ARTICLE IN PRESS
Journal of Theoretical Biology 233 (2005) 25–42
www.elsevier.com/locate/yjtbi
Overgrowth competition, fragmentation and sex-ratio dynamics:
a spatially explicit, sub-individual-based model
Philip H. Crowley, Christopher R. Stieha, D. Nicholas McLetchie
Department of Biology and Center for Ecology, Evolution and Behavior, University of Kentucky, 101 Morgan Building,
Lexington, KY 40506-0225, USA
Received 10 June 2003; received in revised form 10 September 2004; accepted 15 September 2004
Abstract
Sessile organisms that compete for access to resources by overgrowing each other may risk the local elimination of one sex or the
other, as frequently happens within clumps of the dioecious liverwort Marchantia inflexa. A multi-stage, spatially implicit
differential-equation model of M. inflexa growing in an isolated patch, analysed in a previous study, indicated that long-term
coexistence of the sexes within such patches may be only temporary. Here we derive a spatially explicit, sub-individual-based model
to reconsider this interpretation when much more ecological realism is taken into account, including the process of fragmentation.
The model tracks temporally discrete growth increments in continuous space, representing growth architecture and the overgrowth
process in significant geometric detail. Results remain generally consistent with the absence of long-term coexistence of the sexes in
individual patches of Marchantia. Dynamics of sex-specific growth qualitatively resemble those generated by differential-equation
models, suggesting that this much simpler framework may be adequate for multi-patch metapopulation models. Direct competition
between fragmenting and non-fragmenting clones demonstrates the importance of fragmentation in overgrowth competition. The
results emphasize the need for empirical work on mechanisms of overgrowth and for modeling and empirical studies of life history
tradeoffs and sex-ratio dynamics in multi-patch systems.
r 2004 Elsevier Ltd. All rights reserved.
Keywords: Clonal plant populations; Maintenance of sex; Liverworts; Marchantia inflexa; Simulation modeling
1. Introduction
Organisms of many taxa in all five kingdoms compete
for access to limiting resources through a type of
interference known as overgrowth competition (e.g.
Monera: Jarosz, 1996; Jarosz and Kania, 2000; Protista:
Airoldi, 2000; Fungi: Gourbiere et al., 2001; Plants:
Matlack, 2002; Lichens: Hestmark et al., 1997; Armstrong, 2002; Animals: Connell, 1961, Barnes and Dick,
2000; Coral–algal interactions: Lirman, 2001; McCook
et al., 2001; Jompa and McCook, 2002). Overgrowth
competition is the process of attempting to gain and
hold space linked to limiting resources by growing in a
Corresponding author. Tel.: +1 859 257 1996;
fax: +1 859 257 1717.
E-mail address: [email protected] (P.H. Crowley).
0022-5193/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2004.09.017
way that physically restricts a competitor’s access to its
resource supply while increasing access by the growing
individual. Here, space is taken to be a surrogate
limiting resource that cannot be shared at any point,
suggesting that coexistence of competitor clones or
species may be difficult to maintain over a contiguous
area (see Connell, 1961; Buss and Jackson, 1979;
McLetchie et al., 2002).
Organisms that frequently engage in overgrowth
competition are generally capable of sub-dividing into
fully functional, physiologically separate individuals
through modular processes like budding, fission, or
abscission—or through fragmentation (Armstrong,
1979; During, 1990). To date, little effort has been
directed at understanding the population-level implications of fragmentation and its connection to competition
for space. Fragmentation is itself a taxonomically
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P.H. Crowley et al. / Journal of Theoretical Biology 233 (2005) 25–42
widespread phenomenon (Protista: Amsler, 1984; Hooper et al., 1988; Ceccherelli and Cinelli, 1999; fungi:
Leslie and Klein, 1996; lichens: Hale, 1974; Bailey, 1975;
Armstrong, 1979; plants: Room, 1983; Outridge and
Hutchinson, 1990; During, 1990; Ewanchuk and Williams, 1996; animals: Lasker, 1990; Bruno, 1998; Zakai
et al., 2000) based on partitioning of individuals into
totipotent parts that may vary greatly in size. Implications of this process for competitive effectiveness are
difficult to determine, because the underlying mechanisms and spatial ramifications can be complex. Comparisons of fragmentation and overgrowth competition
across taxa in relation to environments and life histories
may be complicated by phylogenetic constraints.
Here, we focus mainly on local populations of a
particular organism, the thalloid liverwort Marchantia
inflexa Nees et Mont, known to compete intensively for
space within patches and to fragment when partially
overgrown by other individuals or other parts of the
same individual (H. During, personal communication;
D.N. McLetchie, unpublished observations). Our previous attempt to model within-patch sex-ratio dynamics
of this intriguing species led to the formulation of a
multi-stage, linear, coupled ordinary differential-equation model, consistent with the simplifying assumption
that each genotype (or sex in this model) could be
represented as arbitrarily small, randomly distributed
increments of biomass (McLetchie et al., 2002). This
avoided explicit representation of fragmentation and of
the structure of individual plants (or ramets) that
actually tend to produce clumped rather than random
distributions of genets within patches in nature. Results
of this previous study indicated that overgrowth
competition made coexistence impossible, at least across
the range of biologically reasonable parameter values we
explored. But we wondered whether our unrealistic
assumptions about the geometry of these plants, their
growth dynamics, and their spatial distribution
might have influenced the model’s behavior and our
conclusions.
In the present study, to address spatial competition
within individual patches of M. inflexa more realistically, we formulated a high-resolution simulation model
focusing on competition between the sexes. Parameter
values were mostly estimated from our own empirical
observations. Our approach involved tracking discrete
growth increments of each individual and two-dimensional spatial relationships among individuals and
among their growth increments, resulting in a spatially
explicit, sub-individual-based (SESIB) model.
This work represents a contribution to our continuing
studies of sex-ratio dynamics in M. inflexa metapopulations. If indeed overgrowth competition precludes
coexistence of sexes in individual patches, then maintaining both sexes in nature seems to hinge on structure
and dynamics at the metapopulation level (McLetchie
et al., 2002). Multi-patch models and empirical studies
of the M. inflexa system are in progress and will
be reported elsewhere (also see Crowley and McLetchie,
2002).
Our goals in the present study were to
(1) derive and examine the behavior of a SESIB patch
model (March) of M. inflexa, incorporating fragmentation;
(2) compare sex-ratio dynamics in the March model
with those obtained for simpler spatially implicit
patch models; and
(3) demonstrate and account for the importance of
fragmentation in overgrowth competition.
We begin the remainder of the presentation with
a conceptual overview of the study system and its main
components, describe some important details of
the model’s structure and function, present results of
the simulations, and consider how the results relate to
the existing literature and possible future directions.
2. Conceptual overview of the study system
M. inflexa is a New World hepatic bryophyte found in
the eastern USA as far north as Tennessee and in and
around the Caribbean basin as far south as Venezuela
(Bischler, 1984). M. inflexa is usually attached to rock or
to bark of fallen trees in moist environments, including
on road-cuts in areas of frequent rainfall and along
permanent streams. Because it tends to occupy discrete
patches vulnerable to elimination by disturbances and
loosely connected by dispersal, M. inflexa populations
typically form metapopulations (McLetchie and Puterbaugh, 2000; McLetchie et al., 2002; see Hanski, 1998,
1999).
M. inflexa grows by extending ribbon-like thalli
horizontally from terminal mericells over its substrate,
which may include other thalli of the same or other
individuals (Fig. 1). (Bryophytes have the single
mericell, in contrast to the multiple cells of angiosperms,
at their meristematic loci (Newton and Mishler, 1994)).
As with many bryophytes, the species is dioecious:
individuals are unisexual and gender is chromosomally
determined (Ramsay and Berrie, 1982; Bischler, 1986).
In a particular season or seasons, both sexes reproduce
asexually by forming cupules from mericells in connection with thallus growth. Cupules eventually produce
and release gemmae, asexual propagules primarily
dispersed by water. In another season or seasons, sexual
reproduction is initiated via formation of umbrella-like
sex structures by mericells, preventing any further
terminal growth of the thallus. Males release spermladen fluids onto the upper surfaces of their sex
structures (antheridiophores); sperm are dispersed by
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27
Since the published reports indicate that this single
surviving sex is most often females, which in Marchantia
typically have slightly higher thallus extension rates
than males (McLetchie et al., 2002; D.N. McLetchie,
unpublished data), we suspect that overgrowth competition may at least partly account for many instances of
single-sex patches and populations. Because of its strong
association with overgrowth competition, the process of
fragmentation is an integral part of the relatively highresolution SESIB model presented here.
Consider a single colonist ramet—a physiologically
independent individual successfully germinating within
a patch. Several key processes and variables interact to
determine contributions to the expected lifetime reproductive success or fitness of the ramet’s clone or genet in
an ephemeral patch system (Fig. 2; see Crowley and
McLetchie, 2002). ‘‘Area held’’ is a state variable
PHYSICAL
DISTURBANCE
+
OVERGROWTH
BY OTHER
GENETS
-
-
D
AREA
HELD
+
FITNESS
A
-
-
+
FRAGMENTATION
Fig. 1. The thalloid liverwort M. inflexa in a tropical rainforest in
Trinidad. (a) A rock, mostly covered by a dense growth of M. inflexa,
adjacent to a stream. White areas are dead tissue, probably killed by
desiccation resulting from intense insolation during a dry interval.
Assemblages of such semi-isolated clumps of M. inflexa on rocks in
and near streams constitute metapopulations (Crowley and McLetchie
2002, McLetchie et al., 2002). (b) Close-up view of overlapping thalli
representing many different independent individuals or ramets.
MORTALITY
+
+
+
NET
PRODUCTIVITY
+
+
THALLUS
EXTENSION
REPRODUCTION
(SEASONAL)
TRADE-OFF
+
+
+
BIFURCATION
B
& SPROUTING
+
C
-
-(
SEX)
water droplets, resulting in possible contact with female
sex structures (archegoniophores) and subsequent fertilization. Fertilized archegoniophores eventually release
wind-dispersed spores, apparently at approximately a
1:1 sex ratio, and these can colonize available substrate.
Some clumps of M. inflexa contain individuals of only
one sex (McLetchie and Puterbaugh, 2000), either
because of (1) unisexual colonization, (2) a disturbance
that happened to remove one sex but not the other, or
(3) the elimination of one sex by the other via
overgrowth competition. In fact, some entire (meta)populations of Marchantia and other bryophytes consist
only of a single sex (Schuster, 1992; Bowker et al., 2000).
NUMBER OF
GROWTH TIPS
Fig. 2. Interactions among key features of individual ramets and their
environment that determine the potential for long-term expansion
(fitness) of their genotype or genet. Positive and negative effects and
the growth-reproduction trade-off in the expending of net productivity
are indicated. Growth, resulting in greater area coverage and more
numerous growth tips, helps generate two positive feedback loops A
and B. The negative feedback loop C and opposing effects of area held
and growth-tip number on splitting and sprouting stabilize the area
held per growth tip. Overgrowth by other genets helps create the
positive feedback loop D. The predominance of these positive feedback
loops underscores the unstable nature of competitive interactions in
this system.
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28
ramet or individual growth increments. Overgrowth by
other ramets can be inhibited by rapid thallus extension,
according to geometric relationships described in the
next section. Overgrowth triggers fragmentation, as
covered tissue (i.e. one or more growth increments) that
falls below the energetic compensation point is eliminated, breaking the ramet into physiologically independent parts (i.e. into new ramets). Fragmentation in turn
can reduce the current rate of thallus extension but helps
protect the ramet as a whole from falling below the
compensation point and becoming moribund, as happens when the model is modified so that overgrowth no
longer elicits fragmentation (see below).
The March model builds thalli as a series of
linked rectangular growth increments of constant width
(Fig. 3), extending from the terminal mericell of the
previous rectangle to a length determined by the thallus
extension rate and the time step duration. Associated
with each increment are state variables and nondynamic (fixed) variables. An increment’s present status,
shown on each increment in the example thallus of Fig.
3, indicates whether it can grow (if located at the end of
the thallus, not overgrown at the tip, and alive, having
not produced a sex structure), engage in asexual or
sexual reproduction (if already in this same reproductive
status—or else as for growth, except with seasonal
restrictions and time limitations), become static
(if it cannot grow but is alive and not engaged
in reproduction), or is dead (killed by disturbance or
by negative energy balance). The number of living
specifying the total amount of ramet surface exposed to
light (i.e. not overgrown), under the plausible assumption that M. inflexa is light-limited in the field. To keep
things manageable, we ignore spatial heterogeneity of
light availability (but see Caldwell and Pearcy, 1994;
Oborny and Kun, 2002). Gross productivity is assumed
proportional to area held; when decremented by
maintenance costs, the resulting net productivity represents the energy available for partitioning into growth
and reproduction. The ‘‘number of growth tips’’ is
another state variable indicating the number of thallusterminal mericells available for growth. Under the
assumption that growth is equally supported at all of
a ramet’s growth tips (as in Oborny and Kun, 2002; see
also Caraco and Kelly, 1991), this support is expressed
by dividing the area held, decremented for metabolic
and reproductive costs, by the number of growth tips.
The resulting quotient is here termed the support ratio s;
which figures prominently in determining details of the
ramet’s demography, described in the next section.
Growth tips, used up through the formation of sex
structures by terminal mericells, are replenished through
bifurcation of mericells and thalli (note the Y-shaped
thallus segments in Fig. 1b) or by occasional sprouting
from thallus segments without terminal mericells.
In the model, as in nature, death of entire ramets can
result from physical disturbances like flooding or
desiccation or from complete overgrowth by other
ramets. More localized physical disturbances and overgrowth may instead simply reduce the area held by a
9
GROW
8
DATA LINKED TO EACH GROWTH INCREMENT
GROW
STATE VARIABLES
6
STATIC
7
SEX
4
5
ASEX
ASEX
3
DEAD
2
STATIC
•
•
•
•
STATUS (GROW, ASEX, SEX, STATIC, DEAD)
NUMBER OF LIVING PROGENY INCREMENTS
AREA HELD (REDUCED WHEN OVERGROWN)
TIMER (CONTROLS TIMED PROCESSES)
FIXED VARIABLES
•
•
•
•
IDENTIFICATION NUMBER (ID)
ID NUMBER OF PROGENITOR INCREMENT
LOCATION COORDINATES
GENOTYPE / GENDER
1
STATIC
Fig. 3. An example cartoon thallus built from rectangular growth increments as if by the March model, with the current status of each increment
indicated. Black dots at the distal ends of the increments indicate the locations of active mericells (increments 8 and 9), a sexual-reproductive
structure (an antheridiophore or archegoniophore—7), asexual-reproductive structures (cupules—4 and 5), and former growth points (1, 2, 3, and 6),
including two bifurcating points (3 and 6). This geometry implies that increments 4 and 5 were formed by bifurcation of the mericell of increment 3 in
one time step. In the next step, increments 6 and 7 grew from their respective progenitors, forming cupules at the former mericell loci of increments 4
and 5. In the final step indicated by the diagram, increments 8 and 9 formed by bifurcation from 6, while increment 7 produced a sex structure from
its mericell. Note that increment 3 is now dead, either because of disturbance or extensive overgrowth; its elimination will fragment the original ramet
into three independent ramets. The data linked to each growth increment that are needed to track the fate of these increments and their ramets are
summarized in the box.
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3. Model details
In this study, we make the simplifying assumption
that productivity per unit area held remains constant
over time and space. Temporal constancy may be a
rough approximation for systems like tropical rainforest
metapopulations in a model with multi-day time steps
that effectively average over time. Spatial constancy is
also imposed primarily for tractability, but we note that
other modeling studies have addressed the role of spatial
resource heterogeneity in clonal plants (e.g. see Oborny,
1991; Cain et al., 1996).
3.1. Vital rates, growth, and overgrowth
pe =
P
r
Slo
Extension Rate, E (cm2/tip/day)
We express thallus extension rate E as an increasing
decelerated function of the support ratio s; asymptotic
E=
Pσ
1 + Pσ / r
0
Support Ratio, σ
to its maximum at r (Fig. 4a). Thus more support for a
growth tip at low s generates a proportional increase
in E, where the productivity P is the constant of
proportionality, but diminishing returns through physiological constraints on growth at higher s causes E to
flatten toward the asymptote (see Caraco and Kelly,
1991). Dividing the extension rate by the support ratio
then generates the per-capita growth rate G, a declining
function of s asymptotic to zero (Fig. 4b). We assume
that the remainder of the productivity that is not
committed to growth is used in reproduction, whenever
this is seasonally appropriate. To generate the additional
growth tips needed to accommodate growth at increasing ramet sizes and to replace growth tips lost to sexual
reproduction, we make the simplifying assumption
that the bifurcation rate is proportional to the support
ratio. The intersecting growth and bifurcation curves
have the effect of stabilizing the ramet’s support ratio
(cf. Sánchez et al., 2004).
Thalli stop growing whenever their tips are overgrown, are removed by disturbance, encounter the patch
boundary, or produce a sex structure. For circumstances
in which s exceeds a threshold value, thalli unable to
grow (except those with an active sex structure) are
assumed to produce a new mericell and thallus by
sprouting (consistent with unpublished observations of
sprouting by D.N. McLetchie).
We assume that thallus growth in the patch can be
represented as two-dimensional and that encounters
between thalli result in one growing over the other
(Fig. 5) but without explicit representation of the third
dimension. Using this two-dimensional approximation,
the area of overlap between growth increments is
subtracted from the area held by the overgrown
increment. Since rectangular growth increments can
potentially overlap each other in many different ways,
the algorithm to calculate the area of overlap, though
Per-Capita Rates (1/day)
‘‘progeny’’—surviving increments that grew directly
from the progenitor increment—and the identity
number of the given increment’s direct progenitor
must be retained so that changing characteristics of
ramets can be tracked and used to drive the dynamics.
Area held by each increment is continually reduced by
partial or complete overgrowth. A timer tracks duration-limited physiological processes, such as gemma
production by a cupule, located at the site of a former
mericell.
To summarize the March model’s depiction of
Marchantia demography, thallus interactions and environmental constraints are expressed through the states
and behavior of growth increments, which, taken
together, determine the characteristics and behavior of
individual ramets and entire genets. The most important
of the relationships underlying these main features are
described in more detail in the next section.
29
BIFURCATION
P
REPRODUCTION
G =E / σ
GROWTH
P
=
1 + Pσ / r
0
Support Ratio, σ
Fig. 4. Key rates for ramets and their dependence on the support ratio s (cost-decremented area held per growth tip). (a) Higher support ratios
increase the extension rate E of thalli at each growth tip. At low support ratios, E is approximately proportional to s, with slope determined by the
productivity parameter P; toward higher support ratios, E increases with diminished returns toward an asymptotic maximum at r. The particular
function used to represent these relationships is commonly employed to express response rates that depend on resource abundance, as in
Michaelis–Menten enzyme kinetics (Michaelis and Menten, 1913) and the type 2 functional response of predators to prey density (Holling, 1959). (b)
Per-capita growth rate G decreases with s as ramet spreading becomes growth-tip limited, and the remainder of the productivity is expended on
reproduction, when this is seasonally appropriate. The rate at which growth tips bifurcate is taken to be directly proportional to the support ratio.
The declining per-capita growth rate and increasing per-capita bifurcation rate tend to stabilize the support ratio.
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30
I
3.2. Reproduction
L2
II
D2
L1
D1
III
IV
D2
L1
L2
L1
D1
∋
θ
L2
L2
V
θ
D2
L1
D1
Fig. 5. Determining which of two growing thalli overgrows the other,
depending on the spatial relationship of their rectangular growth
increments. Growth increments are based on a fixed width w but may
vary in length. For each increment here, each thallus begins growing at
the beginning of the time step along the bold line (the ‘‘base’’ of the
increment) and reaches the other end of the rectangular increment at
the end of the time step. Extension rate during the step is assumed to
be constant, and the thallus that winds up on top is taken to be the one
growing fastest in the direction of the other at the point of contact.
This hinges in some cases on increment lengths L1 and L2, distances
from base to intersection point D1 and D2, and an angular measure of
relative growth direction y. There are five different categories of
overgrowth geometry (identified using Roman numerals in the figure,
next to an example) that can result, and each of these can arise in a
number of different ways (see Appendix A).
straightforward in principle, is rather complex to
implement as a computer program. Determining which
of two thalli overgrows the other can be more
challenging. We assume that the thallus growth increment advancing faster across the other’s boundary at the
initial point of contact obtains the top position (see Buss
and Jackson, 1979). This immediately implies that a
growing thallus always overtops a non-growing increment of another thallus. But if two different growing
tips encounter each other, then relative overgrowth
potential at the contact point depends on the absolute
growth rates, growth directions, and other geometrical
details (Fig. 5 and Appendix A; see Barnes and Dick,
2000 on the relation between the geometry of encounters
and overgrowth). These calculations to identify overgrower and overgrown require the majority of the
processing time when March is implemented as a
computer program.
During the part of the year in which asexual
reproductive structures (cupules or ‘‘cups’’) can be
formed (i.e. during the ‘‘asex season’’), the energy
budget allocated to reproduction is used to produce as
many new cups as possible from growing tips,
except those that bifurcate to form two tips. Cup
formation is linked to thallus extension (Parihar,
1956; Hollensen, 1981); in the model, a cup may
form at the end of a previous growth increment
where a new increment is being generated. There
is a lag from cup formation until the asexual propagules
(gemmae) are ready for release, and an interval and
rate of release by the cup (taken here to be constant),
after which the cup becomes inactive. During each
time step that the cup is releasing gemmae, a fixed
number (decremented to reflect the small chance of
successful germination) are dispersed in a random
direction for a normally distributed distance, having
its mode at zero and mean at 0.798 times the standard
deviation of the dispersal distance (i.e. at the mean
distance from the mode of the normal distribution).
Gemmae that land within the patch on unoccupied
substrate are assumed to germinate. If not overgrown
before achieving the minimum size for ‘‘emergence’’, a
gemma is then recognized as a ramet having the same
genotype as its progenitor and is tracked accordingly
thereafter.
During the sexually reproductive season, the part of
the energy budget not allocated to growth, metabolic
costs, or ongoing reproduction is available to form new
sex structures at tips with viable mericells. There is a lag
between structure formation and maturity, and what
follows depends on the ramet’s gender. Mature male
structures begin producing and dispersing sperm at a
constant rate over a fixed interval, after which the
structure becomes inactive.
Within each time step, for each receptive female
structure, sperm-producing male structures are randomly ordered. Each sperm-producing male structure in
turn then has a probability of fertilizing the female
structure that is a Gaussian function of the distance
between the two structures, with a maximum at distance
zero. This sequence of fertilization attempts continues
until the female structure is fertilized or until all male
structures have taken a turn.
Female structures are receptive to fertilization over a
fixed number of time steps, after which, if not fertilized,
they become inactive. When fertilized, a maturation
interval begins, after which the female structure releases
a fixed number of spores during a single time step and
becomes inactive. Spore dispersal is normally distributed
with modal distance at zero and thus analogous to
gemma dispersal, except that the mean dispersal
distance is much longer for spores. Dispersing spores
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are equally likely to be male or female and are assumed
to initiate ramets of unique genotypes (genets) at
emergence.
3.3. Mortality
Individual increments and entire ramets can die in
either of two ways in the March model—through a
physical disturbance eliminating all increments
within a certain area of the patch, or by falling
into energetic deficit by being almost entirely covered.
There is no mortality directly linked to increment
age. Disturbances are an attempt to represent regions
of local desiccation or scouring by flood water as a
circle centered at a random point in the patch, with
the diameter drawn from the positive half of a
normal distribution centered at zero. Disturbances
are also assumed to be temporally independent.
Space freed by disturbance is then immediately available
for occupancy by adjacent thalli and germinating
propagules.
We assume a fixed metabolic cost per unit area of
living tissue, whether or not this tissue is covered by
other tissue. This implies that once a growth increment
becomes sufficiently covered, its area held becomes
inadequate to compensate for the metabolic cost. In
most of the runs considered here, we assume that
increments in energetic deficit die and are eliminated
from the model at the end of the step in which they
fall into deficit. Since there seems to be a lag of days
to weeks before tissue death from energetic deficit in
M. inflexa (D.N. McLetchie, unpublished observations),
immediate death is a simplification that may slightly
speed up the competitive dynamics in these runs, but
presumably without altering the outcome. When competing clones differ in the time lag from energetic deficit
to death, however, there can be substantial, ecologically
meaningful implications, as demonstrated for one
extreme case below.
Mortality resulting from disturbances and especially
from overgrowth helps prevent a continual increase to
unmanageable levels in the number of growth increments in the model, each with its substantial amount of
associated data (Fig. 3). A complication that arises with
mortality of growth increments, however, concerns the
need to keep track of which growth increments
constitute part of a single physiologically integrated
ramet, even as ramets fragment into smaller ramets. For
example in Fig. 3, the death of increment 3 creates three
new ramets—one containing increments 4, 6, 8, and 9;
another containing five and seven; and the other with
one and two. Recording which increments are tips
and which is the ‘‘base’’ of each intact ramet, along with
the data retained in association with each increment
(Fig. 3), permits the newly formed ramets and their tips
and bases to be determined by ‘‘tip tracing’’. When an
31
increment dies within a ramet, the ramet is tip-traced by
stepping systematically from each tip toward the base,
identifying dead increments and defining new bases
above them and tips below them. This flexible approach
avoids the necessity of retaining considerably more
information in association with each growth increment
and further complicating the redefinition of ramets
resulting from fragmentation (cf. Oborny and Kun,
2002).
4. Simulations
The March model was implemented as a computer
program in MATLAB 6 (see the flow chart, Fig. 6). The
program can illustrate spreading, overgrowth, and
reproduction by clones and disturbances in the patch
as a cartoon and can produce graphs summarizing the
dynamics. Because of the complexity of the geometry
and extensive bookkeeping required to implement the
model, runs typically lasted for hours to days on 2 GHz
microprocessors. Run time increases very rapidly with
patch size—as a power of the patch diameter between 3
and 4. The standard or default parameter set used in the
runs considered in the present analysis is presented in
Table 1.
Four analyses were conducted to address the goals of
the study: five runs to characterize the default condition
and consider whether eliminating the disturbance regime
or changing relative growth rates alters the dynamics of
competition between sexes; a series of runs constituting
a preliminary sensitivity analysis for some key parameters; two runs and related results from a much
simpler model to examine the March model’s relationship to classical differential-equation models of competition; and a run to evaluate the role of fragmentation in
overgrowth competition.
5. Results
Fig. 7 shows an example growth pattern produced at
day 225 from a default run of the March model, with
male and female genets and their reproductive structures
color-coded. Unlike in Fig. 1, disturbances remove
killed increments entirely, but otherwise the simulated
growth pattern is generally similar to the natural ones.
The thallus colors show that the genet distributions,
begun with 10 randomly distributed emerging ramets
per genet, were by this point strongly patchy yet
extensively interwoven. Runs of 10 years or longer were
usually required for one sex (consistently females) to
eliminate the other sex under default conditions, with
the 0.1 m2 patch size.
In the default run of Fig. 8, only a tiny part of
the patch area is still occupied by the soon-to-be
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32
INITIALIZE
TALLY & UPDATE
INCREMEN TS
NEXT
STEP
PROPAGULE
EMERGENCE
ASSESS COSTS
OF REPROD’N
DISTURBANCE
GROW & ADD
REPRODUCTION
GROWTH &
REPRODUCTION
LAST
RAMET?
NEXT
RAMET
RAMET LOOP
START
NO
YES
TIME-STEP LOOP
MORTALITY
NO
DEAD
INT’S?
NEXT
RAMET
YES
TIP-TRACE &
FIND RAMETS
PROPAGULE
GERMINATION
LAST
RAMET?
RAMET LOOP
OVERLAP
ADJUSTMENTS
NO
YES
FERTILIZATION
DRAW PATCH
NO
LAST
STEP?
YES
STOP
OUTPUT DATA
Fig. 6. Flow diagram of the MATLAB computer program used to implement and explore the March model. The simulations are based on a series of
time steps, during each of which new growth increments may be formed, others may be eliminated, and the demography of the entire patch is tracked.
The scheme on the right provides more detail on the growth and reproduction, and mortality components of the basic within-step sequence on the
left.
extinguished males after 10 years. Note here that
females had already gained a considerable advantage
in the first year as the patch filled, and the advantage
gradually widened, despite some abrupt declines in areas
held that are attributable to disturbances. The parameter values for growth and for reproductive costs of
males and females were set to be identical in the default
run, except that unfertilized female sex structures were
taken to be much less expensive to maintain than male
sex structures, which while viable are always actively
producing sperm (Stark et al., 2000). Removing this cost
difference in other runs (not shown) removed most or all
of this otherwise consistent advantage that females had
in competition with males.
In the middle panel of Fig. 8, note the large number of
generally small ramets formed by this intensive overgrowth and fragmentation process. In the bottom panel
are the numbers of propagules released (regardless of
where they landed). Asexual propagules (gemmae) were
released in rough proportion to area held by each genet,
whereas spore production by the patch may have been
as high or slightly higher with a moderately femalebiased sex ratio. Thus sperm were apparently dispersed
widely enough to be in adequate supply, even with less
area held by males and presumably fewer male than
female sex structures.
Our preliminary sensitivity analysis evaluated effects
of parameter changes by factors of two on the sex ratio
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33
Table 1
Parameters of the model
Definition
Default magnitude
Units
Basic spatial and temporal features
Time step duration
Length and width of (square) patch
Days from July 1 until seasonal sex-to-growth transition
Days from July 1 until seasonal growth-to-asex transition
Days from July 1 until seasonal asex-to-sex transition
15a
31.6b
0c
75c
210c
Days/step
cm
Days
Days
Days
Initial abundances
Number of male genets at the beginning of each run
Number of female genets at the beginning of each run
Number of emerging ramets/genet to start each run
1d
1d
10d
Genotype
Genotype
Ramets/genet
Growth, metabolism, and death
Maximum thallus extension rate, growth season
Maximum thallus extension rate, a sex season
Maximum thallus extension rate, sex season
Gross productivity coefficient
Split rate per support ratio
Standard deviation of Gaussian shift in growth direction
Thallus width
Baseline metabolic cost rate
Angle between thalli following split
Minimum support ratio for sprouting
Lag between overgrowth and death of a thallus module
0.020e
0.016f
0.012f
0.200g
0.040g
0.250g
0.500h
0.005i
0.698j
1g
0k
cm2/day/tip
cm2/day/tip
cm2/day/tip
1/day
Tips/cm2/day
Radians
cm
1/day
Radians
cm2/tip
Days
Reproduction (general)
Cost (as lost support area) per active sex structure (~ and #)
Germination-to-emergence lag for sex propagule (~ and #)
Area of growing thallus at emergence of ( via sex or asex)
Minimum area of a ramet for reproduction (sex or asex)
0.015l
45e
0.250m
0.250e
cm2/day
Days
cm2
cm2
Asexual reproduction
Cost (as lost support area) per asex structure
Lag from germination to emergence of asexual propagule
Standard deviation of Gaussian gemma dispersal distance
Duration of asexual propagule release from a cupule
Lag from cup formation to first asexual propagule release
Expected number of asex propagules germinating per cup
0.0015l
15e
30n
60e
0e
1o
cm2/day
Days
cm
Days
Days
Propagule
Sexual reproduction (females and spores)
Cost (as lost support area) per unfertilized ~ sex structure
Standard deviation of Gaussian spore dispersal distance
Interval when a female sex structure is fertilizable
Time from formation until ~ sex structure is fertilizable
Duration of spore release from fertilized ~ sex structure
Time from ~ structure fertilization until first spore release
Expected number of spores germinating per ~ structure
0.003l
300o
15e
0e
15e
60e
1o
cm2/day
cm
Days
Days
Days
Days
Spore
Sexual reproduction (males and sperm)
Interval of sperm release from male sex structure
Time from formation until # sex structure releases sperm
Chance that sperm from adjacent # would fertilize ~
Maximum standard deviation of sperm dispersal distance
Fraction of sx reached in 7 days
30e
15e
0.900e
2.00e
0.400p
Days
Days
Dimensionless
cm
Dimensionless
Disturbance
Time over which the chance of a disturbance is 1/2
Standard deviation of Gaussian disturbance radius
30p
4p
Days
cm
a
Chosen arbitrarily; if modified, then s should be modified in the same direction to maintain a similar growth pattern.
Corresponds to a patch area of approximately 0.1 m2, the maximum size at which individual runs can be completed in a few hours on a
microcomputer.
c
Based on field observations for Marchantia inflexa in Trinidad (see McLetchie et al. 2002).
d
Arbitrary but consistent with expectation of a few colonizing propagules at roughly 1:1 offspring sex ratio.
e
Rough estimate based on greenhouse data (D.N. McLetchie, unpublished) and, where necessary, discretized to fit a 15-day time step (see also
Ramsay and Berrie 1982; McLetchie 1996).
f
Rough guess based on the estimate of the growth-season maximum thallus extension rate.
g
Chosen to generate reasonable growth dynamics generally consistent with observed patterns.
h
Direct measurement; thallus width is consistently within the range 0.45–0.55 cm in this species in the greenhouse and the field (D.N. McLetchie,
unpublished).
i
Rough guess consistent with the estimate of the gross productivity coefficient.
j
Directly measured; mean of 124 split angles in the greenhouse (C. Stieha, unpublished).
k
In the absence of data, setting this to zero allows the computer program to run faster and has little or no effect on the outcome, relative to any
other fixed magnitude.
l
Rough guesses based on plausible relative magnitudes.
m
Arbitrarily chosen size above which origin from sexual vs. asexual propagules is ignored.
n
Rough guess, based on water transport of these hydrophobic asexual propagules.
o
Rough guess (cf. McLetchie et al., 2002).
p
Rough guess (D.N. McLetchie and P.H. Crowley, unpublished).
b
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P.H. Crowley et al. / Journal of Theoretical Biology 233 (2005) 25–42
Fig. 7. Results of a default simulation in a 0.1 m2 patch observed at day
225 of year 1, early in the sexual reproductive season. Male ramets are
shown in blue and females in green. The black diamond-shaped dots
represent former sites of asexual reproduction; purple dots are sites of
active asexual reproduction continuing from the recently ended asexual
reproductive season. Black stars are male sexual structures; red
asterisks are unfertilized female sexual structures; and yellow asterisks
are fertilized female sexual structures. There are small areas of substrate
not yet occupied by ramets at the lower corners of the patch. The light
circular areas of substrate are sites of disturbances that are now
partially overgrown. Though there is no reproduction in progress in
Fig. 1, there are obvious similarities between that figure and this one.
after 10 years. We only sketch these results here for the
strongest responses, as difficulties with the long run
times precluded replication. Sex ratio was considered
sensitive to a factor-of-two parameter shift if the
increase or the decrease (but not both) yielded a malebiased sex ratio (cf. the strongly female-biased default
run in Fig. 8). These shifts were implemented simultaneously for both sexes except as noted. Dividing all
seasonal maximum thallus extension rates (i.e. the
asymptote r in Fig. 4a) by two led to the elimination
of females within 10 years; this parameter shift
corresponds to growth rate inhibition attributable to a
reduction in limiting resources, but field data published
to date fail to show more males at lower light irradiance
(Fuselier and McLetchie, in press), as might be expected
from this result. Moreover, males are strongly favored
when the gross productivity coefficient (P, Fig. 4),
expressing the efficiency with which a given support
ratio produces thallus growth, is increased; increasing
this coefficient shifts more of the available energy
toward reproduction relative to growth, helping to
relieve high male costs of sexual reproduction. Though
sex ratio appears to be insensitive to costs of baseline
metabolism or of gamete-producing sex structures, the
sex ratio favors males when the cost of asexual
reproduction is halved. This result is consistent with
the greater investment in asexual reproduction and thus
the greater benefit from reduced cost for males than for
females (McLetchie and Puterbaugh, 2000); but the
Fig. 8. Results of a default run for a 0.1 m2 patch (female=green, male=blue) and based on the parameter values in Table 1. Females gain a
coverage advantage during the first year (i.e. the first 24 time steps) as the patch fills up. Males are almost completely eliminated after 10 years, the
end of the run. Note the impact of disturbances in reducing area held at various points. As overgrowth intensifies early in the run, the total number of
ramets increases sharply and then fluctuates around 1500. Annual cycles of asexual propagule (gemma) production by each sex alternate with sexual
propagule (spore) production by females, shown in red. Gemma production by each sex is in general accord with relative coverage; because of
relatively efficient sperm transfer and fertilization in this small patch, spore production remains fairly high until males near extinction.
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model was not parameterized to include a difference in
investment between sexes, and this result must therefore
be considered with caution. Finally, it may initially seem
surprising that reducing the cost of unfertilized female
sex structures generates a male-biased sex ratio; here, the
lower cost induces females to produce more of these
structures and thus ultimately incur higher reproductive
costs at the expense of growth when they are fertilized.
Implications of some additional deviations from the
default parameter set are explored in Fig. 9. The top
panel shows that removing disturbance has very little
effect on the outcome or even in the basic dynamics. In
small patches with small-scale disturbances, like those
considered in this analysis, disturbances actually provide
little or no opportunity for propagules to emerge
and survive the onslaught from peripheral thalli
(cf. McLetchie et al., 2002; see also Watson, 1981;
Kimmerer, 1991). Because of this, the reduction in
35
maximum thallus extension rate of males depicted in the
second panel, though permitting additional asexual
reproduction in exchange for reduced growth, further
weakens the males’ ability to compete for space. If
instead, males are given a slight growth advantage
during the season of sexual reproduction (third panel),
this was enough to shift the balance completely in their
favor. Combining the parameter shifts from the second
and third panels in the run depicted in the bottom panel
generated a more balanced competition, though the
trend over the final 4 years of the run seems clearly to
favor females.
The basic patterns observed in Figs. 8 and 9 generally
resemble those resulting from two-species competition
equations similar to the classical Volterra equations in
the absence of stable coexistence (Volterra, 1926;
Hutchinson, 1978). The failure of a Volterra-type model
incorporating both seasonality and disturbances to yield
Fig. 9. Area held vs. time in four different simulations of the March model for a 0.1 m2 patch containing two genets (female=green, male=blue). In
all cases, runs are based on the default parameter set (Table 1), except as indicated below. (a) Here there were no environmental disturbances,
resulting in smoother graphs but with trends similar to those in Fig. 8, in which disturbances were present. (b) In this run, the maximum thallus
extension rate for males during the asexual reproductive season was reduced from 0.016 to 0.015 cm2/day/tip, slightly reducing thallus growth but
allowing a slight compensatory increase in asexual reproduction. Here, males were completely eliminated during year 7. (c) When the maximum
thallus extension rate for males during the sexual reproductive season was increased from 0.012 to 0.013 cm2/day/tip, males soon dominated the
patch, and females appeared likely to face eventual elimination. (d) When the maximum thallus extension rate for males was both reduced from 0.016
to 0.015 cm2/day/tip in the asexual reproductive season and increased from 0.012 to 0.013 cm2/day/tip in the sexual reproductive season, combining
the changes made in (b) and (c), results were somewhat more ambiguous, though females seemed to be gaining the advantage by the end of the run.
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P.H. Crowley et al. / Journal of Theoretical Biology 233 (2005) 25–42
coexistence in previous work (McLetchie et al., 2002)
had led us to expect this outcome, particularly in
patches too small for frequent propagule germination.
The resemblance to the outcome for the simple
two-species system is considered more systematically in
Fig. 10. Handicapping males by reducing their growth
rates slightly made the patterns from 2-year March runs
somewhat easier to visualize (first two panels). This
disadvantage is obviously accelerating the elimination of
males relative to default dynamics (Fig. 8). The more
erratic pattern in the top panel of Fig. 10 was smoothed
out considerably in the middle panel, an otherwise
identical run with disturbances and reproductive seasons
eliminated. These smoothed curves generate a highly
repeatable pattern in which overgrowth proceeds more
rapidly and the males are almost eliminated at the end of
the run. The bottom panel was obtained by constructing
coupled linear ordinary differential equations mathematically similar to the Volterra equations, but based on
the specifics of space-limited growth (see the legend of
Fig. 10 and Crowley et al., in press for details; also see
McLetchie et al., 2002). Dynamics of the coupled linear
differential equations, with parameters chosen to match
those of the non-seasonal, no-disturbance March model
Fig. 10. Comparing clonal dynamics in the March model to a linear model of two-clone overgrowth competition that is mathematically equivalent to
a two-species competition model closely related to the classical Volterra model. (a) Results of a 2-year March run under default conditions, except
that the maximum thallus extension rates for males (blue line) were reduced to 80% of their default values. (b) Another 2-year March run identical to
the one above, except that there are no disturbances or reproductive seasons. (c) Results of simulating two linear, coupled ordinary differential
equations, namely df 1 =f 1 dt ¼ r1 ð1 f 1 f 2 Þ þ ðr1 r2 Þf 2 d 1 and df 2 =f 2 dt ¼ r2 ð1 f 1 f 2 Þ þ ðr2 r1 Þf 1 d 2 where fi is the fraction of the
patch area held by clone i, ri is the per-capita expansion rate, and di is the per-capita death rate. In each equation, the first term on the right-hand side
represents expansion into unoccupied space, the second term is net overgrowth, and the third term is disturbance-induced mortality (Crowley et al., in
press). For this simulation, r1=0.3, r2=0.33, d1=d2=0 (i.e. no disturbances), f1 and f2 were initialized at 0.004 (corresponding to starting conditions
for the March runs), and both areas held and time were rescaled to match the March runs. The ri values were chosen to produce a pattern resembling
the previous panel; their absolute magnitudes are arbitrary under temporal scaling, and their relative magnitudes are consistent with plausible relative
per-capita increase rates in corresponding March run.
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as closely as possible, did achieve a strong resemblance,
though with noticeable differences. The most obvious
differences were that the sexes in the March model
diverged more slowly in coverage as the patch filled but
then closed in more rapidly on extinction of the loser—
in comparison to results for the differential-equation
model parameterized to mimic the initial growth phase
of the March run. We provide an interpretation of this
result in the Discussion.
To test directly whether fragmentation was advantageous in overgrowth competition as represented by the
March model, we conducted a run that included a
fragmenting genet (blue) and a non-fragmenting genet
(green) in the absence of reproductive seasons and
disturbance (Fig. 11). In the top panel, the nonfragmenter gained an advantage in area held as the
patch filled during the first year, at least partly by not
eliminating increments that were mostly but not entirely
overgrown. (Other factors contributing to this difference
are the focus of work in progress based on this model.
During the second year (i.e. between time steps 24 and
49), the non-fragmenting ramets apparently fell below
the compensation point and were completely overgrown
by ramets from the fragmenting clone. The bottom
panel confirms the absence of fragmentation in the nonfragmenting clone (green) and the presence of fragmentation, with subsequent accumulation of large numbers
of ramets, in the fragmenting clone (blue). The number
of fragmenter ramets increased rapidly around the time
37
of patch filling but soon leveled off during an interval in
which new ramets were being overgrown about as
rapidly as they were formed. In the second year, the
number of ramets from the fragmenting clone began to
increase again and overwhelm the increasingly moribund non-fragmenting ramets.
6. Discussion
We have formulated and implemented the SESIB
patch model March and explored some of its major
features. Under standard (default) conditions that did
not include any growth-rate advantage per se, female M.
inflexa consistently vanquished males in overgrowth
competition and dominated the 0.1 m2 patches convincingly within a decade. This is consistent with infrequent-disturbance results obtained by McLetchie et al.
(2002) for somewhat larger patches using a spatially
implicit model of patch dynamics. As previously
postulated (McLetchie et al., 2002), the female advantage seems to result primarily from the higher cost of
mature male sex structures than mature but unfertilized
female sex structures. Results for sexual investment in a
moss are generally consistent with this interpretation
(Stark et al., 2000); other recent estimates of the costs of
sexual reproduction in bryophytes (Bisang and Ehrlen,
2002; Rydgren and Okland, 2002) unfortunately focus
exclusively on females.
Fig. 11. Competition between a clone (blue) that fragmented by immediately eliminating its overgrown increments falling below the compensation
point and a clone (green) that did not fragment. These results are for a 0.1 m2 patch in the absence of disturbances or reproductive seasons, with all
other parameters at default values.
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P.H. Crowley et al. / Journal of Theoretical Biology 233 (2005) 25–42
In the small patches largely invulnerable to intrapatch propagule establishment that are the focus of
the present analysis, spatially and temporally random
disturbances have little effect on the dynamics or
outcome of competitive interactions between genders,
other than the stochastic variation introduced by
these occasional die-backs. In contrast, even shifts
of a few percent in relative growth rates between sexes
in any of the non-reproductive or reproductive growth
seasons were enough to shift the advantage decisively.
The male advantage found by McLetchie et al. (2002)
with a spatially implicit model at high-disturbance
frequencies reflected increased emergence success of
asexual propagules produced in larger numbers by
males. But when the high vulnerability of these new
ramets, obvious in the spatially explicit March model, is
taken into account, this advantage largely disappears.
Large disturbances in especially large patches (say
X10 m2) may provide an exception not investigated
here, in which reproductive propagules can at least
occasionally germinate and establish sufficiently to
withstand overgrowth pressure from the disturbance
periphery.
In the absence of disturbance and seasons, the
dynamics of space coverage by male and female genets
in the March model were fairly well mimicked by a pair
of coupled linear ordinary differential equations that are
mathematically similar to the classical Volterra equations. These equations amount to a greatly simplified
version of the McLetchie et al. (2002) model—without
disturbance, reproduction, or stage structure (see
Crowley et al., in press). But the rate of overgrowth is
slower in the March model while the two sexes are
roughly equal in coverage than in the differentialequation models, because the genet patchiness inherent
in the March model (and presumably in Marchantia
distributions in nature) generates more within-genet
overgrowth. Once one genet is heavily dominant,
however, the other genet is likely to be fragmented into
small, weakly growing ramets surrounded by more
vigorous competitors that may overwhelm them relatively rapidly. These more subtle spatial effects are
difficult or impossible to capture realistically in an
analytical model, providing one rationale for spatially
explicit models like March. Nevertheless, the basic
similarities in dynamics resulting from the two approaches suggest that linear differential-equation systems may be adequate to represent within-patch
dynamics in the context of multi-patch metapopulation
models.
The key conclusion of the McLetchie et al. (2002)
analysis, that coexistence of male and female M. inflexa
in individual patches is only temporary, is consistently
upheld by results for the much more detailed and
realistic representation of overgrowth competition in the
March model. Work in progress focuses more inten-
sively on the coupled two-equation system and some
ecologically meaningful relatives to probe the boundaries of this generalization (Crowley et al., in press).
Other studies are addressing multi-patch systems to
account for long-term coexistence of the sexes through
metapopulation structure and dynamics (Garcı́aRamos et al., in preparation; see also Crowley and
McLetchie, 2002).
The direct competition between fragmenting and nonfragmenting genets demonstrated that non-fragmenters
may achieve a temporary advantage with initial patch
filling (see also Fig. 2 in Oborny et al., 2000), but the
near certainty that non-fragmenting ramets must
eventually fall into energetic deficit by being overgrown
(as in the example run) dooms them to highly probable
elimination. This means that species engaging commonly in overgrowth competition should be capable of
fragmentation, modular abscission, or sloughing of
overgrown tissue. In fact, the March model can be used
to understand the fragmentation process more thoroughly, accounting for the consistent delay in the
sloughing of overgrown tissue by Marchantia and for
the initial advantage of the non-fragmenter in competition with the fragmenter.
Harper and Bell (1979); Bell (1984), and Sutherland
(1990) noted the considerable but largely untapped
potential for simulation models to improve our understanding of the spatial distribution and dynamics of
clonal plant populations. By spreading across space and
separating into independent but genetically identical
parts through asexual propagules or via abscission or
fragmentation, these intriguing organisms raise important questions about spatial ecology and evolution.
Some of the relevant modeling work to date has used a
deterministic (e.g. Room, 1983; Callaghan et al., 1990)
or stochastic (e.g. Remphrey et al., 1983; Callaghan et
al., 1990; Kron and Stewart, 1994) approach to
characterize branching architecture and speculate about
how particular growth ‘‘rules’’ and geometries might be
adaptive. Other studies have specifically addressed
growth form and dynamics as a solution to the problem
of plant foraging across a spatially heterogeneous
resource distribution (e.g. Sutherland and Stillman,
1988; Oborny, 1994; Cain et al., 1996) or of contending
with disturbances distributed randomly in space and
time (Inghe, 1989).
More recent work has begun to evaluate the adaptive
significance of clonal integration (Caraco and Kelly,
1991; Oborny and Kun, 2002) and the implications of
overgrowth competition (McLetchie et al., 2002; Crowley and McLetchie, 2002). Spatially implicit (or pseudospatial) models have permitted spatially distributed
dynamics to be tractably approximated in some cases
(Cain, 1994; McLetchie et al., 2002; Crowley and
McLetchie, 2002). Two-dimensional cellular automata
methodology (Wolfram, 1984), based on variable-state
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nodes in discrete time and space, have been used
successfully to evaluate spatial strategies (e.g. Inghe,
1989; Oborny and Kun, 2002) from a spatially explicit
perspective. Relative to these previous studies and
approaches, the March model provides a particularly
high level of biological realism and spatial resolution as
a spatially continuous representation of local ecological
dynamics in a clonal system.
Competition for space, particularly where holding
space provides access to some demonstrably limiting
resource, is obviously common in nature. Overgrowth
competition is a type of interference that in
effect maintains distinct boundaries separating regions
held by one sessile competitor from those held by
others. Moreover, the means of shifting these boundaries involves the physical expansion of one competitor
such that the other is locally excluded from the
resource supply. This is a process of overtopping in
the typical case when the resource is light or food
particles streaming down from above, though other
geometries having the same basic effect can be imagined,
such as mussels and barnacles that gradually exert
enough lateral force to crush and remove immediate
neighbors (Connell, 1961). In the absence of spaceclearing disturbances, this scenario boils things down to
which competitor can advance the frontier in its
favor, once the patch fills up, a recipe for the eventual
elimination of one or the other in the absence of
frequency dependence. Disturbances, though, allow for
colonization by propagules and uncontested growth
into cleared spaces (spatial exploitation) to become
important (Airoldi, 2000, and references therein). We
emphasize that the patch model of McLetchie
et al. (2002), the March model, and the differential
equations of the Fig. 10 legend all assume that both
uncontested growth and overgrowth are based on the
same growth rates (the ri of the Fig. 10 legend). Since
modifying that assumption in particular ways can
permit stable coexistence of competitors (Crowley et
al., in press), we are currently planning empirical work
to determine whether this mechanism might apply to the
M. inflexa system.
M. inflexa, along with other taxa with separate sexes
that engage in intense clonal overgrowth competition,
provide an unusually clear-cut and extreme example of
male–female conflict (Parker, 1979; Westneat and
Sargent, 1996). Here, natural selection seems to overwhelm sexual selection: the complete loss of access to the
other sex as a result of its local extirpation prevents
sexual reproduction, perhaps (as in Marchantia) without
even eliminating the significant costs of sexual expression. Under the assumptions of the March model,
male–female conflict is ameliorated during the sex
season and even during the asex season, because growth
is assumed to be reduced somewhat to permit significant
amounts of energy to be expended on reproduction.
39
Nevertheless, for one sex or the other (usually males in
M. inflexa) this seems to confer only a temporary stay of
execution, and the fugitive sex seems ultimately to
depend on metapopulation dynamics for its long-term
persistence (McLetchie et al., 2002). Since metapopulations generally contain patches at different stages of
colonization, substrate coverage, and disturbance, even
a few sexually reproducing 2-sex patches may provide
enough spores for both sexes to reach unoccupied
habitat and thus persist.
Gender-specific detection ability has to our knowledge not been demonstrated in bryophytes. But if males
and females could detect each other on thallus contact,
avoiding overgrowth of the opposite sex would nevertheless seem unlikely to be evolutionarily stable, since a
mutant without such avoidance would benefit in
competition for space. If males could detect each other
on thallus contact, both natural and sexual selection
would favor an enhanced overgrowth effort or other
interference effect, but this could also prove to be
evolutionarily unstable if males could arise that were
able to foil detection by mimicking females. More
plausibly, if ramets were able to detect the absence of the
opposite sex (e.g. females cued hormonally that sex
structures remain unfertilized), at least some expenditures could be shifted away from sex expression to asex
(as an alternative means of colonizing other patches) or
to growth. Of course, even without any such detection
ability, natural selection should act to reduce sexual
expression in favor of growth at the patch level; at the
metapopulation level, however, where spore dispersal
and patch colonization are paramount, natural selection
must favor sexuality. An interesting implication worth
pursuing is that the loss of sex by populations may result
primarily from infrequent disturbances or insufficiently
severe disturbances; a benign environment may not
conduce to the long-term maintenance of sexual
reproduction.
The results of this analysis have suggested a number
of issues already under study, including more thorough
consideration of fragmentation, detailed analysis of
simple differential-equation models of overgrowth, and
multi-patch metapopulation models of sex-ratio dynamics. Others worth addressing in future work are
empirical studies of thallus overgrowth and of the
relationship between extension rate and support ratio.
Also, the tradeoffs among growth, asex, and sex in the
March model need to be thoroughly investigated for
larger patches and related to our own empirical results
(McLetchie and Puterbaugh, 2000; D.N. McLetchie, in
preparation). Ultimately, understanding sex-ratio dynamics will hinge on being able to account for
how these life history tradeoffs play out in the multipatch systems characteristic of nature (see the conceptual approaches of Ronce et al., 2000 and Crowley and
McLetchie, 2002).
ARTICLE IN PRESS
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P.H. Crowley et al. / Journal of Theoretical Biology 233 (2005) 25–42
Acknowledgements
We thank Heinjo During and the UK Wort Group
(particularly Linda Fuselier, Gisela Garcı́a-Ramos,
Charles Richardson, Carey Snyder, and Nicole Sudler)
for helpful suggestions during this study and Gisela
Garcı́a-Ramos for insightful comments on an earlier
draft of the manuscript. PHC gratefully acknowledges
the hospitality of Isabelle Olivieri, Ophelie Ronce and
colleagues for hosting his visit to the Institute of
Evolution at the Université de Montpellier, France,
during May, 2002, where this work and many related
ideas were discussed to good effect. We acknowledge
support for this project from NSF grant DEB 9974086.
Appendix A. Which thallus overgrows the other?
Fig. 5 illustrates five general categories of overgrowth
geometry for increments of fixed width w and constant
extension rate during the interval of increment formation. There are several different ways or ‘‘cases’’ by
which each geometric category can arise. Distinguishing
the categories and determining the outcome requires the
algorithm to establish and compare growth orientations
(via the metric y) and lengths L1 and L2 of growth
increments. When sides of the two increments intersect,
the distances D1 and D2 from an intersection point to
the nearest base point of each increment may become
essential to this determination as well. (When there is
more than one side–side intersection, the one associated
with the earliest point of contact during the time step is
the relevant one.) To identify the two thalli and their
growth increments here, we arbitrarily refer to the
increment oriented directly toward the top of the figure
as 1, and the other increment as 2. When the two thalli
are growing in directions within 901 of each other, then
category I (20 distinguishable geometric patterns or
‘‘cases’’) or II (eight cases) applies. In this situation, if
one of the bases is intersected (indicative of category I),
then the increment having the intersected base is
overgrown (two in the example) and the other (1) is
on top. In this particular example, this is because 1
necessarily reaches the base of 2 at some point during
the time step and can simply overgrow it without
opposition. For category II (growth directions within
901 but neither base is crossed), whichever thallus
reaches the initial intersection point first will be overgrown by the later arriving thallus. If D1 =L1 oD2 =L2 ;
then 1 must reach this point first and be overgrown,
because a smaller fraction of the time step would have
passed for 1 to reach the point than for 2 to reach it;
otherwise, ignoring the special case in which the two
fractions are exactly equal, 1 overgrows 2. When the two
thalli are growing in directions that are not within 901 of
each other, then category III (20 cases), IV (five cases),
or V (18 cases) applies. In this situation, the absence of
side–side intersection of the growth increments
implies category III, as does a side–side intersection
in which there is overlap of growth increments
between both closest base corners and the intersection
point (as in the example). Here, an increment in
position 1 would overgrow 2 if and only if
L1 cos y4L2 : This is because, at the point of first
contact during the step, 2 is growing straight over 1,
whereas the vector component of 1’s growth in the
direction of 2 is diminished by the multiplicative factor
cos y. Category IV results from a side–side intersection
in which there is overlap between one closest base point
(on 1 in the example) and the intersection point—but
not between the other base point (on 2) and the
intersection point. In the diagram, 1 overgrows 2 if
and only if D1 =L1 4D2 =L2 and L1 cos y4L2 ; following
logic similar to that for categories II and III. Finally,
category V involves a side–side intersection in which
there is overlap between neither closest base corner
and the intersection point. Here, there are two ways 1
could wind up on top: either {D1 =L1 4D2 =L2 and
L1 4L2 cos y} or {D2 =L2 4D1 =L1 and L1 cos y4L2 }.
Otherwise, 2 is on top.
References
Airoldi, L., 2000. Effects of disturbance, life histories, and overgrowth
on coexistence of algal crusts and turfs. Ecology 81, 798–814.
Amsler, C.D., 1984. Culture and field studies of Acinetospora crinita
(Carmichael) Sauvageau (Ectocarpaceae, Phaeophyceae) in North
Carolina, USA. Phycologia 23, 377–382.
Armstrong, R.A., 1979. Growth and regeneration of lichen thalli with
the central portions artificially removed. Environ. Exp. Bot. 19,
175–178.
Armstrong, R., 2002. The effect of rock surface aspect on growth, size
structure and competition in the lichen Rhizocarpon geographicum.
Environ. Exp. Bot. 48, 187–194.
Bailey, R.H., 1975. Ecological aspects of dispersal and establishment
in lichens. In: Brown, D.H., Hawksworth, D.L., Bailey, R.H.
(Eds.), Lichenology: Progress and Problems. Academic Press,
London, UK.
Barnes, D.K.A., Dick, M.H., 2000. Overgrowth competition in
encrusting bryozoan assemblages of the intertidal and infralittoral
zones of Alaska. Mar. Biol. 136, 813–822.
Bell, A.D., 1984. Dynamic morphology: a contribution to
plant population ecology. In: Dirzo, R., Shanikhan, J. (Eds.),
Perspectives on Plant Population Ecology. Sinauer Associates,
Sunderland, MA.
Bisang, I., Ehrlen, J., 2002. Reproductive effort and cost of sexual
reproduction in female Dicranum polysetum. Bryologist 105,
384–397.
Bischler, H., 1984. Marchantia L. The New World species. Bryophytorum Biobliotheca 26.
Bischler, H., 1986. Marchantia polymorpha L. s. lat. Karyotype
analysis. J. Hattori Bot. Lab. 60, 105–117.
Bowker, M.A., Stark, L.R., McLetchie, D.N., Mishler, B.D., 2000. Sex
expression, skewed sex ratios, and microhabitat distribution in
the dioecious desert moss Syntrichia caninervis (Pottiaceae). Am.
J. Bot. 87, 517–526.
ARTICLE IN PRESS
P.H. Crowley et al. / Journal of Theoretical Biology 233 (2005) 25–42
Bruno, J.F., 1998. Fragmentation in Madracis mirabilis (Duchassaing
and Michelotti): how common is size-specific fragment survivorship in corals? J. Exp. Mar. Biol. Ecol. 230, 169–181.
Buss, L.W., Jackson, J.B.C., 1979. Competitive networks: nontransitive competitive relationships in cryptic coral reef environments.
Am. Nat. 113, 223–234.
Cain, M.L., 1994. Consequences of foraging in clonal plant species.
Ecology 75, 933–944.
Cain, M.L., Dudle, D.A., Evans, J.P., 1996. Spatial models of foraging
in clonal plant species. Am. J. Bot. 83, 76–85.
Caldwell, M.M., Pearcy, R.W. (Eds.), 1994, Exploitation of Environmental Heterogeneity by Plants. Academic Press, Inc., San Diego,
CA.
Callaghan, T.V., Svensson, B.M., Bowman, H., Lindley, D.K.,
Carlsson, B.A., 1990. Models of clonal plant growth
based on population dynamics and architecture. Oikos 57,
257–269.
Caraco, T., Kelly, C.K., 1991. On the adaptive value of physiological
integration in clonal plants. Ecology 72, 81–93.
Ceccherelli, G., Cinelli, F., 1999. The role of vegetative fragmentation
in dispersal of the invasive alga Caulerpa taxifolia in the
Mediterranean. Marine Ecol. Progr. Ser. 182, 299–303.
Connell, J.H., 1961. The influence of interspecific competition and
other factors on the distribution of the barnacle Chthalamus
stellatus. Ecology 42, 710–723.
Crowley, P.H., Davis, H.M., Ensminger, A.L., Fuselier, L.C., Jackson,
J.K., McLetchie, N.M., 2005. A general model of local competition
for space. J. Theor. Biol., in press.
Crowley, P.H., McLetchie, D.N., 2002. Trade-offs and spatial lifehistory strategies in classical metapopulations. Am. Nat. 159,
190–208.
During, H.J., 1990. Clonal growth patterns among bryophytes.
In: van Groenendael, J.deKroon,H. (Ed.), Clonal Growth in
Plants. SPB Academic Publishing, The Hague, The Netherlands,
pp. 153–176.
Ewanchuk, P.J., Williams, S.L., 1996. Survival and re-establishment of
vegetative fragments of eelgrass (Zostera marina). Can. J. Bot. 74,
1584–1590.
Fuselier, L., McLetchie, D.N., 2004. Microhabitat and sex distribution
in Marchantia inflexa, a dioicous liverwort. Bryologist 107,
345–356.
Gourbiere, F., van Maanen, A., Debouzie, D., 2001. Associations
between three fungi on pine needles and their variation along a
climatic gradient. Mycol. Res. 105, 1101–1109.
Hale, M.E., 1974. The Biology of Lichens. Arnold, London, UK.
Hanski, I., 1998. Metapopulation dynamics. Nature 396, 41–49.
Hanski, I., 1999. Metapopulation Ecology. Oxford University Press,
Oxford, UK.
Harper, J.L., Bell, A.D., 1979. The population dynamics of growth
form in organisms with modular construction. In: Anderson, R.M.,
Turner, B.D., Taylor, L.R. (Eds.), Population Dynamics. Blackwell, Oxford, UK.
Hestmark, G., Schroeter, B., Kappen, L., 1997. Intrathalline and sizedependent patterns of activity in Lasallia pustulata and their
possible consequences for competitive interactions. Funct. Ecol. 11,
318–322.
Hollensen, R.H., 1981. A gemmiparous population of Marchantia
polymorpha var. aquatica in Cheboygan county, Michigan.
Michigan Bot. 20, 189–191.
Holling, C.S., 1959. Some characteristics of simple types of predation
and parasitism. Can. Entomol. 91, 385–398.
Hooper, R.G., Henry, E.C., Kuhlenkamp, R., 1988. Phaeosiphoniella
cryophila gen. et sp. nov., a third member of the Tilopteridales
(Phaeophyceae). Phycologia 27, 395–404.
Hutchinson, G.E., 1978. An Introduction to Population Ecology. Yale
University Press, New Haven, CN.
41
Inghe, O., 1989. Genet and ramet survivorship under different
mortality regimes—a cellular automata model. J. Theor. Biol.
138, 257–270.
Jarosz, J., 1996. Do antibodies and compounds produced in vitro by
Xenorhabdus nematophilus minimize the secondary invasion of
insect carcasses by contaminating bacteria? Nematologica 42,
367–377.
Jarosz, J., Kania, G., 2000. The question of whether gut microflora of
the millipede Ommatoiulus sabulosus could function as a threshold
to food infections. Pedobiologia 44, 705–708.
Jompa, J., McCook, L.J., 2002. Effects of competition and herbivory
on interactions between a hard coral and a brown alga. J. Exp.
Mar. Biol. Ecol. 271, 25–39.
Kimmerer, R.W., 1991. Reproductive ecology of Tetraphis pellucida II.
Differential fitness of sexual and asexual propagules. Bryologist 94,
284–288.
Kron, P., Stewart, S.C., 1994. Variability in the expression of a
rhizome architecture model in a natural population of Iris
versicolor (Iridaceae). Am. J. Bot. 81, 1128–1138.
Lasker, H.R., 1990. Clonal propagation and population dynamics of a
gorgonian coral. Ecology 71, 1578–1589.
Leslie, J.F., Klein, K.K., 1996. Female fertility and mating type effects
on effective population size and evolution in filamentous fungi.
Genetics 144, 557–567.
Lirman, D., 2001. Competition between macroalgae and corals: effects
of herbivore exclusion and increased algal biomass on coral
survivorship and growth. Coral Reefs 19, 392–399.
Matlack, G.R., 2002. Exotic plant species in Mississippi, USA: critical
issues in management and research. Nat. Areas J. 22, 241–247.
McCook, L.J., Jompa, J., Diaz-Pulido, G., 2001. Competition between
corals and algae on coral reefs: a review of evidence and
mechanisms. Coral Reefs 19, 400–417.
McLetchie, D.N., 1996. Sperm limitation and genetic effects on
fecundity in the dioecious liverwort Sphaerocarpos texanus. Sex.
Plant Reprod. 9, 87–92.
McLetchie, D.N., Puterbaugh, M.N., 2000. Population sex ratios, sexspecific clonal traits and tradeoffs among these traits in the
liverwort, Marchantia inflexa. Oikos 90, 227–237.
McLetchie, D.N., Garcı́a-Ramos, G., Crowley, P.H., 2002. Local sexratio dynamics: a model for the dioecious liverwort Marchantia
inflexa. Evol. Ecol. 15, 231–254.
Michaelis, L., Menten, M.L., 1913. Die Kinetik der Invertinwirkung.
Biochem. Z. 49, 333–369.
Newton, A.E., Mishler, B.D., 1994. The evolutionary significance of
asexual reproduction in mosses. J. Hattori Bot. Lab. 76, 127–145.
Oborny, B., 1991. Criticisms on optimal foraging in plants: a review.
Abstr. Bot. 15, 67–76.
Oborny, B., 1994. Growth rules in clonal plants and predictability of
the environment: a simulation study. J. Ecol. 76, 807–825.
Oborny, B., Kun, A., 2002. Fragmentation of clones: how
does it influence dispersal and competitive ability? Evol. Ecol. 15,
319–346.
Oborny, B., Kun, A., Czaran, T., Bokros, S., 2000. The effect of clonal
integration on plant competition for mosaic habitat space. Ecology
81, 3291–3304.
Outridge, P.M., Hutchinson, T.C., 1990. Effects of cadmium on
integration and resource allocation in the clonal fern Salvinia
molesta. Oecologia 84, 215–223.
Parihar, N.S., 1956. Bryophyta. Indian Universities Press, Allahabad,
India.
Parker, G.A., 1979. Sexual selection and sexual conflict. In: Blum,
M.S., Blum, N.A. (Eds.), Sexual Selection and Reproductive
Competition in Insects. Academic Press, New York, NY,
pp. 123–166.
Ramsay, H.P., Berrie, G.K., 1982. Sex determination in bryophytes.
J. Hattori Bot. Lab. 52, 255–274.
ARTICLE IN PRESS
42
P.H. Crowley et al. / Journal of Theoretical Biology 233 (2005) 25–42
Remphrey, W.R., Neal, B.R., Steeves, T.A., 1983. The morphology and growth of Arctostaphylos uva-ursi (bearberry): an
architectural model simulating colonizing growth. Can. J. Bot.
61, 2451–2458.
Ronce, O., Perret, F., Olivieri, I., 2000. Landscape dynamics and
evolution of colonist syndromes: interactions between reproductive
effort and dispersal in a metapopulation. Evol. Ecol. 14, 233–260.
Room, P.M., 1983. ‘Falling apart’ as a lifestyle: the rhizome
architecture and population growth of Salvinia molesta. J. Ecol.
71, 349–365.
Rydgren, K., Okland, R.H., 2002. Ultimate cost of sporophyte
production in the clonal moss Hylocomium splendens. Ecology 83,
1573–1579.
Sánchez, J.A., Lasker, H.R., Nepomuceno, E.G., Sánchez, J.D.,
Woldenberg, M.J., 2004. Branching and self-organization in
marine modular colonial organisms: a model. Am. Nat. 163,
E24–E39.
Schuster, R.M., 1992. Volume VI. The Hepaticae and Anthocerotae of
North America. Field Museum of Naural History, Chicago, IL.
Stark, L.R., Mishler, B.D., McLetchie, D.N., 2000. The cost of
realized sexual reproduction: assessing patterns of reproductive
allocation and sporophyte abortion in a desert moss. Am. J. Bot.
87, 1599–1608.
Sutherland, W.J., 1990. The response of plants to patchy environments. In: Shorrocks, B., Swingland, I.R. (Eds.), Living in a Patchy
Environment. Oxford University Press, Oxford, UK, pp. 45–61.
Sutherland, W.J., Stillman, R.A., 1988. The foraging tactics of plants.
Oikos 52, 239–244.
Volterra, V., 1926. Variazioni e fluttuazioni del numero d’individui in
specie animali conviventi. Mem. R. Acad. Naz. dei Lincei (ser. 6) 2,
31–113.
Watson, M.A., 1981. Chemically mediated interactions among juvenile
mosses as possible determinants of their community structure.
J. Chem. Ecol. 7, 367–376.
Westneat, D.F., Sargent, R.C., 1996. Sex and parenting: the effects of
sexual conflict and parentage on parental strategies. TREE 11,
87–91.
Wolfram, S., 1984. Universality and complexity in cellular automata.
Physica D 10, 1.
Zakai, D., Levy, O., Chawick-Furman, N.E., 2000. Experimental
fragmentation reduces sexual reproductive output by the
reef-building coral Pocillopora damicornis. Coral Reefs 19,
185–188.