Dynamics at the Horsetooth Vol. 2, 2010
Tamarisk biocontrol: implications for riparian community composition
D. A. Auerbach
Graduate Degree Program in Ecology
Colorado State University
daa@colostate.edu
Report submitted to Prof. P. Shipman for Math 540, Fall 2010
Abstract: The question of tamarisk biocontrol by the Diorhabda beetle and the consequences for
cottonwood, an important native riparian tree are examined using a simple Lotka-Volterra formulation
for the interaction between a specialist predator and two competing resources. The behavior of the
system relative to ecologically plausible parameterizations is explored, and the expected outcomes are
interpreted in terms of these species. The model suggests that although tamarisk is likely to persist
even under intense predation because of the character of its competitive relationship with cottonwood,
the biocontrol effort may realistically result in lower equilibrium densities for the non-native and higher
for the native. Finally, inconsistencies between the model and the naturally occurring community are
briefly described in terms of opportunities for further research.
Keywords: predator-prey, competition, riparian community, biocontrol
1. Introduction
Mathematical representations of community interaction have been central to the conceptual development
of ecology (Kingsland 1995), and efforts to characterize the behavior of multi-trophic systems continue
to refine our basic understanding of progressively more complex assemblages (Chesson and Kuang
2008). Such models may also yield important insight into various pressing management questions, for
example illustrating the likely outcomes for biocontrol programs intended to restrict or eradicate
invasive species. Along rivers of the arid and semi-arid Western United States, the detrimental effects
of widespread establishment and floodplain dominance by tamarisk (hybrids of several species of genus
Tamarix) following its intentional introduction as an erosion control measure led to the search for, study
of and eventual release of an appropriate biocontrol agent (Shafroth et al. 2005, Hultine et al. 2010).
The recent release of beetles of several species of the genus Diorhabda (primarily D. elongata) has
resulted in extensive defoliation of large, monotypic tamarisk stands in some areas little to no damage in
others (D. Auerbach pers. obs., W. Williams pers. comm.). The effect of tamarisk biocontrol on the
composition of Western floodplain forests, particularly the degree to which it may create favorable
opportunities for native cottonwood trees (primarily Populus deltoides and P. fremontii) remains an
open question. Numerous stakeholders have an interest in this ongoing, politically charged ecological
story (Hultine et al. 2010), yet, to my knowledge, no published model of the interaction currently exists.
Study of a vastly simplified portrayal of the tamarisk-cottonwood-Diorhabda assemblage (TCD) will
not generate quantitatively meaningful predictions for the transient or equilibrium densities of these
community members, but may usefully distinguish qualitative outcomes and may suggest productive
directions for both further modeling and empirical research. Although future work may extend the
analysis to the regional-scale, introducing the complexity of between-site movements in a dendritic river
network (sensu Campbell-Grant et al. 2007), here I consider the simpler case of within-floodplain
dynamics. Thus, the model describes only local birth and death process, requiring the assumption that
immigration and emigration do not significantly influence behavior. Stage-structure and temporally
variable environmental forcing (e.g. high and low flows and temperatures) are likely important
considerations in the population dynamics of all three of the species of interest (Lewis et al. 2003, Lytle
Tamarisk biocontrol
D.A. Auerbach
and Merritt 2004), but homogenous populations subject only to intrinsic biological constraints are
assumed for the purposes of developing a first approximation from which more complicated models
structures can build. Note that “species” is used here for convenience, despite the complexity of the
respective taxonomies, and that “predator” and “consumer” are used interchangeably following the
arguments of Holt and Lawton (1994).
The paper is organized as follows: section 2 provides a brief review of the encompassing generalistconsumer-two-resource system, section 3 describes the reduced version of this system appropriate to the
issue of tamarisk biocontrol and explores the behavior of this system under different competitive
domains and levels of predation, and section 4 concludes with a discussion of the management relevance
of the mathematical results and suggestion for further research.
2. A generalized Lotka-Volterra formulation
While the question of consumer mediated competitive coexistence has received considerable empirical
and theoretical attention (reviewed by Holt and Lawton 1994 and Chase et al. 2002), much of this has
focused on the situation in which a generalist consumer (or guild of consumers) attacks competing
resource populations. Takeuchi and Adachi investigated a general formulation of this system:
dR1
= R1(b1 − R1 − αR 2 − εC)
dt
dR 2
= R 2(b 2 − βR 2 − R 2 − µC)
dt
dC
= C ( −b 3 + εdR1 + µdR 2)
dt
Where R1 and R2 describe the densities of competitors with intrinsic growth rates bi and interspecific
density effects α and β. A shared consumer C attacks R1 and R2 at rates ε and µ, with conversion
€
efficiency d and intrinsic growth
rate b3. Their analysis of the predator’s effect on the system
distinguished three conditions for the underlying competitive dynamics: weak competition leading to
stable coexistence of R1 and R2, stronger competition implying dependence on initial conditions, and
competitive dominance by R1 or R2.
The first situation corresponds to the general coexistence criteria suggested by Chesson (2000), whereby
negative intraspecific density effects are greater than interspecific ones. Takeuchi and Adachi observed
that introduction of a generalist predator afforded additional possibilities for coexistence in the second
and third situations when it would be otherwise impossible. When competition is strong but neither
resource species maintains an overall advantage (such that initial conditions determine which will
exclude the other to arrive at a locally stable equilibrium under two-way competition), the relationship
between the attack rate parameters ε and µ determines the system’s behavior, which can include a Hopf
bifurcation between stably cycling coexistence and collapse to a single resource species. In the third
circumstance, depending on the strength of competition relative to predator preference, a consumer can
potentially counteract the dominance of one resource species to support either stable three-way
coexistence or, following another Hopf bifurcation, periodic and eventually chaotic cycles. These
authors also noted, however, that the latter “spiral chaos” form of coexistence (described earlier in
Vance 1978 and Gilpin 1979) was, “nonsense from the biological point of view since the population
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D.A. Auerbach
densities of [the] three species become nearly equal to zero in the evolution of the system” (Takeuchi
and Adachi 1983).
3. The specialist predator system
These results provide guidance for the simplified TCD system in which individuals of Diorhabda sp. are
specialized to attack only tamarisk and therefore fail to persist in its absence. Removing terms
containing µ then yields the reduced equations for a specialist consumer and two competing resource
species, when the first is subject to predation:
dR1
= R1(b1 − R1 − αR 2 − εC)
dt
dR 2
= R 2(b 2 − βR 2 − R 2)
dt
dC
= C ( −b 3 + εdR1)
dt
Without making any assumptions regarding the strength of competition between tamarisk and
cottonwood, this system has ecologically plausible fixed points corresponding to extinction of all three
species (0,0,0), persistence of€either tamarisk (+,0,0) or cottonwood (0,+,0), coexistence of both trees
but not the beetle (+,+,0), coexistence of tamarisk and the beetle (+,0,+), and coexistence of all three
species (3+).
The Jacobean for this system is given by:
⎡b1 − 2R1 − αR 2 − εC
−αR1
−εR1 ⎤
⎢
⎥
J = ⎢
−βR 2
b 2 − βR1 − 2R 2
0
⎥
⎢⎣
εdC
0
εdR1 − b 3 ⎥⎦
Unless the intrinsic growth parameters are zero the three-way extinction fixed point is unstable by
definition (three real eigenvalues, two positive). If tamarisk is the only surviving species at equilibrium,
€
then its stable density
and the evaluated Jacobean are:
R1* = b1
⎡ −b1 −αb1
−εb1 ⎤
⎢
⎥
J (+, 0, 0) = ⎢ 0 b 2 − βb1
0 ⎥
⎢⎣ 0
0
εdb1 − b 3 ⎥⎦
The top row of J(+,0,0) is negative by definition, and the terms on the diagonal are either negative or zero
if this fixed point is biologically plausible. Indeed, the local stability conditions for this fixed point
€
given by Takeuchi and Adachi
(1983) are those that produce negative values on the diagonal and
consequently three negative real eigenvalues (numerical exploration not shown). In accordance with
biological expectations, this indicates that the persistence of only tamarisk in this system requires both
that cottonwood be competitively subordinate (β>b2/b1) and that Diorhabda be an ineffective consumer
(such that its intrinsic decline is greater than its gains through predation).
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D.A. Auerbach
Similarly, if only cottonwood is to stably persist then tamarisk must be a sufficiently bad competitor that
its intrinsic growth is weaker than the negative effects due to cottonwood (making the upper left element
and real parts of eigenvalues negative).
R2* = b 2
⎡b1 − αb 2 0
0 ⎤
⎢
⎥
J (0, +, 0) = ⎢ −βb 2
−b 2 0 ⎥
⎢⎣ 0
0 −b 3 ⎥⎦
Stable coexistence of the two trees (+,+,0) in the absence of the beetle has the following form
€
b1 − αb 2
1 − αβ
b 2 − βb1
R2* =
1 − αβ
R1* =
This equilibrium is globally stable (by Theorem 5 of Takeuchi and Adachi 1983) when competition is
bilaterally weak, with the condition α<b1/b2 and β<b2/b1 (and αβ<1). If the increasing strength of
bilateral competition means that the€product of α and β is greater than one, then this equilibrium may
still be locally stable if both α>b1/b2 and β>b2/b1 and depending on the relative strength of dominance
and predation (see below).
Both tamarisk and cottonwood are “weedy” early successional species, adapted to take advantage of
fresh, bare alluvium such as that deposited following floods (via a suite of “fugitive” or “r-selected”
traits such as numerous small seeds and rapid seedling growth). A variety of hypotheses have been
proposed for the dominance of tamarisk on many Western floodplains (e.g. Figure 1 in Merritt and Poff
2010), but recent research indicates that flow alteration resulting from water infrastructure has combined
with tamarisk’s longer period of seed viability to favor it in scramble competition for establishment
space (Merritt and Poff 2010). Without certainty regarding the exact mechanisms of interspecific
interaction, it is nonetheless plausible to parameterize the system such that the tree species have identical
intrinsic growth rates (b1 = b2) while experiencing different levels of competition.
It is plausible that tamarisk is competitively superior to cottonwood, but that competition is relatively
weak, so that α< β<1, and cottonwood is not excluded in the absence of Diorhabda (H1). Alternatively,
conditions could be such that a competitively superior tamarisk excludes cottonwood in the absence of
the beetle (α<b1/b2<1 and β>b2/b1>1, H2). Although perhaps less likely given the intensity of
disturbance on Western floodplains, tamarisk could also be weakly superior under stronger overall
competition, so that α< β but α>b1/b2>1 and β>b2/b1>1 (H3). Assuming convenient values for the
intrinsic change in the Diorhabda population and its resource assimilation (b3=d=1), it is then possible
to focus on the effects of changing the intensity of predation (variation in the attack rate ε) under each of
these hypotheses.
The fixed points at (+,0,+) and E3 are relevant to this question. In the absence of cottonwood, the
conditions for the coexistence of tamarisk and the beetle at equilibrium are:
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D.A. Auerbach
b3 1
C˙ = 0 ∩ C > 0 ⇒ −b 3 + εdR1 = 0 ⇒ R1* =
=
εd ε
b3
1 ⎛
b 3 ⎞ 1 ⎛ 1 ⎞
R˙ 1 = 0 ∩ R1 > 0 ⇒ b1 − − εC = 0 ⇒ C * = ⎜ b1 − ⎟ = ⎜1 − ⎟
εd
ε ⎝
εd ⎠ ε ⎝ ε ⎠
and therefore
b3
ε>
⇒ ε >1
b1d
This can be interpreted simply (and in accordance with the classic formulation of a two-species
predator-prey subsystem) as the situation in which gains to the consumer (attack rate x assimilation
efficiency €
x resource growth) are greater than its intrinsic decline. The Jacobean evaluated at this point
is
⎡ b 3
⎢ − εd
⎢
J (+, 0, +) = ⎢ 0
⎢
⎢ db1 − b 3
⎣
ε
b3
εd
b3
b2 − β
εd
−α
0
b 3 ⎤ ⎡ 1
1
−
−α
⎥
⎢
d
ε
ε
⎥ ⎢
1
0 ⎥ = ⎢ 0
1− β
ε
⎥ ⎢
1
0 ⎥ ⎢1 −
0
⎦ ⎣ ε
−
⎤
−1⎥
⎥
0 ⎥
⎥
0 ⎥
⎦
Again, the top row is negative by definition, and the lower left entry is positive by the conditions for the
plausible existence of the fixed point. The absence of cottonwood implies the central element is
€ the product of the competitive effect of tamarisk on cottonwood (β) and its predationnegative when
controlled density are greater than cottonwood’s intrinsic growth (here β>ε). Numerical evaluation
indicates that if (+,0,+) exists then the real parts of all three eigenvalues are negative (e.g. for b1 = b2 =
b3 = d =1, β = 1.5, ε = 1.25, and αH2=0.5 or αH3=1.25 the eigenvalues are λ1= -0.2, λ+/-= -0.4±0.2i).
Coexistence of all three species (E3) has the following equilibrium densities:
b3 1
=
εd ε
b3
β
R2* = b 2 − β = 1 −
εd
ε
⎞ 1 ⎛
⎞
1 ⎛
b3
1
C * = ⎜b1 − αb 2 − (1 − αβ)⎟ = ⎜1 − α − (1 − αβ)⎟
⎠ ε ⎝
⎠
ε ⎝
εd
ε
R1* =
With the positivity conditions:
€
ε >0
ε>β
1 − αβ
ε>
1−α
€
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Tamarisk biocontrol
And JE3 evaluates as:
D.A. Auerbach
⎡
⎤
1
α
⎢
−
−
−1⎥
ε
ε
⎢
⎥
⎛
⎞
⎛
⎞
β
β
⎢
JE 3 =
β⎜ −1⎟
⎜ −1⎟ 0 ⎥
⎢
⎥
⎝ ε ⎠
⎝ ε ⎠
⎢⎛
⎥
⎞
1
0
0 ⎥
⎢⎜1 − α − (1 − αβ)⎟
⎠
ε
⎣⎝
⎦
Again, numerical evaluation indicates that if E3 exists (i.e. if the positivity conditions are met), the real
parts of the eigenvalues are negative. The inference that these fixed points are stable if they are possible
is further supported by€numerical solutions of the system. Figure 1 illustrates how increasing the value
of the parameter controlling the attack rate of the specialist predator alters outcomes in relation to the
strength of competition and the degree of superiority of the consumed resource. Under sufficiently weak
competition that the resources coexist independently of the consumer (H1), an increase in predation
pressure allows the E3 fixed point to emerge from the (+,+,0) equilibrium as ε exceeds the quotient of
(1-αβ)/(1-α) and the consumer is able to balance its intrinsic decline. Increasing ε well beyond the range
shown in Figure 1 does not appear to produce any new dynamics, though numerical errors are produced
at values much larger than shown. Finally, it is perhaps worth noting that the E3 fixed point can
correspond to higher equilibrium densities for either resource.
If the superior resource excludes the unconsumed resource in the absence of predation (H2), then
increasing predation can generate both the (+,0,+) fixed point and three way coexistence. Consumersuperior resource coexistence occurs as ε exceeds the ratio of its intrinsic decline to the product of
assimilation efficiency and the resource growth rate (here ε>1). When ε > β, or predation pressure is
greater than the competitive effect of the stronger resource on the weaker, E3 is possible and stable. In
contrast, this transition is not possible under stronger competition even if the superior resource would
exclude the weaker in the absence of predation (such that αβ>1 and α>1 but α<β). As shown in the
lower panel of Figure 1, the system instead jumps from the (+,0,+) fixed point to the (0,+,0) equilibrium,
as the combination of predation and competition cause the superior resource population to oscillate close
enough to zero that it crashes. Additionally, this behavior is sensitive to the consumer’s initial density,
with initial values greater than approximately 0.4 leading to the loss of the range of ε values that can
support the (+,0,+) fixed point. These results are summarized in Table 1 and Appendix A contains
phase portraits for these qualitatively different regions.
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Figure 1: Bifurcations as a function of variation in the attack rate parameter ε under three hypothesized
competitive domains. R1=tamarisk, R2=cottonwood, C=Diorhabda. Initial densities of all three species = 0.3.
(H1) When resources coexist in the absence of the consumer, increasing predation pressure can shift the system
to E3 from (+,+,0). (H2) If exclusion occurs under weak competition (+,0,0), an increase in predation can
sustain both consumer and superior resource (+,0,+) or E3 if ε>β. (H3) While outcomes are similar under
strong competition and weak predation, as ε exceeds a critical value at β when both α>b1/b2>1 and β>b2/b1>1
the system shifts abruptly to the fixed point at (0,+,0) corresponding to exclusion by the “weaker” competitor.
In this case, consumer and consumed oscillate to collapse of the resource. Note that the dynamics in the first
two panels are insensitive to initial conditions, but that higher initial consumer densities can cause the loss of
C-R1 coexistence in the H3 case. See text for details.
4. Discussion
Although the system describing the dynamics of a specialist predator consuming one of two competing
resources appears to lose most of the mathematically interesting behavior associated with the
encompassing generalist predator system (i.e. the limit cycles and spiral chaos described in Takeuchi
and Adachi (1983) and Bazykin (1998)) it nonetheless has potentially compelling implications for the
future of floodplain forests in the Western U.S. If the system analyzed here were a reasonable proxy for
the interactions of tamarisk, cottonwood and Diorhabda, the results summarized in Table 1 suggest that
biocontrol is unlikely to eliminate tamarisk unless the least intuitive scenario of strong competition
holds. While this is certainly not out of the question, empirical observations of the prolonged cooccurrence of both tree species (i.e. H1) and monotypic tamarisk stands (i.e. H2) are more common.
Furthermore, the trees compete primarily for establishment surfaces as seeds and for light as seedlings
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rather than for the supply of nutrients as adults. Spatial heterogeneity in appropriate geomorphic forms,
the inter- and intra-annual sequence of flood flows, and the relatively nutrient, light and water rich
conditions in riparian zones all combine to reduce the ecological plausibility of the strong competition
scenario and consequently the possibility of beetle-mediated tamarisk eradication. Although not
described in detail because it is even less intuitive, the model also predicts that even very low levels of
predation suffice to advantage cottonwood and lead to the elimination of tamarisk in the event that
tamarisk and cottonwood are competitively equivalent and strongly competing.
Perhaps more hopeful to managers interested less in the eradication of tamarisk than the maintenance of
natives such as cottonwood (though the argument could apply to several other native species in decline),
is the model’s prediction of considerably reduced equilibrium densities for tamarisk and stable
coexistence with an otherwise excluded competitive subordinate given sufficient consumption by
Diorhabda. Before any such conclusion can be made with confidence, however, it will be necessary to
examine the effects of including alternative predator responses (e.g. saturation related to the beetles’
numerical and functional responses), population stage-structure (e.g. seedlings and adults and the known
differences in consumptive capacity of the three beetle instars and adults), and time lags (e.g. tamarisk’s
ability to survive multiple rounds of defoliation and perhaps recover following the departure or death of
beetles). Moving beyond the species considered here, the ultimate state of riparian forest is also
certainly influenced by other competing trees (particularly Russian Olive, Elaeagnus angustifolia) as
well as other herbivores (beaver, cattle) and higher trophic levels (e.g. spiders that consume Diorhabda,
sensu Chesson & Kuang 2008). As a further frontier, Cuddington and Yodzis (2002) demonstrated how
a complexly branching habitat structure could reduce the interaction rates between predators and prey
and consequently alter outcomes. Clearly, the dynamics examined here provide at most a first glimpse
of the natural system’s full complexity, yet this initial indication provides a enticing sense of the rewards
of attempting to capture these dynamics.
Competitive
domain
Predation intensity
Low
Intermediate
High
H1
(+,+,0)
E3
E3
H2
(+,0,0)
(+,0,+)
E3
H3
(+,0,0)
(0,+,0)
(0,+,0)
Table 1: Qualitative outcomes of the specialist predator, 2 resource system according to
three scenarios for the strength and form of competition and variation in the degree of
predation on the consumed resource.
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Bibliography
D.A. Auerbach
Bazykin, A.D. 1998. Nonlinear dynamics of interacting populations. World Scientific. Singapore.
Campbell-Grant, E.H., Lowe, W.H., and W.F. Fagan. 2007. Living in the branches: population
dynamics and ecological processes in dendritic networks. Ecol. Lett. 10:165-175.
Chase et al. 2002. The interaction between predation and competition: a review and synthesis. Ecol.
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Chesson, P. 2000. Mechanisms of maintenance of species diversity. Ann. Rev. Ecol. Syst. 31:343-366.
Cuddington, K., and P. Yodzis. 2002. Predator-prey dynamics and movement in fractal networks. Amer.
Natur. 160:119-134.
Chesson, P. and J. Kuang. 2008. The interaction between predation and competition. Nature. 456:235238.
Holt, R.D. and J.H. Lawton. 1994. The ecological consequences of shared natural enemies. Ann. Rev.
Ecol. Syst. 25:495-520.
Hultine et al. 2010. Tamarisk biocontrol in the western United States: ecological and societal
implications. Front. Ecol. Environ. 8:467-474.
Kingsland, S.1995. Modeling Nature. University of Chicago Press. Chicago.
Lewis, P. A. et al. 2003. Biology of Diorhabda elongata deserticola (Coleoptera: Chrysomelidae), an
Asian leaf beetle for biological control of saltcedars (Tamarix spp.) in the western United States.
Biological Control 27:117-147.
Lytle D.A. and D.M. Merritt. 2004. Hydrologic regimes and riparian forests: a structured population
model for cottonwood. Ecology 85:2493–2503.
Merritt, D.M. and N.L. Poff. 2010. Shifting dominance of riparian Populus and Tamarix along gradients
of flow alteration in western North American rivers. Ecol. Applic. 20:135-152.
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Appendix A
D.A. Auerbach
Example trajectories corresponding to qualitative variation in the behavior of the specialist predator, two
resource system under three competitive domains and varying levels of predation pressure. For all three
cases b1 = b2 = b3 = d =1. In A1 and A2 α=0.5 and β = 0.5 and 1.5 respectively. In A3, α=1.25 and β =
1.5. As described above, the first two competitive scenarios are insensitive to initial conditions, while
the last (H3) loses the possibility of consumer-superior resource coexistence at higher initial consumer
densities. In effect, this appears to yield the predation-driven collapse of the consumed resource more
readily. Regardless, the “coexistence” window under these circumstances is somewhat suspect to the
extent that it involves periods of very low consumer density.
Figure A1: H1, competition weak, resources coexist in absence of predator.
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Figure A2: H2, competitive exclusion by the consumed resource in the absence of the consumer;
competitive effects on the superior resource are weak.
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Figure A3: H3, competitive exclusion by the consumed resource in the absence of the consumer;
competitive effects on the superior resource are strong.
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