within the metapopulation.
For isolated patches under small disturbances but
invulnerable to extinction (default), spore recruitment
influenced the sex ratio somewhat differently (Fig. 3c).
Males were eliminated from patches for lower establishment rates and larger uncovered thresholds, including default conditions. The female competitive advantage
dominated the re-establishment of males from spores
except under weak constraints on spore establishment
and uncovered threshold.
© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society,
Journal of Ecology
95, 937–950

The spore dispersal distance affected sex ratio depending on the recruitment of spores in two alternative ways
(Fig. 4). (i) For unrestricted recruitment, the male proportion ranged from 0.36 to 0.12 with increasing
dispersal distance. This decline in male fraction with
dispersal is the result of decreasing the proportion of
spores landing in the source patch and thus reducing
the impact of sexual reproduction. (ii) For moderate
uncovered thresholds only very short dispersal distances
resulted in loss of males (or the metapopulation went
extinct, depending on gemma dispersal rates). But over
a wide range of spore dispersal (including default conditions), the male proportion was relatively constant
around 0.128. Note that for large dispersal there is no
effect of spore limitation on establishment as a reduced
number of spores stay on the originating patch [ f (0) in
Fig. 4], and so additional restrictions for establishment
are superfluous. These results indicate that some limitation of spore recruitment in the source patch largely
account for the metapopulation sex ratio.
Simulations altering gemma dispersal distance had
little impact on sex ratio. Even high distances of gemma
dispersal were unable to create much of a gender bias,
945
Persistence of sexes
in metapopulations
Fig. 4 Spore dispersal and metapopulation sex ratio, showing the effect of increasing dispersal with mode of spore recruitment.
Dispersal is described by single or combined exponential functions for spore density (dsp, in units of m–2) along the one2
2
dimensional array of patches dsp ( x ) = p [1/(2πλ1 )] exp( − | x |/ λ ) + (1 − p )[1/(2πλ 2 )] exp( −| x |/ λ 2 ), where p is the proportion of
short-distance dispersers, (1 – p) is the proportion of long-distance dispersers and the λi are the corresponding dispersal
parameters (m). The magnitude of dispersal was estimated as λ′ = pλ1 + (1 – p)λ2. The legend illustrates some of the dispersal
functions on a log scale for the vertical axis. f (0) is the proportion of spores falling in the source patch. Other parameters are
at default values.
as males have only a modest advantage in gemma
production on a per-unit basis.
   
In the absence of whole-patch extinction and for low
to moderate within-patch disturbance frequency, the
metapopulation consisted only of females (Fig. 5a);
but at higher disturbance frequency (≥ 0.4 month–1
patch–1), male proportion increased in a sigmoid fashion from strong female bias to strong male bias. Both
results are consistent with the dynamics of isolated,
immortal patches (McLetchie et al. 2002). In the presence of patch extinction, lower disturbance frequencies
produced female-biased sex ratios, and higher disturbance frequencies yielded male-biased sex ratios.
Even in the absence of within-patch disturbances,
asynchronous patch extinctions can sustain coexistence of the sexes.
© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society,
Journal of Ecology
95, 937–950
 
Sex ratio depended strongly on growth rates (Fig. 5b).
At maximum growth rates achieved in a glasshouse,
males were eliminated from the metapopulation; at
growth rates below 20% of the glasshouse level, the
model yielded highly male-biased metapopulations.
The sensitivity analysis demonstrated much stronger
responses of sex ratio to non-reproductive stages (i.e. to
G2 and G5) than to the other Gi and the Ai (Table 3).
Proportionally increasing both G2 and G5 intensified
competition and increased female advantage, whereas
proportionally decreasing both G2 and G5 decreased
the female growth advantage, accounting for the
pattern in Fig. 5(b).
Increased transition rates from non-reproductive
stages to sexual stages (T42 and T75) favoured males
(Table 3). This increased the proportion of individuals
expressing sex during the sex season, reducing the
female growth advantage. A higher fertilization rate
(M7) shifted the advantage to males even more strongly
by keeping a larger proportion of females from growing
without inhibiting males. Reducing the transition
rate from fertilized to non-reproductive females (T51)
similarly favoured males.
Sensitivities to modifications of the sex-specific gemma
production rates A3 and A6 reflect the importance of
the male advantage in this mechanism for within-patch
spreading. Eliminating asexual reproduction or assuming equal production of asexual propagules by males
and females (A3 = A6 in Table 3) leads to a declining
male proportion in the metapopulation, with males
946
G. García-Ramos
et al.
Discussion
Fig. 5 Disturbance, extinction and growth rate on sex ratio.
(a) Effects of within-patch disturbance frequency and patch
extinction frequency on metapopulation sex ratio. (b) Relative
magnitude of M. inflexa vegetative growth rates and the
resulting metapopulation sex ratio. Other parameters are at
default values.
persisting for periods ranging from several hundred to
a few thousand years. This indicates that metapopulation
dynamics extend the coexistence of males and females
by extending low-density conditions that reduce the
impact of competition, while patches are continually
being formed at 1 : 1 sex ratio. When males have a greater
asexual production of propagules (as in the default
scenario), the metapopulation dynamics lead to
permanent coexistence.
  
© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society,
Journal of Ecology
95, 937–950
High temporal variation in sex ratio was typical of
small metapopulations of a dozen patches. These fluctuations were reduced for a greater number of patches,
reflecting the spatial averaging effect of larger assemblages. Sex ratio was sensitive to patch size. Large patch
sizes extended the filling and sex elimination patch
cycle, maintaining males in the patch for a longer period.
The sex ratio was insensitive to the ratio of male to
female spores at the initiation of the metapopulation.
In none of the simulations at the metapopulation level
did males persist without females.
To our knowledge, this study provides the first demonstration of how metapopulation structure can facilitate
coexistence of males and females, even under intense
competition. By focusing on the liverwort Marchantia
inflexa, featuring sexual dimorphism with asymmetric
overgrowth competition and asexual reproduction,
we address a particularly strong challenge to the maintenance of sexual reproduction in a metapopulation
system under empirical study. In essence, when Marchantia
patches linked by spore dispersal undergo asynchronous cycles of extinction that permit re-establishment
of both sexes, the two sexes generally coexist in the
whole system – even when one sex is consistently being
lost locally.
Our results indicate that this conclusion holds for at
least two very different spatial distributions of patches,
over a wide range of sexual recruitment and over at
least a factor of four in the magnitudes of most model
parameters, and even for systems consisting of only a
few equal-sized patches. The female-biased metapopulation sex ratios produced by the model are generally
consistent with the strong female bias in sex expression
observed at field sites in Trinidad. A patch-assemblage
of M. inflexa occurring at Quare River in Trinidad
reported a male proportion of 0.24 (year 1999, 358
plants sampled from 11 of 70 patches, with patch male
proportion ranging from 0.03 to 0.68 and spatial standard deviation of ±0.20; D.N. McLetchie, unpublished
data). This estimation, derived from a single sampling
effort, was higher than the temporal averaged metapopulation male proportion (0.128), while the spatial
standard deviation for both theoretical and field data
shared similar large variation (±0.20). In future models,
some of the assumptions could be relaxed to allow for
spatial and temporal autocorrelation in patch disturbance and extinction, to deal more realistically with
spatial distributions of the sexes within patches (Crowley
et al. 2005b), and to consider the implications of variation
in distances and patch sizes within the metapopulation
(C.R. Stieha, unpublished data).
The basic features of ecological sex ratio dynamics
that favour coexistence of the sexes at the metapopulation
level are those that permit sexual recruitment sufficient
to ameliorate the dominant biasing mechanism favouring one sex over the other. Across the range of disturbance frequencies capable of opening habitat areas
sufficiently to allow spore establishment and growth to
maturation by individuals of both sexes, metapopulation structure and dynamics can facilitate coexistence
of the sexes in many systems (cf. maintenance of true
males in hermaphrodite-dominated populations; Pannell
2000). Coexistence of males and females generally
depends on interacting processes at different spatial
scales (e.g. see Crowley & McLetchie 2002; with males
and females as competing clones). Within patches,
these include local disturbances, clonal expansion,
fertilization, spore dispersal and establishment of new
947
Persistence of sexes
in metapopulations
Table 3 Sensitivity analysis for the model. Change refers to default values in Table 2
Parameter†
DEFAULT
Life-history
G 2, G 5
G 3, G 6
T23, T56
T24, T57
T32, T65
T42, T75
T51
M7
Change
Male proportion§
0.128
×2; ÷2
×2; ÷2
×2; ÷2
×2; ÷2
×2; ÷2
×2; ÷2
×2; ÷2
=0
×2; ÷2
×5; ÷5
=0
= 0.085
= 1.112
×2; ÷2
×2; ÷2
=0
×2; ÷2
=0
×10; ÷10
×100; ÷100
0.1–1.0 (∝ A3, A6, Gi)
0; 0.913
0.143; 0.108
0.113; 0.136
0.105; 0.100
0.264; 0.078
0.887; 0.089
0.090; 0.513
0
0.265; 0.093
0.905; 0.098
0 (3/10)‡, 0.055 (7/10)
0 (4/35), 0.060 (31/35)
0 (3/40), 0.061 (37/40)
0.226; 0.090
0.768; 0.069
0
0.068; 0.221
0.447
0.135; 0.123
0.133; 0
Fig. 5(b)
×2; ÷2 (∝ K )
×4; ÷4 (∝ K )
10
50
100
0.133; 0.106
0.209; 0.103
0.133 ± 0.049 SD, [0.050–0.255]**
0.130 ± 0.026 SD, [0.093–0.240]
0.130 ± 0.020 SD, [0.094–0.209]
Dispersal and establishment
λi
q
r
0.7–30
0.0–0.8
10– 2000, unlimited
Fig. 4
Fig. 3(a–c)
Fig. 3(a–c)
Disturbance and extinction
pd
pe
0.0–0.7
1/10–1/120, no extinction
Fig. 5(a)
Fig. 5(a)
A3 = A6
A 3, A 6
A3
A6
F1
rg¶
Patch system
4h2
n
Initial conditions (male/female spores in unit per patch)
20/80, 50/50, 80/20
0.120, 0.128, 0.138
†Parameters from Table 2 not explicitly analysed here include F1 and K (these are proportional to parameters that were analysed,
as indicated), x* and z.
‡Proportion of simulations resulting in loss or maintenance of males.
§Male proportions were also obtained for each of the indicated changes for a single isolated, immortal patch (see also McLetchie
et al. 2002). In those runs, males were eliminated from the single patch where there was a female bias in the corresponding
metapopulation, and females were eliminated from the single patch where there was a male bias in the corresponding
metapopulation. From 12 to 177 years were required for each elimination, depending on the parameter change.
¶
Ratio of field growth rate to overall glasshouse growth rate (see Appendix S1).
**
95% confidence interval, and SD standard variation within simulations.
© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society,
Journal of Ecology
95, 937–950
individuals; among patches, these processes are dispersal of propagules and patch extinction.
Our results emphasize the role of asexual reproduction
in promoting coexistence of sexes and sexual reproduction.
In the Marchantia system, asexual reproduction favours
the retention of males through sexual dimorphism in
the production of asexual propagules, which balances
the dimorphism in growth rates. In species without
asexual reproduction, sexual dimorphism in growth
could be balanced by another dimorphism, such as a
tolerance to dry conditions, favouring males.
A key feature of metapopulation structure and
dynamics in the field may be resource heterogeneity.
For instance, field observations of M. inflexa indicate
that sex tends to be expressed in patches exposed to
high light intensities (McLetchie & Puterbaugh 2000).
This suggests that some patches may act as sources and
others simply as sinks for dispersing spores. Moreover,
one sex could be favoured under some field conditions
and the other sex favoured in others, leading to a
mosaic of patches dominated by one sex or the other.
This spatial heterogeneity could provide additional
948
G. García-Ramos
et al.
© 2007 The Authors
Journal compilation
© 2007 British
Ecological Society,
Journal of Ecology
95, 937–950
opportunities for maintaining both sexes at the metapopulation level and could account for the exclusively
male patches that are sometimes observed in the field
(D.N. McLetchie, unpublished data) but were not
produced by our simulations. Studies in seed plants
have documented spatial segregation of the sexes
along environmental gradients (Grant & Mitton 1979;
Bierzychudek & Eckhart 1988; Williams 1995; Eppley
2001; Bertiller et al. 2002). In Marchantia inflexa patch
sex ratio varies spatially (Schuster 1992; McLetchie &
Puterbaugh 2000), but correlation with environmental
variation has not been detected (Fuselier & McLetchie
2004). Our results suggest, however, that spatial heterogeneity, although a likely contributor to coexistence
of the sexes, is unlikely to be a requirement for coexistence
in nature.
For plants, the metapopulation concept has taken
hold more gradually than for animals, primarily
because of missing or inadequate data on dispersal,
colonization, establishment and extinction (Ehrlén
& Eriksson 2003). Additional field work can greatly
strengthen our understanding of the M. inflexa
metapopulation. Of particular importance is the
quantitative documentation of field sex ratio and the
loss of one or both sexes at the patch and metapopulation level, propagule dispersal within and between
patches, and patch extinction and recolonization rates.
Moreover, our modelling results were particularly
sensitive to our glasshouse data and how they might
translate into field growth rates (Fig. 5b), which thus
merit additional attention. Important aspects to be
addressed for predicting a metapopulation sex ratio in
dioecious systems with strong competition among
sexes include the existence of sexual dimorphisms in
different aspects of life-history, as well as determining
the dispersal abilities of propagules and their opportunities
for recruitment.
Metapopulation dynamics have been invoked to
account for coexistence of species and maintenance
of diversity. In metapopulations, interactions among
competition, disturbance and dispersal can maintain
species regionally that are eliminated locally in the
presence of superior competitors (Caswell & Cohen
1991). A trade-off between competitive and colonizing
ability has been suggested as a mechanism that promotes species coexistence in these patchy environments
(Levins & Culver 1971). Nonetheless, many studies
explicitly considering competition and dispersal have
showed that a trade-off between colonization and competition is not needed for coexistence (Bolker & Pacala
1999). In this respect, Higgins & Cain (2002) found
coexistence of species in a spatially distributed system
even when the inferior competitor was also an inferior
colonizer. Similarly, within a species in our model, one
sex may be a locally inferior competitor but manage to
coexist in the metapopulation without an advantage
over the other in colonizing newly available patches
given that spores have an even sex ratio. Notice that
males, through asexual reproduction, have advantages
at low density, a condition generally associated with
patch filling and, hence, post-colonization.
Sexual systems range from only males and females
(dioecy) to only hermaphrodites. Between these extremes,
there are mixtures of mating systems involving unisexual
individuals and hermaphrodites. Metapopulation models
have been used to explain the maintenance of gynodioecy,
a mating system in which females coexist with hermaphrodites (McCauley & Taylor 1997; Pannel 1997; Couvert
et al. 1998). In this system, gender has a complex inheritance often determined by epistatic interactions between
nuclear and cytoplasmic genes. Therefore, these metapopulation models are mainly concerned with retaining genetic variation for gender determination. These
models address how this population structure retains
genes under frequency-dependent selection (McCauley
& Taylor 1997) and the effect of the colonization on
the genetic variation for sex determination (Couvert
et al. 1998). These studies suggest that metapopulation
dynamics are limited for explaining gynodioecy. By
contrast, the present study addresses a different sexual
system consisting of males and females, and the concern
for sex coexistence is framed as a population problem,
dealing with sexual dimorphism, asexual reproduction,
competition and the issue of recruitment of different
propagules. In this study, males and females are maintained in the metapopulation for a broad range of situations. As such, the metapopulation perspective obviously
opens many new avenues for addressing competition,
coexistence and spatially distributed reproduction.
As we mentioned previously, Maynard Smith (1978)
pointed out that, because sexual and asexual propagules in species that produce both are often adapted
to different ecological situations, the mixed mode of
reproduction could readily be maintained by environmental heterogeneity. In our ecological metapopulation
model, the key advantage of sexually produced offspring is the ability to colonize unpredictably available
new habitat. While this colonization ability is essential
for maintaining both sexes in disturbed systems, the
final result depends on the combinations and levels
of sexual dimorphism in key ecological traits. Extensive sexual dimorphism in Marchantia inflexa, typical
of clonal organisms (Popp & Reinartz 1988; Williams
1995; McGovern 2002), generates asymmetric competition capable of preventing local coexistence of the
sexes. However, at the larger scale of a metapopulation,
this level of dimorphism and the broad dispersal of
sexually produced propagules allow the persistence
of both sexes, and thus the retention of sexual reproduction. The metapopulation favoured the coexistence
of sexes by initiating patches at 1 : 1 sex ratio and by
intervals of reduced growth competition in which the
male advantage in production of asexual propagules
predominates. We therefore conclude that – even with
biasing mechanisms like clonal competition at work
– metapopulation processes may greatly extend the
range of conditions for coexistence of the sexes where
such coexistence is locally impossible.
949
Persistence of sexes
in metapopulations
Acknowledgements
We thank the Wildlife Section of the Forestry Division
of The Republic of Trinidad and Tobago for collection
and export permits; the Water and Sewage Authority in
Trinidad for allowing access to the research site; Linda
Fuselier, Isabelle Olivieri, Charles Richardson, Ophélie
Ronce, Nicole Sudler and Carey Sydney for discussions
of the ideas presented here; the Department of Agronomy at the University of Kentucky for the use of glasshouse space; and the US National Science Foundation
(DEB 9974086-DNM [PI] and PHC [co-PI]; and DEB0415339-GGR[PI] and PHC [co-PI]) and National
Institutes of Health (AI060468-02-GGR [PI] and PHC
[co-PI]) for grant support.
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Supplementary material
The following supplementary material is available for
this article:
Appendix S1 Estimating the default parameters values
of Table 2.
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