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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


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.


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).


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.


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.


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


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.

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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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