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vol. 166, no. 6 the american naturalist december 2005
When Can Herbivores Slow or Reverse the Spread of an
Invading Plant? A Test Case from Mount St. Helens
William F. Fagan,^1 Mark Lewis, 2 Michael G. Neubert,^3 Craig Aumann, 1 Jennifer L. Apple, 4 and John G. Bishop 4
- Department of Biology, University of Maryland, College Park, Maryland 20742;
- Departments of Biological Sciences and Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2E1, Canada;
- Biology Department, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543;
- School of Biological Science, Washington State University, Vancouver, Washington 98686
Submitted February 1, 2005; Accepted July 26, 2005; Electronically published October 4, 2005
Online enhancements: appendixes.
abstract: Here we study the spatial dynamics of a coinvading consumer-resource pair. We present a theoretical treatment with ex- tensive empirical data from a long-studied field system in which native herbivorous insects attack a population of lupine plants re- colonizing a primary successional landscape created by the 1980 vol- canic eruption of Mount St. Helens. Using detailed data on the life history and interaction strengths of the lupine and one of its her- bivores, we develop a system of integrodifference equations to study plant-herbivore invasion dynamics. Our analyses yield several new insights into the spatial dynamics of coinvasions. In particular, we demonstrate that aspects of plant population growth and the intensity of herbivory under low-density conditions can determine whether the plant population spreads across a landscape or is prevented from doing so by the herbivore. In addition, we characterize the existence of threshold levels of spatial extent and/or temporal advantage for the plant that together define critical values of “invasion momen- tum,” beyond which herbivores are unable to reverse a plant invasion. We conclude by discussing the implications of our findings for suc- cessional dynamics and the use of biological control agents to limit the spread of pest species.
Keywords: biocontrol, Filatima , integrodifference equation model, Lupinus lepidus , primary succession, spatial spread.
Am. Nat. 2005. Vol. 166, pp. 669–685. 2005 by The University of Chicago. 0003-0147/2005/16606-40873$15.00. All rights reserved.
The spatial spread of invading species has been the subject of extensive ecological research since Skellam (1951) ap- plied Fisher’s (1937) model of gene flow to characterize the “wave of advance” of an invading species. Invasion dynamics warrant such attention because understanding how species spread spatially is of tremendous practical importance. Whether in characterizing risks from ex- panding pest species (Hajek et al. 1996; Sharov et al. 2002), understanding the success or failure of biological control agents (Louda et al. 1997; Fagan et al. 2002), or predicting the resurgence of native species reintroduced within their historical ranges (Lubina and Levin 1988; Lensink 1997; Caswell et al. 2003), understanding the dynamics of spatial spread has proved important in many areas of applied ecology. Since Skellam’s initial effort, empiricists and theoreti- cians have demonstrated that biological invasions are con- siderably more complex than Skellam’s single-species de- terministic model suggests. Characteristics of invading species, such as density dependence (Veit and Lewis 1996) and life history (van den Bosch et al. 1990; Neubert and Caswell 2000), and characteristics of the environments through which invasions are occurring, such as landscape heterogeneity (Shigesada et al. 1986; Andow et al. 1990) and temporal stochasticity (Neubert et al. 2000), have re- ceived attention by ecologists. In addition, theoretical at- tention to the “dispersal kernels” that characterize the probability of individuals dispersing particular distances has identified the importance of long-distance dispersal to the velocity of spatial spread of an invading species (Kot et al. 1996; Clark et al. 1998; Caswell et al. 2003). Species interactions can mediate the spatial dynamics of invading species. For example, studying two species of squirrels in Britain, Okubo et al. (1989) demonstrated how invasion models could be used to understand the spatial displacement of one species by a competitor. Parker (2000) and Parker and Haubensak (2002) demonstrated the po- tential for mutualistic pollinator species to influence in- vasion success of nonnative shrubs and established that such influences can be strongly context dependent. Dun- bar (1983) modeled the dynamics of a consumer invading
670 The American Naturalist
a resource species at equilibrium, and Sherratt et al. (1995) explored how coinvading predator and prey species can generate spatiotemporal chaos in the invasion’s wake. Pe- trovskii et al. (2002) studied how predator-prey interac- tions in concert with Allee effects can generate “patchy” invasions, where the distribution of invasive species across a landscape is heterogeneous in both time and space. Working with a two-species partial differential equation model, Owen and Lewis (2001) provided the first analysis of conditions under which a coinvading consumer can alter the velocity of spatial advance of a resource species. For that model, only under very stringent conditions can the consumer actually reverse an invasion by a resource species. In particular, for reversal to occur, the resource species must possess a strong Allee effect such that its population growth rate is negative at low densities (Owen and Lewis 2001). Here we expand on the general theoretical efforts of Owen and Lewis (2001) to study the spatial dynamics of a specific, empirically motivated case of coinvasion by a consumer-resource pair recolonizing a primary succes- sional landscape. Using detailed data on the life history and interaction strengths of a lupine plant and one of its herbivores, we develop a system of integrodifference equa- tions to study plant-herbivore invasion dynamics. We demonstrate that the intensity of herbivory under low- density conditions can determine whether the plant pop- ulation spreads across a landscape or is prevented from doing so by the herbivore. In addition, we characterize the existence of threshold levels of spatial extent and/or tem- poral advantage for the plant that together define critical values of “invasion momentum,” beyond which herbivores are unable to reverse a plant invasion. We conclude by discussing the implications of our findings for successional dynamics and the use of biological control agents.
Overview of the Field System
The 1980 eruption of Mount St. Helens (Washington) be- gan with the largest landslide in recorded history, which was followed by a devastating lateral volcanic blast, py- roclastic flows, and lahars that together created 60 km 2 of primary successional habitat referred to as the “pumice plain and debris avalanche” (hereafter “pumice plain”). Since first colonizing the pumice plain in 1981 (Wood and del Moral 1987), the prairie lupine Lupinus lepidus var. lobbii , a native nitrogen-fixing legume, has spread across this landscape. The interactions between this plant and its associated specialist herbivores afford a unique opportu- nity to study the effects of consumer-resource dynamics on primary succession. Contrary to the prevailing view that insects have little influence on primary succession (e.g., McCook 1994; Walker and del Moral 2003), large-
scale demographic studies (Bishop and Schemske 1998; Bishop 2002; Bishop et al. 2005) and small-scale removal experiments (Fagan and Bishop 2000) demonstrate that insect herbivory has strongly affected the abundance, de- mography, and spatial structure of colonizing prairie lupines. Like many invading species (Mack 1981; Petrovskii et al. 2002; Sharov et al. 2002), the lupine population exhibits a patchy spatial structure. Patches within the “edge” region are small (less than a few tens of meters in diameter) and young and are located 0.2–3 km from the position of the initial 1981 colonization event. Edge patches possess less than 15% lupine cover and are separated by a matrix of mostly bare rock featuring very low lupine densities (typ- ically far less than 1 plant m^2 , with less than 1% cover). Edge patches are often short-lived and exhibit a high turn- over rate (del Moral 2000 a , 2000 b ). On the other hand, in “core” regions, where lupine densities are far higher (hundreds to more than a thousand plants per square meter, with more than 20% cover from lupines) and patches are hundreds of meters in diameter and more than a decade old (Fagan and Bishop 2000; Bishop 2002; Fagan et al. 2004; Bishop et al. 2005). Although the patchy nature of recolonization and bursts of lupine recruitment mean that there is not really a distinct border between core and edge regions, edge and core regions provide a conspicuous visual contrast. Several lepidopteran species feed on lupines and can inflict severe damage. Such herbivory has occurred for more than a decade (fig. 1; Fagan and Bishop 2000; Bishop et al. 2005) and exhibits striking inverse density depen- dence. High densities of lepidopterans and their associated damage are consistently restricted to areas of low lupine density, specifically edge areas and the outer margins of core patches (Fagan et al. 2004). The identity of herbivore- affected patches shifts over time in a spatial mosaic as lupine patches form, grow, and fail. Evidence suggests that top-down and bottom-up mech- anisms may interact synergistically to produce this inverse density–dependent herbivory (Fagan and Bishop 2000; Fa- gan et al. 2004; Bishop et al. 2005). Our focus here, how- ever, is on the dynamic consequences of this density de- pendence rather than its specific mechanism. By inducing net negative population growth at the fringes of the ex- panding lupine population, intense herbivory confined to the edge region reduces the rate of spatial spread for lupine (Fagan and Bishop 2000) and increases patch turnover (del Moral 2000 a , 2000 b ). In some locations, we have even observed patch “shrinkage” due to intense herbivory at patch edges that suppresses lupine recruitment (Bishop et al. 2005). Because lupines make diverse and important contributions to the mechanics of local primary succession at Mount St. Helens (Kerle 1985; Titus and del Moral 1998;
672 The American Naturalist
Figure 2: Life cycle and phenology of Lupinus and Filatima. Breakpoints between the winter season (stage I of the model) and the growing season (stage II) are approximate.
An Integrodifference Model of Consumer-Resource Invasion Dynamics
To match both the perennial life history of Lupinus lepidus and the univoltine life cycle of the leaf tiers, we developed a model that possessed a tripartite stage structure for the lupines (seed/seedling/adult) but modeled explicitly only the larval stage of the herbivores. To take full advantage of long-term spatial data on the lupine–leaf tier interac- tion, including data collected by other researchers for other purposes, we adopt the somewhat unconventional practice of modeling plant-herbivore dynamics via a mixture of “proportion cover” for adult plants and herbivore damage (i.e., a rescaling of plant ecologists’ traditional “percent cover” measure) and densities for seeds and seedlings. We model the spatial dynamics of herbivores and plants in one spatial dimension, denoting position by x (cm) and time by t (years). Our four state variables are p (^) t ( x ), a
dimensionless measure defined as the proportion of the surface of a site covered by undamaged plants; s (^) t ( x ), the lupine seed density (cm^1 ); j (^) t ( x ), the lupine seedling den- sity (cm^1 ); and lt ( x ), the density of larval leaf tiers (cm^1 ). We use integrodifference equations (e.g., Kot et al. 1996) to model dispersal of lupine seeds and, via assumptions about moth dispersal and oviposition, the redistribution of the next generation of feeding caterpillars. Integrodif- ference equations take the general form
N t 1 ( x ) p k ( x y ) f ( N t ( y )) dy , (1)
where N is the density of a species, f ( N ( y )) is a function describing the recruitment of the species at position y , and k ( x y )is a “redistribution kernel” defining how individ- uals produced at positions y get redistributed to position
Herbivore Effects on an Active Plant Invasion 673
x. These kernels can take a wide variety of shapes (Neubert et al. 1995), but in the context of spatially spreading pop- ulations, particular interest is focused on kernels with broad tails that allow for long-distance dispersal. Although long-distance dispersal occurs occasionally in lupines (e.g., the first recolonizing lupine seed is estimated to have dis- persed more than 2 km from the nearest seed source; Bishop et al. 2005), long-distance dispersal appears to be far more frequent among herbivores. Adult Filatima moths are weak and reluctant fliers, but if they are disturbed and if they then manage to ascend through the boundary layer near the ground surface, wind currents can carry them many tens of meters in just a few seconds. When windblown, these ∼6-mm-long moths routinely outpace field crews at- tempting to track their long-distance movements. In con- trast, redistribution of feeding caterpillars is highly localized and is negligible on the scale of the pumice plain. Lacking detailed data on the dispersal of either the plant or the herbivores, we assume that the lupines and the leaf tiers each have Laplace dispersal kernels. Thus, for the plants,
a (^) aF x y F k (^) p( x y ) p e , (2) 2
where a (cm^1 ) is the reciprocal of the mean seed dispersal distance. Likewise, for the moths,
b (^) bF x y F k (^) m( x y ) p e , (3) 2
where b (cm^1 ) is the reciprocal of the mean moth dis- persal distance. These functional forms are consistent with a diffusive form of one-dimensional dispersal in which there is a constant probability per unit time of settling (Broadbent and Kendall 1953; Neubert et al. 1995). (See app. A in the online edition of the American Naturalist for a brief discussion of dimensionality and invasion dy- namics.) The kernel k then describes the expected locations of settled individuals that started at point y.
Stage I: Overwinter Dynamics
During stage I (see fig. 2), four different aspects of the plant herbivore dynamics occur between time t and the end of the stage, denoted t. First, lupine seed production is determined by local conditions set by plant crowding (as measured by proportion cover) and the density of Filatima caterpillars, both measured at the end of the pre- ceding growing season. As mentioned above, one of Fi- latima ’s most important effects on lupine demography is a major reduction in seed output in patches with extensive herbivory (Fagan and Bishop 2000; Bishop 2002). We rep-
resent this effect on seed production at location y by discrete-time host-resource dynamics,
f ( p ( y ), l ( y )) p b p ( y ) e^ b l^1^ t ( y ) (4) p t t 0 t
(Hassell 1978), where b 0 is the seed density produced at 100% lupine cover in the absence of Filatima larvae and b 1 determines how steeply the density of lupine seeds pro- duced decays with increases in herbivore density (Bishop 2002). Seeds produced in a given year are redistributed ac- cording to a dispersal kernel k p from equation (2). We assume that after this initial redistribution, seeds do not disperse any further (e.g., seeds in the seed bank remain in place). A fraction js of all seeds survive the winter. A fraction g of the surviving just-dispersed seeds and an equal fraction of the surviving seed bank seeds germinate in late spring. Thus, the combined effects of seed pro- duction, redistribution, mortality, and germination yield the seedling density at the start of the growing season,
j t t( x ) p j gs [ k p ( x y ) f p ( pt ( y ), l t ( y )) dy s t ( x ) .] (5)
Second, seeds that do not germinate, excluding the frac- tion ( 1 js) that die during the winter, enter the seed bank, yielding
st (^) t( x ) p j (^) s(1 g)
# [ k p( x y ) f p( pt ( y ), l t ( y )) dy s t ( x ) .] (6)
Third, lupine plants existing at the end of the preceding growing season also suffer overwinter mortality, which re- duces the undamaged proportion cover by a factor jp, yielding
pt (^) t( x ) p j (^) p p t ( x ). (7)
Fourth, Filatima moths are produced in proportion to the local larval density at the end of the preceding growing season, scaled by the fraction of larvae that survive diapause (jl). To simplify the modeling, we assume that oviposition by Filatima moths depends on the proportion lupine cover that was present at a particular site at the end of the pre- ceding growing season. This assumption matches recent dis- coveries about the moths’ biology. Filatima females lay eggs in leaf axils as lupine leaves are flushing out in spring. Lupine cover in spring and fall are highly correlated because these leaf tiers cause little direct lupine mortality and instead affect the lupine population through reduction of seed set (Fagan
Herbivore Effects on an Active Plant Invasion 675
tion, we can write the density of larvae at the end of the growing season as
l p f( p^ ∗^ , T ) F ( p ∗ , l ). (13) t 1 t t
Having determined how much damage the caterpillars would cause, we can update lupine cover from step 1 to the end of the growing season using
p p p^ ∗^ F ( p ∗ , l ). (14) t 1 t t
Finally, we assume no mortality of seeds in the seed bank during the growing season, giving
s (^) t 1 p s (^) t t. (15)
In summary, our final system of equations thus becomes
pt (^) t( x ) p j (^) p p t ( x ),
j t t( x ) p j gs [ k p( x y ) f p( pt ( y ), l t ( y )) dy s t ( x ) ,]
s t t( x ) p js (1 g) [ k p( x y ) f p( pt ( y ), l t ( y )) dy s t ( x ) ,]
l t t( x ) p v k m( x y ) pt ( y ) l t ( y ) dy
(16a)
for stage I and
p^ ∗ ( x ) p B ( p ( x ), T ) min {1 B ( p ( x ), T ), gj ( x )}, t t t t t t l ( x ) p f( p ( x ), T ) F ( p^ ∗ ( x ), l ( x )), t 1 t t t t st (^) 1 ( x ) p s (^) t t( x ),
p ( x ) p p^ ∗^ ( x ) F ( p ∗ ( x ), l ( x )) t 1 t t (16b)
for stage II. Initial conditions for this system were a lo- calized (100-m radius) lupine colonization event, followed several years later by moth colonization. In all numerical solutions, the initial moth colonization was confined to the same 100-m-radius site that was the source of the lupine colonization. (This strategy accords with the actual invasion dynamics that occurred at Mount St. Helens, where, perhaps because of topographic effects on wind currents and the deposition of airborne particles [includ- ing insects; Edwards and Sugg 1993], the core region was the site of several “first introductions.” Naturally, if we allowed the moths to colonize throughout the lupine pop-
ulation immediately upon arrival, the dynamics would be different and the advantage would tend to shift to the herbivore.) We use the parameter t 0 (years) to control the duration of the time lag between the start of the plant and herbivore invasions. Using parameter estimates discussed in the next section, we solved equations (16) numerically in Matlab until t p 250 years on a linear domain scaled to represent 50 km of invasible space. Thus, the lower and upper limits of integration for all of the integrals in equa- tions (16) are 0 and 50 km; this domain was effectively unbounded with respect to the parameter conditions we considered. For computational stability, we used 8, (i.e., 2^13 ) computational nodes and assumed that if the maximum lupine percent cover ( p ) fell below 0.000001, then both species became extinct.
Parameterizing the Lupine- Filatima Model The full model in equations (16) involves 18 parameters, which are summarized in table 1. We could estimate 14 of the parameters from a combination of field surveys, field experiments, and laboratory experiments, as dis- cussed below. The four exceptions involved the dispersal parameters (a and b) and two parameters related to sur- vivorship in the early life stages of Filatima (jl and je ). We treated these exceptions as follows. Because we lacked detailed information on dispersal of lupine seeds and ovipositing Filatima moths but believed moths to disperse farther on average than lupine seeds, we set the distance decay parameter for the seeds steeper than that for the moths. We used a p 0.0035(cm^1 ) for the seeds and b p 0.0005(cm^1 ) for the moths. These parameter values convert into mean dispersal distances of ∼2.9 and 20 m for the lupines and moths, respectively. (See “Discussion” for more information regarding these assumptions.) For the unknown larval parameters, we as- sumed j (^) l p 0.75 and je p 1 , implying that larval mor- tality does occur during the winter but that all eggs laid successfully hatch into larvae. Data from a related gelechiid moth species, Chionodes psiloptera , from eastern Wash- ington, give j (^) e 1 0.9(Oetting 1977). Because jl and je enter the same composite parameter (v in eq. [8]), we are left with three parameters that did not have direct empirical support (v, a, and b). We established the beginning and end of the growing season as June 15 and August 31, respectively, as approx- imations for the dynamics witnessed over the last decade at Mount St. Helens. These choices yield t p June 15and T p 75 days. Remaining parameters were estimable from empirical data or the literature. For example, the param- eters b 0 and b 1 , which together determine lupine seed pro- duction as functions of undamaged lupine cover and Fi- latima larval density (eq. [4]), were obtained by nonlinear
676 The American Naturalist
Table 1: Model parameters with definitions, dimensions, estimated values, and sources Parameter Definition Estimate Data source f m (eggs per female) Fecundity Range: 30–147 Oetting 1977; J. Apple and J. G. Bishop, unpublished data jl Overwinter larval survival Guess:. je Egg hatching success rate Guess: 1 v (larvae per female) Product of moth fecundity and egg and larval survivorship ( f mjejl/2)
Calculated
c (cm^2 ; mean SE) Number of caterpillars produced per cm 2 of plant eaten
.42 .04 Fagan et al. 2004
m 0 (day^1 ) Mortality of larvae independent of percent cover
Point estimate: .0086 Fagan et al. 2004
m 1 (day^1 ) Mortality of larvae dependent on undamaged percent cover
Point estimate: .054 Fagan et al. 2004
f (cm^2 ) Number of mature caterpillars produced at a given level of plant cover ( ce (m^0 ^ m^1 p^^ tt)^ T )
Calculated
b 0 (cm^2 ; mean SE)
Seed density produced at 100% lupine cover in the absence of herbivores
.75 .45 Bishop 2002
b 1 (cm 2 ; mean SE) Decay of seed density with herbivore density 139 95 Bishop 2002 jp Overwinter plant survivorship .80; range: .75–.84 Bishop 2002 js (mean SE) Seed survivorship .22 .14 Bishop 1996 g (mean SE) Seed germination fraction .77 .03 Bishop 1996 r (day^1 ; mean SE) Vegetative growth rate of plants Current: .006.004.002; historic: .026.006.
Fagan and Bishop 2000; Bishop 2002 a (cm 2 ; mean SE) Damage cover saturation variable 2.09 .33 Fagan and Bishop 2000 g (cm 2 ; mean SE) Average end-of-season size of seedlings 2.98 .17 Bishop 2002 a (cm^1 ) Distance decay of lupine seed dispersal .0035 Fagan and Bishop 2000 b (cm^1 ) Distance decay of dispersal by ovipositing moths
.
T (days) Duration of growing season 75 t Time of germination June 15 t 0 (years) Time lag between the start of the lupine and Filatima invasions
8; range: 5–12 Bishop 1996; Bishop et al. 2005
least squares fits of the function f p( p , l ) to a large de- mographic data set of lupine fecundity estimates (Bishop 1996, 2002; Bishop and Schemske 1998). For the purposes of this calculation, fecundity was determined at the level of the patch, not the plant, yielding 355 unique patch # yearcombinations spanning five field seasons. Seeds produced but subsequently destroyed by insect seed predators were excluded from the calculations. In 86 of the patch # yearcombinations, no seeds were produced; virtually all of these cases of reproductive failure involved patches with low lupine density but high Filatima density. The same long-term study yielded data on the fraction of seeds that successfully overwinter, the overwinter persis- tence of lupine proportion cover, and the fraction of seeds germinating, providing estimates for js, jp , and g, re- spectively (Bishop 1996, 2002). The estimate of per capita fecundity of female Filatima moths, f (^) mp 30 , was obtained from field-caught females that were enclosed in chambers with potted plants on
which to oviposit. Data from a related species, Chionodes psiloptera , suggest that egg laying in this group of moths may actually be a multiday, multi-eggmass process (Oet- ting 1977), and therefore our data may be a substantial underestimate because we did not know the age and ovi- position history of our field-caught females. For Chionodes psiloptera , per capita fecundity was 147 (range 126–184). Note, however, that uncertainty in this parameter is sub- sumed in the composite parameter v, which is already “tunable” because we lack estimates for its other com- ponents (eq. [8]). Multiple methods are available for estimating the veg- etative growth rate of lupine (eq. [9]). One approach is to calculate r p ln (SA /SA )/E B T , where SAB and SAE rep- resent lupine surface area at the beginning and end of the growing season, respectively. Making this calculation using data from low-density lupine patches to which a broad- spectrum insecticide had been applied at the beginning of the 1995 season yields an upper estimate of r p 0.
Figure 3:
Time course of the
Lupinus
model in equations (16).
a
,^ b
, Summary of the dynamics for a situation in which the coinvasion continues to expand spatially.
c ,
d
, Summary of
the spatial extent of the lupine and herbivore populations, respectively, for a situation in which inverse density–dependent herbivory causes the coinvasion to collapse. In
a and
c , the black curve
representing
F
, the proportion of the lupine surface area damaged by herbivores, is stacked on top of a white curve representing t
p
, the undamaged lupine surface area. In t
b
and
d , the density
of Filatima
caterpillars (
l ) is given. All parameters are as in table 1, except that for t
a and
b
,^
, whereas for
c and
d ,^
. Herbivores are instantiated into the model at
years.
f^
p
95
f^
p
100
t^
p
8
m^
m^
0
Herbivore Effects on an Active Plant Invasion 679
Figure 4: Forwardmost spatial position of high densities of ( a ) Lupinus and ( b ) Filatima over the time course of the invasion as a function of r , the lupine’s intrinsic rate of vegetative growth (rescaled by a factor of 100). Estimates of r from the mid-1990s, earlier in the invasion’s history (rescaled r 1 2.5[day^1 ]) were sufficient to cause the rapid expansion of the lupine population. More recently obtained estimates of r , which are lower (rescaled r p 0.6[day^1 ]) and are estimated in the absence of Filatima herbivory but in the presence of feeding damage by other insect herbivores, can cause spatial collapse of the lupine invasion (other parameters as in table 1). To avoid local artifacts or transients, we smooth the surface plot by presenting the spatial location of those densities corresponding to 95% of the species’ respective peak densities.
which herbivory is exactly sufficient to “stall out” the co- invasion, resulting in a stationary patch of lupines (with intense herbivory at its outer margin) that neither expands nor contracts. One such stationary patch results when f m is increased from its baseline value to ∼97 (app. B in the online edition of the American Naturalist ). Changes in other pa- rameters (e.g., r , a ) can accomplish the same effect. Visualizing the spatiotemporal effects of changes in the model’s parameters is a challenging task. One solution is to track the predicted location of particular densities of the lupine and herbivore populations. We opted to follow the time evolution of the forwardmost locations of the lupine and herbivore populations corresponding to 95% of the species’ respective peak densities at each point in a model run. Because Filatima survivorship decreases with increasing lupine density (eq. [12]), Filatima densities tend to be higher under conditions of low lupine density. Thus, the spatial peak in the Filatima population tends to occur farther out along the invasion’s trajectory than the spatial peak in the lupine population (i.e., peak herbivore densities are right- shifted relative to maximum lupine cover in fig. 3 and app. B). Manipulating the vegetative growth rate parameter r provides special insight into the plant-herbivore dynamics because estimates of this parameter are context dependent.
In the absence of herbivory by Filatima , estimates of r from the mid-1990s, earlier in the recolonization process, were about four times as high as comparable measures obtained more recently ( r p 0.026vs. 0.006 day^1 ). The lower estimates of r derive from plots in which Filatima was absent but where several other insect species, repre- senting a range of herbivore guilds, were present. Together with our best estimates of all other parameters (table 1), the distinction between the old and new estimates of r is sufficient to cause a qualitative shift in the dynamics of the coinvasion from rapid expansion to spatial collapse (fig. 4). Thus, the synergistic effects of several insect her- bivores will likely be critical to determining the long-term success of the lupine invasion, with only a portion of the story being captured in the model we present here. The timing of herbivore colonization relative to the start of the lupine invasion also is critical to long-term expansion or collapse of the coinvasion. From our best empirical es- timates of the model’s parameters, if the herbivores are allowed to colonize shortly after the start of the plant in- vasion, they can reduce the plants’ population growth rate sufficiently to collapse the plant population backward on itself (fig. 5; t (^) 0! 9 years). In contrast, if the herbivores arrive later, after the lupine population has had more time to establish, then the herbivores have less effect on the lupine
Herbivore Effects on an Active Plant Invasion 681
Figure 6: Qualitative dynamics of the lupine-herbivore model in r # t 0 parameter space. Three classes of long-run dynamics are possible. For slow lupine growth rates with short lag times before herbivore arrival, we observed contracting patch dynamics and collapse of the plant-herbivore system (as in fig. 3 c , 3 d ). For faster lupine growth rates, we observed forward-spreading coinvasion dynamics (as in fig. 3 a , 3 b ). For intermediate lupine growth rates but short delays in herbivore arrival, a single stationary patch emerged with coexisting lupines and herbivores (as in app. B). We fixed the boundaries of this region of parameter space as those parameter combinations for which the invasive front of the lupine population moved by less than 0.5 m year^1. From table 1, the best estimate and range of the delay t 0 and the best estimate (1 SE) of r are overlaid in gray.
Discussion
The lupine- Filatima system at Mount St. Helens appears perched very close to the dynamic threshold separating con- ditions for spatial advance and spatial retreat of the plant- herbivore coinvasion. To the extent that our mathematical representation captures the main components of the plant- herbivore coinvasion, our analyses suggest that destructive herbivory by Filatima is quite capable of demonstrably slow- ing the rate of spatial spread by the lupine population. In- deed, we know from field observations and experiments (Bishop 1996, 2002; Fagan and Bishop 2000; Fagan et al. 2004; Bishop et al. 2005) that Filatima causes tremendous reductions in lupine seed production, and we have witnessed herbivore-induced patch shrinkage that represents invasion failure on a small scale (Bishop et al. 2005). However, our mathematical analyses here suggest that even these prodi- gious effects are insufficient to engender a long-term, broad- scale spatial contraction of the lupine population, but the herbivores’ effects are close to doing so. If all other param- eters of the model are held at our best estimates (table 1),
relatively small changes in any of several parameters reflect- ing life-history traits or attributes of the plant-herbivore interaction (e.g., r , a , f m) would allow a modified Filatima - like herbivore to reverse the invasion of a lupine-like plant (fig. 3 c , 3 d ; figs. 4, 5). Interestingly, something as simple as the timing of herbivore arrival would also switch the system between spread and collapse of the coinvasion (fig. 6). In this discussion, we focus first on the specific implications of rate of spread of the plant-herbivore coinvasion for pri- mary succession at Mount St. Helens and then on the more general issue of invasion timing.
Insect Herbivory at Mount St. Helens: Inverse Density Dependence and Its Successional Effects
The lupine- Filatima system at Mount St. Helens deviates from conventional wisdom about terrestrial plant-herbi- vore interactions in several important respects. The long- term pattern of inverse density–dependent herbivory (fig.
- constitutes perhaps the most striking exception to the
682 The American Naturalist
predictions that the major theories of plant-herbivore in- teractions would make about the system. Both the resource concentration hypothesis (Root 1973) and plant apparency theory (Feeny 1976; Rhoads and Cates 1976) would predict greater levels of herbivory in the core region, where lupine patches exist at higher density, offer much greater biomass, and are far more persistent. All the more surprising is that the conspicuous inverse density–dependent herbivory is not restricted to feeding by Filatima. Remarkably, the ac- tivities of at least four different herbivore species (repre- senting three distinct feeding guilds) are effectively con- fined to low-density areas of lupine, where they cause local spatial collapse of lupine patches, a shifting mosaic of small patches, and extreme fluctuation in percent cover (del Mo- ral 2000 b ; Bishop 2002; Bishop et al. 2005). Inverse density dependence appears critical to the coinvasion dynamics in this system because the herbivores’ enhanced performance under conditions of low lupine density induces a strong Allee effect for the plant, a trait that has been shown nec- essary to reverse a traveling wave of invasion by a resource species in partial differential equation models (Owen and Lewis 2001). Negative relationships between host density and herbivory are known in other plant-herbivore systems, but the underlying mechanisms in any particular case re- main obscure (Thompson and Price 1977; Courtney and Courtney 1982; Courtney 1986; Kunin 1999). Revealing the mechanisms that underlie the persistent patterns of herbivory observed at Mount St. Helens will require ex- perimental manipulation, such as our ongoing studies that factorially manipulate soil nutrients and natural enemies. It is worth comparing results from this model of lupine spatial spread with the far slower spread speeds presented in earlier work. In Fagan and Bishop (2000), we adopted a reaction diffusion framework where the diffusion coef- ficient was estimated from the recruitment locations of first-generation offspring of lupines growing under ultra- low densities (so parentage could be assured). This ap- proach essentially provided an estimate only of “local” spread because our searches for recruiting plants were re- stricted to ∼20-m radii around parent plants. Thus, the model of Fagan and Bishop (2000) yielded information on how quickly an outlying patch of lupine should expand, approximately 4 m year^1. In contrast, the current model, which yields spread speeds of ∼40 m year^1 (fig. 3 a , 3 b ), adopts an integrodifference equation framework with as- sumed rather than estimated exponential kernels. The co- efficient governing the lupine’s kernel is based on our knowledge that long-distance seed dispersal events (e.g., 1 100 m) do occur but are relatively rare (Wood and del Moral 2000). The framework we present here is thus a model of population-level rather than patch-level spread. Although our model’s predictions (e.g., spread over ∼ 25 years of ∼1,000 m) are likely conservative because we lack
good measures of very long distance dispersal, the general results match up well with information gleaned from a recent survey of the entire pumice plains region (J. G. Bishop and J. Apple, unpublished data). Understanding the broad importance of lupine’s herbivore-mediated absence from so much of this land- scape requires an overview of lupines’ role in successional dynamics at Mount St. Helens. Soon after lupines’ initial colonization of the volcanically devastated pumice plain, ecologists predicted that lupines would have major effects on the successional development because they are nitrogen fixers, have a temporal advantage, and can colonize a great diversity of sites. These predictions about the importance of lupine to local community development have proved accurate. For example, lupines accelerated soil develop- ment through direct nutrient and organic matter input, trapping of windblown debris and propagules, attraction of insects that ultimately die in situ, and attraction of animal dispersal vectors that transport seeds and micro- organisms (Morris and Wood 1989; del Moral and Rozzell, forthcoming). Soils under lupines have much higher levels of total N, organic matter, and microbial activity than do adjacent bare areas (Halvorson et al. 1991, 1992; Halvorson and Smith 1995; Fagan et al. 2004). Furthermore, exper- iments demonstrate a net positive effect of lupines on growth of ruderal plant species (Morris and Wood 1989; Titus and del Moral 1998), although these species may have to wait for lupines to die to take advantage of par- ticular sites. Recent surveys demonstrated that percent cover of other plant species was higher within lupine patches than outside them and that species composition differed substantially inside and outside lupine patches (del Moral and Rozzell, forthcoming). Indeed, nutrient- responsive species such as the grass Agrostis pallens are so common in core-type lupine patches that the region was, at one point, predicted to transition successionally into high-elevation grasslands, although grass-eating herbivores (elk and grasshoppers) have put an end to this idea (C. Crisafulli, personal communication). The presence of lupines is clearly important to com- munity development at Mount St. Helens. By extension, then, insect herbivory that has prevented or at least delayed lupine colonization of large portions of the pumice plain has indirectly determined the course of primary succession by hindering the early stages of soil formation and chang- ing the floristic trajectory. This conclusion, that insect her- bivores are helping to shape the pace and pattern of pri- mary succession at Mount St. Helens, runs counter to the general theory of primary succession (Walker and del Mo- ral 2003; see also app. D in the online edition of the American Naturalist ). We suspect that insect herbivores may be more important to succession in disturbed land- scapes than is commonly appreciated. The lesson of this
684 The American Naturalist
successfully locate, attack, and have substantial negative ef- fect on small patches of its host plant. If the lupine were a weed whose spread we sought to limit, early introduction of Filatima as a biocontrol agent would clearly be a key to the success of our management actions. This would be true whether the goal was to collapse the weed population or, more plausibly, to slow the spatial spread rate of the weed (fig. 5). The unfortunate reality, of course, is that it may be difficult to convince managers to expend resources on an invading species that is not yet a pest.
Conclusions
Our model suggests that chance events have played a strong role in shaping the successional trajectory of Mount St. Helens. In particular, the timing of colonization by lupine relative to the onset of herbivory appears to have been crit- ical to the initial success and subsequent spread of the co- invasion. We believe that the lupine- Filatima system makes a good case for the importance of insect herbivory during succession, but when and where herbivory occurs (i.e., early vs. late and low- vs. high-density areas) may be at least as important as how much herbivory occurs.
Acknowledgments
We are indebted to M. Owen for crucial insights, helpful discussions, double-checking the Matlab code, and unflag- ging interest in the havoc that little bugs can cause for purple plants. This effort would not have been possible without the support of the National Science Foundation through its Mathematical Biology and Ecological Studies programs (awards NSF OCE-9973212, DEB-9973518, DEB-0235692, and DEB-0089843). The University of Maryland, Washing- ton State University–Vancouver, and the Banff International Research Station for Mathematical Innovation and Discov- ery provided additional support. M.L. acknowledges sup- port from Mathematics of Information Technology and Complex Systems Canada, a Natural Sciences and Engi- neering Research Council operating grant, a Contract Re- search Organisation grant, and a Canada Research Chair. For identifying the lupine moths, we are indebted to D. Adamski ( Filatima ), L. Crabo ( Euxoa ), and A. Solis ( Staudingeria ).
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Associate Editor: Benjamin M. Bolker Editor: Donald L. DeAngelis
