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Ecological Applications, 19(2), 2009, pp. 387–397� 2009 by the Ecological Society of America
Complex population dynamics and control of the invasive biennialAlliaria petiolata (garlic mustard)
ELEANOR A. PARDINI,1,3 JOHN M. DRAKE,2 JONATHAN M. CHASE,1 AND TIFFANY M. KNIGHT1
1Department of Biology, Washington University, St. Louis, Missouri 63130 USA2Odum School of Ecology, University of Georgia, Athens, Georgia 30602 USA
Abstract. Controlling species invasions is a leading problem for applied ecology. Whilecontrolling populations expanding linearly or exponentially is straightforward, intervention insystems with complex dynamics can have complicated, and sometimes counterintuitive,consequences. Most invasive plant populations are stage-structured and density-dependent—a recipe for complex dynamics—and yet few population models have been created to explorethe effects of control efforts on such species. We examined the demography of the invasivebiennial plant Alliaria petiolata (garlic mustard) on the front of its spread into a natural areaand found evidence of strong density dependence in vital rates of first-year rosette andsecond-year adult stage classes. We parameterized a density-dependent, stage-structuredprojection model using field-collected data. This model produces two-point cycles withalternating years in which adults vs. rosettes are more prevalent. Such population dynamicsmatch observations in natural populations, suggesting that these complicated populationdynamics may result from deterministic rules. We used this model to evaluate simulatedmanagement strategies, including herbicide treatment of rosettes and clipping or pulling ofadult plants. Management of A. petiolata by inducing mortality of either rosettes or adultswill not be effective at reducing population density unless the induced mortality is very high(.95% for rosettes and .85% for adults) and repeated every year. Indeed, induced mortalityof rosettes can be counterproductive, causing increases in the stationary distribution of A.petiolata density. This species is typical of many invasive plants (stage-structured, short-lived,high fertility) and exhibits common forms of density dependence. Thus, the managementimplications of our study should apply broadly to other species with similar life histories. Wesuggest that management should focus on managing adults rather than rosettes, and oncreating efficient control in targeted areas of the population, rather than spreading lessefficient efforts widely.
Key words: Alliaria petiolata; density dependence; garlic mustard; invasive plant; management options;matrix population model; oak–hickory forest, St. Louis, Missouri, USA; population dynamics; populationgrowth rate; stage structure.
INTRODUCTION
Understanding the dynamics of natural populations is
one of the central problems in population ecology. Since
the seminal work of May (1974), extensive theoretical
and empirical research in population ecology has
demonstrated that simple deterministic rules, such as
stage structure and density dependence, can lead to
complex population dynamics, including regular cycles
and chaos (e.g., Dennis et al. 1995, Costantino et al.
1997, Cushing et al. 1998). Understanding whether the
observed fluctuations in population dynamics are caused
by deterministic rules is of practical significance since it
informs us about the mechanistic processes that
determine species numbers. For invasive species, under-
standing the underlying population dynamics allows one
to predict the amount of mortality that must be induced
in the population to achieve management goals (Hast-
ings and Botsford 2006, Whittle et al. 2007). Further,
population models can elucidate the parameter space in
which management would be ineffective or even
counterproductive (Buckley et al. 2001, Davis et al.
2006), which might help explain frustrating outcomes of
management efforts and help with the planning of
management so that goals can be achieved.
Invasive plant species seem particularly prone to
complex population dynamics since these organisms
exhibit multiple life stages and often achieve high
densities at which severe density dependence is expected.
For example, density-dependent fertility and survival
have been reported in numerous weedy plant species
(e.g., Palmblad 1968, Symonides 1983, Thrall et al. 1989,
Watkinson et al. 1989, Matthies 1990), and oscillatory
or cyclic plant-population dynamics expected to result
from such density dependence have been observed in
some plant populations (Symonides 1983, Symonides et
Manuscript received 5 May 2008; revised 15 May 2008;accepted 19 May 2008; final version received 12 June 2008.Corresponding Editor: R. A. Hufbauer.
3 E-mail: [email protected]
387
al. 1986, Crone and Taylor 1996, Gonzalez-Andujar et
al. 2006).
Density dependence in the underlying vital rates of an
invasive species can have implications for the manage-
ment of these species. Management may consist of
habitat alterations, physical removal, chemical applica-
tions, or introductions of biological control agents.
When density dependence occurs, management efforts
that result in mortality can be compensated for by
higher survival and/or fertility rates of remaining
individuals (Buckley et al. 2001). Indeed, when density
dependence is not properly understood, management
actions can be counterproductive (Buckley and Metcalf
2006). Several authors have called for a more thorough
understanding of density dependence at different life
stages (Mortimer et al. 1989, Watkinson et al. 1989,
Gillman et al. 1993, Gonzalez-Andujar and Hughes
2000). This is especially relevant for weedy invasive
species where control efforts are often implemented at
only one life-history stage (e.g., adult plants). To predict
the outcomes of these management actions, it is
necessary to create a model that exhibits the complex
dynamics observed in nature and relates these to the
underlying mechanisms of discrete stage structure and
density dependence.
Complex dynamics resulting from density-dependent
vital rates might explain the persistence of invasive plant
populations despite extensive control efforts in some
instances (Gonzalez-Andujar 1996). For example, Buck-
ley and colleagues (2001) demonstrated that density-
dependent flowering and fertility could produce damped
oscillations in the invasive plant Tripleurospermum
perforatum, and simulations revealed that, due to
overcompensatory density dependence, complete eradi-
cation of this plant would be extremely difficult. Here,
we create and parameterize a stage-structured and
density-dependent population model for an invasive
plant that is characterized by density dependence in
multiple life stages.
In this study we consider the exotic biennial Alliaria
petiolata (garlic mustard; see Plate 1), which invades
relatively pristine communities (Cavers et al. 1979,
Nuzzo 1999) and can achieve exceptionally high
densities (Anderson et al. 1996, Nuzzo 1999). We focus
on A. petiolata because it is a particularly noxious
invasive species in many areas in the United States.
Many states in the Midwestern and Northeastern
United States rank A. petiolata as a class A noxious
weed or prohibited invasive species (USDA NRCS 2007)
because of its ability to dominate the understory of
forested areas and reduce native-plant abundance
(Anderson et al. 1996, Nuzzo 1999, Stinson et al.
2007). The presence of density-dependent vital rates in
this species has been suggested in the literature, but
results have been mixed (Trimbur 1973, Meekins and
McCarthy 2000, 2002, Smith et al. 2003), and the
implications of density dependence on its population
dynamics, and therefore on management, have not
previously been investigated.
Effective management of A. petiolata includes remov-
ing rosette and/or adult individuals prior to seed
production every year until the seed bank is exhausted
(Baskin and Baskin 1992, Drayton and Primack 1999,
Nuzzo 2000). Management options include burning,
herbicides, clipping, and pulling, but none of these
control measures are 100% effective. Burning and
herbicide application generally occur in the fall or early
spring, and therefore focus on the rosette stage, whereas
clipping and pulling are usually employed on adult
plants. Prescribed burning has yielded inconsistent
results: repeated fires have been found to reduce
population size in some cases (Nuzzo 1991, Nuzzo et
al. 1996, Schwartz and Heim 1996), but low-intensity
fires may increase population size because fire may not
kill the root crowns of the plants and because fire
removes the litter layer, which may enhance seedling
survival (Nuzzo et al. 1996). Applications of herbicides
during the growing season have been shown to reduce
rosette cover by 90–95% (Nuzzo 1994). Cutting or
pulling adult A. petiolata plants can result in 71–99%
mortality (Nuzzo 1991). Cutting is most effective when
plants are in full bloom and/or have developed siliques;
plants cut earlier in the flowering period may have
sufficient resources to produce additional flower stems
from buds or crown roots (Pardini et al. 2008).
However, fruiting plants that are cut or pulled will
produce viable seeds, thus they must be removed from
the site to prevent seed input (Solis 1998, Pardini et al.
2008).
We collected empirical data on survivorship and
fertility of plants in different life stages and densities
from a recent incursion of A. petiolata into an oak–
hickory forest near St. Louis, Missouri, USA. We tested
for evidence of density dependence in two life-stages
(rosette and adult). We then used these data to
parameterize a density-dependent, stage-structured pro-
jection model to explore scenarios for population
dynamics of A. petiolata. The dynamic model yields
regular population cycles. To investigate potential
effectiveness of management, we determined the mor-
tality required at different life stages for effective
management of this species, defined as a stable
population at low density.
METHODS
Study species and site
Alliaria petiolata is an obligate biennial herb of the
mustard family (Brassicaceae) with a three-stage life
cycle (Fig. 1): seeds, rosettes, and adults (SRA). Seeds
germinate in early spring and form basal rosettes by
midsummer. Rosettes overwinter and then become adult
plants in the spring of their second year. Adult plants
flower, set seed, and subsequently die in midsummer.
Seeds require cold stratification over winter and then
ELEANOR A. PARDINI ET AL.388 Ecological ApplicationsVol. 19, No. 2
either germinate in the following spring or remain
dormant in the soil (Cavers et al. 1979).
Robust adult plants can produce hundreds of small,
white flowers. Flowers contain nectar, and are pollinated
primarily by small bees and flies. Flowers that are not
insect pollinated automatically self-pollinate, and seed
production is similar for both self-pollinated and cross-
pollinated flowers (Anderson et al. 1996, Cruden et al.
1996). Thus, Allee effects due to insufficient pollination
are unlikely for this species. The majority of seeds fall
within a few meters of the adult plant and germinate the
following spring. However, seeds can remain viable in
the seed bank for at least 10 years (Nuzzo 2000; V.
Nuzzo and B. Blossey, personal communication).
Our study was conducted on Washington University’s
Tyson Research Center (hereafter ‘‘Tyson’’), west of St.
Louis, Missouri, USA (388300 N, 908300 W). Tyson is a
;800-ha field station, approximately 85% of which is
second-growth oak (Quercus spp.) and hickory (Carya
spp.) deciduous forest. Alliaria petiolata likely first
invaded the site in 2000 and has continued to spread
despite annual control efforts (D. Larson and D.
Schilling, personal communication). Tyson provides an
ideal location for this study because it has only recently
been invaded by A. petiolata, and the area invaded has
relatively homogeneous environmental conditions so
that areas with low A. petiolata densities likely result
from dispersal limitation and not poor microsite
conditions.
SRA model for garlic mustard population dynamics
The stage-specific demography of A. petiolata de-
scribed above and in Fig. 1 can be expressed as a set of
difference equations for the abundance of seeds,
rosettes, and adults:
Stþ1 ¼ vð1� g1ÞAt f ðAtÞ þ Stð1� g2Þ
Rtþ1 ¼ vg1s1At f ðAtÞ þ g2s1St
Atþ1 ¼ Rts2ðRt;AtÞs3ðRtÞ
ð1Þ
where v is seed viability, g1 is the proportion of viable
seeds that germinate following a single winter stratifica-
tion, f is fertility, g2 is the proportion of viable seeds that
germinate out of the seed bank, s1 is the early
survivorship of recently germinated individuals until
early May, s2 is summer survivorship of rosettes between
early May and August, and s3 is winter survivorship of
rosettes between August and early May. Several vital
rates are represented as functions because they can be
density dependent: f(At) is the possibly density-depen-
dent per capita fertility of adults, s2(Rt, At) is the
possibly density-dependent summer survivorship of
rosettes and s3(Rt) is the possibly density-dependent
winter survivorship of rosettes. Note that although in
principle v, g1, g2, and s1 may be density dependent, we
have included these parameters as constants in our
model (see Demographic data, below).
Demographic data
We obtained estimates of seed survival and germina-
tion (used for parameters v, g1, and g2) from several
published and unpublished sources including studies
conducted in Illinois, Kentucky, and New York (Baskin
and Baskin 1992, Anderson et al. 1996, Davis et al. 2006;
V. Nuzzo and B. Blossey, unpublished data). To
parameterize our model, we used the mean values of
FIG. 1. Life cycle of Alliaria petiolata, where each arrow represents a one-year transition from May to May. Fertility, f, is theaverage number of seeds produced per reproductive plant, which can be a function of adult density. Seeds are viable at a rate v, andsurviving seeds either germinate the following year with probability g1 or enter the seed bank with probability 1� g1. Seeds in theseed bank either germinate or remain in the seed bank with probabilities of g2 and 1� g2, respectively. Seeds that germinate surviveto the rosette stage with probability s1. Rosettes survive from early May to August with probability s2 (which can be a function ofboth adult and rosette densities). Rosettes survive from August to early May with a probability s3 (which can only be a function ofrosette density). Drawing credit: Erin Mitchell.
March 2009 389DYNAMICS AND CONTROL OF GARLIC MUSTARD
the available estimates (v ¼ 0.8228; g1 ¼ 0.5503; g2 ¼0.3171). For more details about the source of these data
and the natural variation in these estimates, see
Appendix A. These values, particularly g1 and g2, vary
a great deal in nature, and therefore we explored the
sensitivity of our model results to these parameters using
bifurcation graphs to vary v over a range of 0.7–1.0, g1over a range of 0.1–0.9, and g2 over a range of 0.01–0.6
(see Appendix B ).
It is unknown whether seed viability declines over time
for seeds in the seed bank, so we examined how different
assumptions about seed mortality would affect the
longevity of seeds in the seed bank and projected
population dynamics. We find that for seed mortality
rates ranging between 0% and 20%, the longevity of seeds
in the seed bank is between 10 and15 years. This longevity
corresponds to results found by V. Nuzzo (unpublished
data and ongoing study), in which A. petiolata seeds
survive and germinate out of the seed bank for at least 10
years. For values of seed mortality ranging from 0%
through 20%, the projected population dynamics and
bifurcation results (simulating management) were quali-
tatively similar. Therefore, the model and results we
present here do not incorporate a decay in seed viability
(seeds only leave the seed bank by germinating).
To estimate early survivorship from germination to
the rosette stage (parameter s1) we established 40 1–m2
plots within the same population at Tyson, but at a later
date (2006). Within these plots, we tagged 469 recently
germinated individuals in early April 2006 and censused
them for survival in mid May 2006 (s1¼ 0.131) (Pardini
et al. 2008). Because these plots did not span the range
of possible densities, we were not able to test for density
dependence in s1. We explored the sensitivity of our
model to variation in this parameter over a range of
values, 0.1–0.9 (Appendix B).
We collected data on survivorship and fertility at a
range of plant densities to estimate functions for s2, s3,
and f. A large, very dense area of A. petiolata occurs at
Tyson, reflecting the core population at the site; plant
density declined with distance from that core, most
likely as a result of dispersal limitation because plants
that occurred on the periphery of the core were very
successful (see Results: Density dependence, below). We
selected 34 1-m2 square plots that varied in rosette
density, and were separated from each other by at least
10 m. In August 2003 we counted the number of A.
petiolata rosettes in each plot, which ranged from 1 to
278 rosettes (N ¼ 1795 total rosettes). We re-censused
all of the plots in early May 2004 to determine the
number of the previous year’s rosettes that survived
and bolted into the flowering adult stage (parameter
s3). We censused the number of new rosettes present in
each plot in early May 2004 (N ¼ 1346 new rosettes),
and re-censused in August 2004 to determine their
survivorship (parameter s2). In June–July 2004 we
counted the number of siliques (seed pods) per
reproductive plant. We collected 58 siliques from
plants in the area but outside of our plots, counted
the number of seeds within, and calculated the average
number of seeds per silique. The variation among
plants in their total seed production appeared to be
primarily through production of siliques, and not the
number of seeds per silique (mean ¼ 15.8 seeds, range:
8–21 seeds), and so we estimated the number of seeds
produced per plant (parameter f ) as the product of
each plant’s silique production and the average number
of seeds per silique.
Density dependence
We used scatterplots to explore whether individual
survivorship was density-dependent. Fig. 2A, B show
observed summer and winter rosette survivorship
(parameters s2 and s3) in each plot i as a function of
the total density of individuals in the plot (Riþ Ai) and
as a function of the density of rosettes (Ri) in the plot,
respectively. Confidence intervals (95%) for the survi-
vorship were obtained from the likelihood function for
the observed number of survivors (Vi) from the binomial
distribution with estimated mean si ¼ Vi/Ri. Clearly,
survivorship decreases with individual density, though
the uneven coverage across the range of densities
somewhat obscures the pattern and the confidence
intervals are wide.
To obtain density-dependent equations for the dy-
namical model, we fit linear and logistic regression
models to these data. First, for s2 we considered that
individual survivorship could be a function of rosette
density (R), adult density (A), an interaction between the
two (U¼A3R), and/or total density (T¼AþR). Using
variable significance for model selection, we determined
summer survivorship, s2, was best described by a logistic
regression:
s2 ¼ 1=ð1þ eb2Utþb1Ttþb0Þ: ð2Þ
Values designated by b here and in the following
equations represent regression parameter coefficients
and are reported in Table 1. As winter survivorship can
only be dependent on rosette density (adults are not
present in the population from August through early
May), the model for s3 is simpler. A log-log plot of s3suggested that a simple linear regression on a log-log scale
was adequate (Fig. 2B inset). Thus, s3 was modeled as
s3 ¼ e½b1logeðRþ1Þ�: ð3Þ
To test for density dependence in fertility, we regressed
individual seed production ( f ) on adult density, after
natural log-transforming seed production to stabilize the
variance and to linearize the approximately exponential
decrease evident in Fig. 2C. That natural log-transformed
observations of fertility were well fit by a linear model
(Fig. 2C inset) suggests that a two-parameter exponential
form is appropriate here, namely,
f ¼ eb1Aþb0 : ð4Þ
ELEANOR A. PARDINI ET AL.390 Ecological ApplicationsVol. 19, No. 2
FIG. 2. Survivorship and fertility of Alliaria petiolata. (A) Summer survivorship of A. petiolata rosettes (s2) as a function of thetotal density of garlic mustard (density of rosettes and adults [no. individuals/m2]). (B) Winter survivorship (s3) as a function ofrosette density, with the inset showing the data on a log–log scale. For (A) and (B) the confidence intervals (95%) for survivorshipwere obtained from the likelihood function for the observed number of survivors (Vi) from the binomial distribution with estimatedmean si ¼ Vi/Ri. (C) Observed fertility ( f ) of adult garlic mustard plants as a function of adult density, with the inset showingnatural log-transformed individual fertility regressed against adult density.
March 2009 391DYNAMICS AND CONTROL OF GARLIC MUSTARD
Parameter estimates for a dynamical model
Substituting the regression equations obtained in this
section into the dynamical system described above (Eq.
1) yields the following nonlinear discrete-time model for
A. petiolata dynamics:
Stþ1 ¼ vð1� g1ÞAteb1Atþb0 þ Stð1� g2Þ
Rtþ1 ¼ vg1s1Ateb1Atþb0 þ g2s1St
Atþ1 ¼Rte
b1logeðRtþ1Þ
1þ eb2Utþb1Ttþb0 ð5Þ
where the regression parameters designated by b are
given in Table 1. Time-series predictions of A. petiolata
population dynamics were obtained by iterating this
model using MATLAB (2007). All iterations were
started with 10 seeds and no rosettes or adults.
Simulating the effects of management
To explore the potential effect of management, we
simulated induced mortality of adults (e.g., applying
herbicide or hand-pulling adults in the spring) or of
rosettes (e.g., applying herbicide in the fall). We
examined the effects that induced mortality of adults
and rosettes would have on the population size of A.
petiolata by multiplying At or Rt, respectively, by a
factor corresponding to the desired level of control.
RESULTS
Density dependence
We find unambiguous evidence of density dependence
in each stage. Parameter estimates of the fitted functions
are provided in Table 1.
Garlic mustard population dynamics
Time series of Alliara petiolata based on the SRA
(seeds, rosettes, and adults) model are illustrated in Fig. 3,
for parameter estimates of v (seed viability)¼ 0.8228, g1(proportion of seeds that germinate following a single
winter)¼ 0.5503, g2 (proportion of seeds that germinate
out of the seed bank) ¼ 0 .3171, and s1 (survivorship of
recently germinated individuals until early May)¼ 0.131,
as described above (seeMethods: Demographic data). The
phase portraits for each stage after initial transients (not
shown) suggest that at the estimated parameter values A.
petiolata dynamics exhibit two-point oscillatory dynam-
ics. Bifurcation diagrams illustrate that these dynamics
are fairly robust to different parameter values for v, g1,
and g2 (Appendix B); while the size of the attracting set
may vary, our model consistently produces stable, cyclical
dynamics. Across the range of published values for v and
g2, our model produces stable cycles on a set of two-, four-,
six-, or eight-point attractors. For g1, two-point cycling is
produced except at high values where dynamics are
chaotic. The model results are slightly more sensitive to
the value of s1, which over a wide range of values produces
stable cycles or chaos (Appendix B). Our overall model
results remain fairly robust however, because there is a
wide range of values for s1 that produce stable cycles.
Simulating the effects of management
We present results of induced rosette and adult
mortality as bifurcation diagrams showing the attracting
set of population densities (individuals/m2) (Fig. 4).
Note that results assume management is applied on an
annual basis and graphs display equilibrium population
TABLE 1. Fitted functions for summer survivorship (s2), winter survivorship (s3), and fertility ( f )of Alliaria petiolata obtained from field data collected in St. Louis, Missouri, USA.
Parameter Regression type Variable� Coefficient, b P value R2
Summer survivorship, s2 logistic intercept �0.1560 0.4305 0.75U 0.0016 0.0438T �0.0664 0.0003
Winter survivorship, s3 linear R þ 1 �0.2890 ,0.0001 0.89Fertility, F linear intercept 7.4890 ,0.0001 0.79
A �0.0389 ,0.0001
� R¼ rosette density; A¼ adult density; U¼R 3 A, interaction term; T¼Rþ A, total density.
FIG. 3. A density-dependent model for garlic mustardpopulations (Eq. 1) suggests that managed populations mightexhibit complex dynamics. The parameterized model (Eq. 5)without management displays two-point cycling for each stage(seeds, rosettes, adults).
ELEANOR A. PARDINI ET AL.392 Ecological ApplicationsVol. 19, No. 2
densities. When ,20% of adult A. petiolata plants are
killed each year, A. petiolata population dynamics are a
two-point cycle, and the maximum density achieved
during this cycle decreases with increasing mortality.
When a moderate amount of adults are killed each year
(20–80%), the attracting sets are much more complicat-
ed, exhibiting both low- and high-dimensional cycles
and possible chaos (Fig. 4A). Moderate levels of
management are expected to increase the complexity of
the population dynamics, and at some levels (40–70%)
increase the density of the population. Management
efficiency of 85% adults killed is required to produce
simple, noncyclic dynamics in which the total popula-
tion density decreases (Fig. 4A).
Our model suggests that induced mortality of rosettes
will be largely ineffective at reducing the density of
individuals in the population. Total density increases
over a large range of induced rosette mortality (Fig. 4B).
Predicted dynamics include cycling for low to moderate
levels of mortality (0–70%) and chaotic dynamics for
moderately high values of mortality (70–80%). High
levels of rosette mortality produce simple dynamics but
cause a dramatic increase in equilibrium population
density; a strong reduction in density is not produced
until induced rosette mortality is very high (.95%) (Fig.
4B).
These results for induced adult and rosette mortality
are remarkably robust to different combinations of seed
parameter estimates (Appendix B for v, g1, g2; results
not shown for combinations). We conclude, therefore,
that over a broad range of seed viability and germina-
tion, management actions increasing rosette mortality
may exacerbate population growth of invasive A.
petiolata by inducing complex population cycles and
increasing population densities, and that induced
mortality must be very high to reduce population
density.
DISCUSSION
Our results show that for the invasive plant Alliaria
petiolata (garlic mustard) stable population cycles are
produced from a model with a structured population
with density dependence in multiple vital rates and
stages. Our qualitative model results (i.e., stable
population cycles) are robust to a wide range of seed
bank and germination parameters, which are known to
vary a great deal in nature (see Appendices A and B).
Further, we performed exploratory analyses to see if any
one of the three density-dependent vital rates was most
important in determining the projected population
cycling. We examined the projected dynamics of our
model in which one or more of the three vital rates were
systematically changed from density dependent to
density independent, for all combinations of potentially
density-dependent vital rates (results not shown). We
did not find evidence that any one density-dependent
vital rate contributed disproportionately to our projec-
tions. One intriguing feature of our model is that it
shows that years with high adult abundance are followed
with years of low adult abundance but high rosette
abundance. Interestingly, this pattern has been observed
in numerous A. petiolata populations across its range in
North America (Baskin and Baskin 1992, McCarthy
1997, Nuzzo 1999, Meekins and McCarthy 2002,
Winterer et al. 2005). Thus, it is possible that this
observed dynamic variation is generated by determinis-
tic rules. This pattern is likely to occur almost anywhere
that densities become high; thus it is probably more
likely to occur in dense, core populations than in
smaller, low-density, satellite populations. Further,
population cycling may be less pronounced in poor
microsite conditions where reproductive output is lower
and does not exhibit density dependence (Smith et al.
2003).
FIG. 4. Bifurcation diagrams for a model of garlic mustardpopulation dynamics with management (induced mortality)applied to (A) adults or (B) rosettes. The bifurcation parameter(x-axis) represents a hypothetical reduction in survivorshipbased on a control practice (such as hand-pulling or herbicide)and ranges from 0% to 100%. The y-axis is the total populationdensity of adults and rosettes (R þ A) at the equilibria. Whilecomplete effectiveness (100%) in control reduces the equilibri-um population density to zero, incomplete effectiveness canexhibit a range of patterns, including two-point cycles, complexcycles, and chaos.
March 2009 393DYNAMICS AND CONTROL OF GARLIC MUSTARD
Our model incorporated density dependence in
multiple vital rates, and yet we suspect that our model
is still simplistic in its incorporation of density depen-
dence. For example, we assumed that early survivorship
(s1) is a constant, although it is likely to be density-
dependent because we know that early seedling survi-
vorship is increased when neighboring adults are
removed by pulling or clipping (Winterer 2005, Pardini
et al. 2008). However, we note that because early
seedling survivorship is already quite low (see also
Trimbur 1973, Cavers et al. 1979, Anderson et al. 1996,
Byers and Quinn 1998), incorporating density depen-
dence in this function is unlikely to cause large changes
in the projected population dynamics. Further, A.
petiolata is known to be allelopathic (Prati and Bossdorf
2004, Aminidehaghi et al. 2006, Cipollini and Gruner
2007), and the chemicals exuded from the roots of plants
may have a long half-life, and reduce the fitness of future
PLATE 1. (Upper left) Alliaria petiolata individuals growing at high density tend to have fewer stems, produce fewer seeds, andhave a lower probability of survival. (Upper right) Isolated A. petiolata individuals can have multiple stems and produce thousandsof seeds; the photo shows a single individual. Upper photo credits: T. M. Knight. (Lower left) Flowering adults in a denseinfestation of A. petiolata at Darst Bottoms, Missouri Department of Conservation, Missouri, USA. (Lower right) First-yearseedlings and second-year rosettes of A. petiolata immediately following seed germination in the spring, when seedling density canbe exceptionally high. Lower photo credits: E. A. Pardini.
ELEANOR A. PARDINI ET AL.394 Ecological ApplicationsVol. 19, No. 2
A. petiolata, creating delayed density dependence.
However, despite the potential simplicities in our model,
we note that our projections do seem to reflect the
densities and patterns that are observed in this and in
other A. petiolata populations, which include rosette
densities of about 20–350 rosettes/m2 (Meekins and
McCarthy 2002, Smith et al. 2003, Winterer et al. 2005)
and a few higher estimates of about 400–2000 roset-
tes/m2 (Anderson et al. 1996; E. A. Pardini, B. J. Teller,
and T. M. Knight, unpublished data); and adult densities
of about 10–150 plants/m2 (Anderson et al. 1996, Byers
and Quinn 1998, Nuzzo 1999, Meekins and McCarthy
2002, Smith et al. 2003, Rebek and O’Neil 2006).
Complex population cycling in this species may be
interrupted by stochastic events such as environmental
variation. For example, a severe late-summer drought
might cause high density-independent mortality that
virtually eliminates most rosette A. petiolata individuals.
The following year, in a population that would have
contained many second-year, adult plants would consist
only of seedlings germinated from seed in the seed bank.
We observed such a drought event at Tyson Research
Center (Missouri, USA) in fall 2007, which has resulted
in two sequential years with high rosette and low adult
abundances. Interestingly, extreme environmental vari-
ation that affects a wide regional area may contribute to
the observed regional synchrony of population cycling
among rosette and adults years (McCarthy 1997).
However, because the presence of the seed bank allows
the abundance of the population to quickly rebound, we
expect that a population would rapidly return to cycles
after an extreme environmental disturbance.
The management implications of complex population
dynamics of invasive species that result from stage-
structured density dependence are profound, though
rarely considered (but see Myers et al. 1989, Buckley et
al. 2001, Buckley and Metcalf 2006). Our model focuses
on reducing the number of individuals in the population
through management and not on reducing the biomass
of the population. In density-dependent populations,
biomass and individuals cannot be considered separate-
ly, since decreasing the number of individuals will allow
per capita biomass to increase. Our model demonstrates
that when A. petiolata management is less than
completely efficient, release from density dependence
can allow for increased survival and increased fertility of
the remaining individuals. In our model, a control
efficiency of .85% removal of adults in every year is
required to successfully reduce the population density of
A. petiolata. Preliminary results of experimentally
managed plots confirm these predictions (Winterer
2005, Pardini et al. 2008). Further, moderate levels of
induced mortality of both rosettes and adults can cause
complex dynamics in which population density may
fluctuate unpredictably among a wide range of densities
(e.g., 200–550 individuals/m2 for intermediate levels of
adult mortality). Importantly, management of rosettes
can be counterproductive across a wide range of induced
mortalities, because increasing rosette mortality poten-
tially increases the equilibrium population densities of
A. petiolata.
The complicated population dynamics we observe in
A. petiolata are likely typical for short-lived organisms
with stage structure and density dependence. In addition
to our study, others have found evidence that density-
dependent vital rates can lead to oscillations in short-
lived plants (Symonides et al. 1986, Thrall et al. 1989,
Crone and Taylor 1996, Buckley et al. 2001) and in
laboratory insects (e.g., Costantino et al. 1997). Further,
Neubert and Caswell (2000a) explored the effects of
density dependence and stage structure among organ-
isms with disparate life histories by altering parameters
in a two-stage discrete-time model. They found that
semelparous life histories were more likely to have
complicated and unstable dynamics than iteroparous
ones. The results and management implications for A.
petiolata should apply to other invaders with similar life
histories that are short lived and often reach high
densities in which strong density dependence might
occur (e.g., Carduus nutans, Centaurea spp., Cirsium
vulgare, Heracleum mantegazzianum, Microstegium vim-
ineum, Verbascum thapsus).
Our study combined with the results of other studies
suggests that management strategies targeting adults are
superior to strategies targeting rosettes. Although the
majority of A. petiolata management has been focused
on killing adults before they set seed (typically through
cutting or manually pulling individuals; Nuzzo 1991)
some management has also been directed towards killing
rosettes, typically with herbicides (Nuzzo 1991, 1994), a
strategy often employed by natural-areas managers (D.
Larson and G. Raeker, personal communication). Ex-
perimental control efforts with herbicides has yielded
mixed results: treating rosettes in the fall reduces A.
petiolata density and cover in the following spring
(Nuzzo 1994, Carlson and Gorchov 2004) but has little
effect on A. petiolata recruitment over several years
(Hochstedler et al. 2007, Slaughter et al. 2007). Our
analyses suggest that over a relatively wide range of
intermediate levels of rosette mortality, as might often
be achieved with herbicides, control efforts can actually
increase the overall density of A. petiolata. In addition,
herbicides may harm co-occurring native species (but see
Carlson and Gorchov 2004). Thus, we recommend
managers avoid the use of herbicide on rosette plants
and instead focus management on adult plants. Our
recommendation to focus management on a single life
stage (adults) of an invasive-plant population agrees
with the recommendation of a recent theoretical study
that takes a different approach (e.g., density-indepen-
dent population growth is assumed; Hastings and
Botsford 2006), suggesting that this recommendation
might apply to a variety of invasive plant species.
While our stage-structured model is not spatially
explicit, it has implications for achieving effective
management in space. That our model shows inefficient
March 2009 395DYNAMICS AND CONTROL OF GARLIC MUSTARD
management efforts do not reduce population densities
suggests efforts should be focused on inducing 100%
mortality in a targeted area of a population rather than
spreading efforts widely. It may be a general feature that
management of nascent foci, or satellite populations, is
critical for controlling the spread of invasive plants
(Moody and Mack 1988). Alliaria petiolata spreads in a
demic pattern through the establishment of multiple
satellite populations that eventually coalesce (McCarthy
1997). Because of their prime location in space, these
satellite individuals are expected to contribute dispro-
portionately to the expansion of the population in
density-independent models (Neubert and Caswell
2000b). In these smaller satellite populations, managers
might be able to achieve the high mortality efficiency
(ideally 100% mortality of rosettes and/or adults every
year) that our model suggests is necessary to successfully
manage A. petiolata.
Recently, Simberloff (2003) suggested that control of
invasive species does not necessarily require detailed
knowledge of the underlying population biology of the
species themselves. Instead, he suggested that simple
control efforts would often be successful. While we agree
that control efforts cannot necessarily wait for the best
management options to be considered by scientists, we
also warn that not understanding the underlying
population dynamics—such as density dependence of
invasive species in our case—can sometimes lead to
ineffective or even harmful management strategies.
ACKNOWLEDGMENTS
We thank the staff at Washington University’s TysonResearch Center for logistical support and B. J. Teller and P.Van Zandt for assistance with data collection. We thank ErinMitchell for the drawings in Fig. 1. This manuscript wasimproved by comments from Yvonne Buckley and JessicaMetcalf. We acknowledge funding from the National ResearchInitiative of the USDA Cooperative State Research, Educationand Extension Service, grant number 05-2290, the NationalCenter for Ecological Analysis and Synthesis (Santa Barbara,California), the University of Georgia Odum School ofEcology, and Washington University’s Tyson Research Center.The opinions expressed in this publication are those of theauthors and do not necessarily reflect the views of the U.S.Department of Agriculture.
LITERATURE CITED
Aminidehaghi, M., A. Rezaeinodehi, and S. Khangholi. 2006.Allelopathic potential of Alliaria petiolata and Lepidiumperfoliatum, two weeds of the Cruciferae family. Journal ofPlant Diseases and Protection 10:455–462.
Anderson, R. C., S. S. Dhillion, and T. M. Kelley. 1996.Aspects of the ecology of an invasive plant, garlic mustard(Alliaria petiolata), in central Illinois. Restoration Ecology 4:181–191.
Baskin, J. M., and C. C. Baskin. 1992. Seed germinationbiology of the weedy biennial Alliaria petiolata. NaturalAreas Journal 12:191–197.
Buckley, Y. M., H. L. Hinz, D. Matthies, and M. Rees. 2001.Interactions between density-dependent processes populationdynamics and control of an invasive plant species, Tripleur-ospermum perforatum (scentless chamomile). Ecology Letters4:551–558.
Buckley, Y. M., and J. Metcalf. 2006. Density dependence ininvasive plants: demography, herbivory, spread and evolu-tion. Pages 109–123 in M. W. Cadotte, S. M. McMahon, andT. Fukami, editors. Conceptual ecology and invasionbiology: reciprocal approaches to nature. Springer, Dor-drecht, The Netherlands.
Byers, D. L., and J. A. Quinn. 1998. Demographic variation inAlliaria petiolata (Brassicaceae) in four contrasting habitats.Journal of the Torrey Botanical Society 125:138–149.
Carlson, A. M., and D. L. Gorchov. 2004. Effects of herbicideon the invasive biennial Alliaria petiolata (garlic mustard)and initial responses of native plants in a southwestern Ohioforest. Restoration Ecology 12:559–567.
Cavers, P. B., M. I. Heagy, and R. F. Kokron. 1979. Biology ofCanadian weeds. 35. Alliaria petiolata (M. Bieb.) Cavara andGrande. Canadian Journal of Plant Science 59:217–229.
Cipollini, D., and B. Gruner. 2007. Cyanide in the chemicalarsenal of garlic mustard, Alliaria petiolata. Journal ofChemical Ecology 33:85–94.
Costantino, R. F., R. A. Desharnais, J. M. Cushing, and B.Dennis. 1997. Chaotic dynamics in an insect population.Science 275:389–391.
Crone, E. E., and D. R. Taylor. 1996. Complex dynamics inexperimental populations of an annual plant, Cardaminepensylvanica. Ecology 77:289–299.
Cruden, R. W., A. M. McClain, and G. P. Shrivastava. 1996.Pollination biology and breeding system of Alliaria petiolata(Brassicaceae). Bulletin of the Torrey Botanical Club 123:273–280.
Cushing, J. M., R. F. Costantino, B. Dennis, R. A. Desharnais,and S. M. Henson. 1998. Nonlinear population dynamics:models, experiments, and data. Journal of TheoreticalBiology 194:1–9.
Davis, A. S., D. A. Landis, V. Nuzzo, B. Blossey, E. Gerber,and H. L. Hinz. 2006. Demographic models inform selectionof biocontrol agents for garlic mustard (Alliaria petiolata).Ecological Applications 16:2399–2410.
Dennis, B., R. A. Desharnais, J. M. Cushing, and R. F.Costantino. 1995. Nonlinear demographic dynamics: math-ematical models, statistical methods, and biological experi-ments. Ecological Monographs 65:261–281.
Drayton, B., and R. B. Primack. 1999. Experimental extinctionof garlic mustard (Alliaria petiolata) populations: Implica-tions for weed science and conservation biology. BiologicalInvasions 1:159–167.
Gillman, M., J. M. Bullock, J. Silvertown, and B. C. Hill. 1993.A density-dependent model of Cirsium vulgare populationdynamics using field-estimated parameter values. Oecologia96:282–289.
Gonzalez-Andujar, J. L. 1996. High control measures cannotproduce extinction in weed populations. Ecological Modeling91:293–294.
Gonzalez-Andujar, J. L., C. Fernandez-Quintanilla, and L.Navarrete. 2006. Population cycles produced by delayeddensity dependence in an annual plant. American Naturalist168:318–322.
Gonzalez-Andujar, J. L., and G. Hughes. 2000. Complexdynamics in weed populations. Functional Ecology 14:524–526.
Hastings, A., and L. W. Botsford. 2006. A simple persistencecondition for structured populations. Ecology Letters 9:846–852.
Hochstedler, W. W., B. S. Slaughter, D. L. Gorchov, L. P.Saunders, and M. H. H. Stevens. 2007. Forest floor plantcommunity response to experimental control of the invasivebiennial, Alliaria petiolata (garlic mustard). Journal of theTorrey Botanical Society 134:155–165.
MATLAB. 2007. Matlab, version 7.4 (R2007a). The Math-Works, Natick, Massachusetts, USA.
Matthies, D. 1990. Plasticity of reproductive components atdifferent stages of development in the annual plant Thlaspiarvense L. Oecologia 83:105–116.
ELEANOR A. PARDINI ET AL.396 Ecological ApplicationsVol. 19, No. 2
May, R. M. 1974. Biological populations with nonoverlappinggenerations: stable points, stable cycles, and chaos. Science186:645–647.
McCarthy, B. C. 1997. Response of a forest understorycommunity to experimental removal of an invasive nonin-digenous plant (Alliaria petiolata, Brassicaceae). Pages 117–130 in J. O. Luken and J. W. Thieret, editors. Assessment andmanagement of plant invasions. Springer-Verlag, New York,New York, USA.
Meekins, J. F., and B. C. McCarthy. 2000. Responses of thebiennial forest herb Alliaria petiolata to variation inpopulation density, nutrient addition and light availability.Journal of Ecology 88:447–463.
Meekins, J. F., and B. C. McCarthy. 2002. Effect of populationdensity on the demography of an invasive plant (Alliariapetiolata, Brassicaceae) population in a southeastern Ohioforest. American Midland Naturalist 147:256–278.
Moody, M. E., and R. N. Mack. 1988. Controlling the spreadof plant invasions: The importance of nascent foci. Journal ofApplied Ecology 25:1009–1021.
Mortimer, A. M., J. J. Sutton, and P. Gould. 1989. On robustweed population models. Weed Research 29:229–238.
Myers, J. H., C. Risley, and R. Eng. 1989. The ability of plantsto compensate for insect attack: why biological control ofweeds with insects is so difficult. Pages 67–73 in E. S.Delfosse, editor. Proceedings of the Seventh InternationalSymposium on the Biological Control of Weeds. PlantPathology Research Institute, Ministry for Agriculture andForestry Practices, Rome, Italy.
Neubert, M. G., and H. Caswell. 2000a. Density-dependentvital rates and their population dynamic consequences.Journal of Mathematical Biology 41:103–121.
Neubert, M. G., and H. Caswell. 2000b. Demography anddispersal: Calculation and sensitivity analysis of invasionspeed for structured populations. Ecology 81:1613–1628.
Nuzzo, V. 2000. Element stewardship abstract for Alliariapetiolata (Alliaria officinalis) Garlic Mustard. The NatureConservancy, Arlington, Virginia, USA. hhttp://tncweeds.ucdavis.edu/esadocs/documnts/allipet.pdf i
Nuzzo, V. A. 1991. Experimental control of garlic mustard[Alliaria petiolata (Bieb.) Cavara & Grande] in northernIllinois using fire, herbicide and cutting. Natural AreasJournal 11:158–167.
Nuzzo, V. A. 1994. Response of garlic mustard (Alliariapetiolata Bieb [Cavara and Grande]) to summer herbicidetreatment. Natural Areas Journal 14:309–310.
Nuzzo, V. A. 1999. Invasion pattern of the herb garlic mustard(Alliaria petiolata) in high quality forests. Biological Inva-sions 1:169–179.
Nuzzo, V. A., W. McClain, and T. Strole. 1996. Fire impact ongroundlayer flora in a sand forest 1990–1994. AmericanMidland Naturalist 136:207–221.
Palmblad, I. G. 1968. Competition in experimental populationsof weeds with emphasis on regulation of population size.Ecology 49:26–34.
Pardini, E. A., B. J. Teller, and T. M. Knight. 2008.Consequences of density dependence for management of a
stage-structrued invasive plant (Alliaria petiolata). AmericanMidland Naturalist 160:310–322.
Prati, D., and O. Bossdorf. 2004. Allelopathic inhibition ofgermination by Alliaria petiolata (Brassicaceae). AmericanJournal of Botany 91:285–288.
Rebek, K. A., and R. J. O’Neil. 2006. The effects of natural andmanipulated density regimes on Alliaria petiolata survival,growth and reproduction. Weed Research 46:345–352.
Schwartz, M. W., and J. R. Heim. 1996. Effects of a prescribedfire on degraded forest vegetation. Natural Areas Journal 16:184–191.
Simberloff, D. 2003. How much information on populationbiology is needed to manage introduced species? Conserva-tion Biology 17:83–92.
Slaughter, B. S., W. W. Hochstedler, D. L. Gorchov, and A. M.Carlson. 2007. Response of Alliaria petiolata (garlic mustard)to five years of fall herbicide application in a southern Ohiodeciduous forest. Journal of the Torrey Botanical Society134:18–26.
Smith, G. R., H. A. Dingfelder, and D. A. Vaala. 2003. Effectof plant size and density on garlic mustard reproduction.Northeastern Naturalist 10:269–276.
Solis, K. 1998. Update: new results indicate flowering garlicmustard should be bagged and destroyed (Wisconsin).Restoration and Management Notes 16:223–224.
Stinson, K., S. Kaufman, L. Durbin, and F. Lowenstein. 2007.Impacts of garlic mustard invasion on a forest understorycommunity. Northeastern Naturalist 14:73–88.
Symonides, E. 1983. Population size regulation as a result ofintra-population interactions. I. Effect of density on survivaland development of individuals of Erophila verna (L) C.A.M.Polish Journal of Ecology 31:839–882.
Symonides, E., J. Silvertown, and V. Andreasen. 1986.Population cycles caused by overcompensating densitydependence in an annual plant. Oecologia 71:156–158.
Thrall, P. H., S. W. Pacala, and J. A. Silander. 1989. Oscillatorydynamics in populations of an annual weed species Abutilontheophrasti. Journal of Ecology 77:1135–1149.
Trimbur, T. J. 1973. An ecological life history of Alliariaofficinalis, a deciduous forest ‘‘weed.’’ Ohio State University,Columbus, Ohio, USA.
USDA NRCS [U.S. Department of Agriculture, NaturalResource Conservation Service]. 2007. Alliaria petiolata(Bieb.) Cavara & Grande. The Plants Database. NationalPlant Data Center, Baton Rouge, Louisiana, USA. hhttp://plants.usda.govi
Watkinson, A. R., W. M. Lonsdale, and M. H. Andrew. 1989.Modeling the population dynamics of an annual plantSorghum intrans in the wet-dry tropics. Journal of Ecology77:162–181.
Whittle, A. J., S. Lenhart, and L. J. Gross. 2007. Optimalcontrol for management of an invasive plant species.Mathematical Biosciences and Engineering 4:101–112.
Winterer, J., M. C. Walsh, M. Poddar, J. W. Brennan, andS. M. Primak. 2005. Spatial and temporal segregation ofjuvenile and mature garlic mustard plants (Alliaria petiolata)in a central Pennsylvania woodland. American MidlandNaturalist 153:209–216.
APPENDIX A
A table showing estimates of seed viability and germination parameters reported in published sources and used to parameterizedynamic SRA model (Ecological Archives A019-017-A1.
APPENDIX B
Bifurcation analyses showing sensitivity of SRA model to parameter estimates (Ecological Archives A019-017-A2).
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