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Modeling the interactions between an exotic
invasive aquatic macrophyte (Myriophyllum
spicatum L.) and a native biocontrol agent
(Euhrychiopsis lecontei Dietz).
Kyle Miller, Heath Garris and Lara
RoketenetzIntegrated Biosciences, University of
Akron
Introduction ECOLOGICAL BACKGROUND Invasive
species are non-indigenous plants or animals that
become established in a natural area and pose
risks to the existing ecosystem through loss of
local biodiversity, changes in overall species
composition and loss of recreational value.
Eurasian watermilfoil (Myriophyllum spicatum L.),
native to Europe and Asia, is an invasive
submerged aquatic plant found throughout most of
the United States and portions of Canada. Since
its introduction in the 1940s it has become one
of the most noxious aquatic weeds in North
America (Smith and Barko, 1990).
Eurasian watermilfoil forms
dense ?oating mats that can reduce native
macrophyte populations, as well as associated
invertebrate and ?sh populations. Historic
control of this invasive plant have included
chemical, mechanical, and physical methods such
as herbicide application, raking, suction
harvesting and bottom barriers. Researchers have
been investigating the ability of the native
milfoil weevil (Euhrychiopsis lecontei Dietz) to
control populations of Eurasian watermilfoil..
Bolstering populations of native host-speci?c
predators for biological control is a relatively
novel concept. MATHEMATICAL MODELING
BACKROUND Owing to the relative importance and
impact of M. spicatum as a noxious weed, many
articles have been published describing
mathematical growth models (Herb and Stefan,
2006 Titus et al., 1975). However, little work
exists concerning mathematical models of the
milfoil weevil E. lecontei. These sources of
uncertainty have resulted in a lack of successful
prediction of treatment efficacy in natural lake
systems. Despite this, there has been some work
done on characterizing life cycle, development
time, and survival rates of E. lecontei (Sheldon
and OBryan, 1996 Mazzei et al., 1999). Using
this information it was possible to construct an
age-structured population model based on
development index. The M. spicatumE. lecontei
interaction has been characterized as follows. It
has been shown that E. lecontei lays its eggs on
the meristems of M. spicatum. Damage caused by
larval tunneling destroys biomass and interrupts
gas exchange within M. spicatum reducing
buoyancy. Finally, it has been shown that M.
spicatum translocates energy reserves into its
roots late in the growth season. It has been
suggested that the larval tunneling also
interrupts this process, thereby reducing the
fecundity of M. spicatum in the following
season. Model Development MERISTEM MODEL
DEVELOPMENT The number of meristems on a plant
can be modeled in the following way. Consider a
plant of initial length PP0 and increment plant
length using a branching pattern. From this
illustration it is clear that the number of
meristems is equal to 2P-1. Therefore we conclude
it is reasonable to model meristem count as an
exponential function of plant length or biomass.
A SIMPLIFIED ARGUMENT The following is a brief
mathematical argument regarding expected behavior
of this predator prey system. In simple terms,
growth of M. spicatum could be considered as
follows. where P(t) is the total
plant mass at time t, g and d are constants
representing the rate of growth and damage
respectively, and L(t) represents the larval
population. Considering that E. lecontei egg
laying behavior is dependent on available
meristems, a best case scenario may be that the
larval population equals the number of meristems.
That is, the plant system is totally saturated
with larva. Thus it could be argued that L(t) is
a function of P(t) L(t) f(P(t)) f0 ?P(t)
using an exponential function to model meristem
count as a function of plant mass. The above
equation becomes If we plot dP(t)/dt as a
function of P(t) as in figure 4 it can be seen
that the system has two critical points. One is
stable, the other unstable. This indicates that
for any initial condition P(0) gt a the system
will tend toward b. This simple argument shows
that utilization of E. lecontei should be
expected to serve as a method of control rather
than eradication of M. spicatum. Figure
4. dP(t)/dt vs. P(t) - indicating stable and
unstable critical points under the simplified
argument. FULL MODEL DEVELOPMENT Prey -
Myriophyllum spicatum Growth Model The M.
spicatum - E. lecontei interaction was modeled
using a predator-prey style system of
differential equations. An existing growth model
for submerged macrophytes presented by Herb and
Stefan (2006) was used as the foundation for the
M. spicatum portion of the system. This model was
then modified to incorporate the impact of the E.
lecontei population. The resulting derivation is
shown below. Net biomass production within the
water column is represented by with biomass
P, growth rate ?, loss due to respiration ?,
mechanical loss ?, irradiance I, and temperature
T. Irradiance attenuates exponentially though the
water column blocked by turbid water and biomass
according to Beers law. Taking ? to be an
exponential function of temperature and defining
total biomass W allows the model to be simplified
under the assumptions of constant temperature and
biomass density throughout a partitioned water
column. The complete M. spicatum growth model
is obtained using a partition size of 2
representing canopy and sub-canopy layers,
incorporating terms for use of stored energy, and
adding elements describing damage by E. lecontei
larva.
Predator - Euhrychiopsis lecontei Growth
Model While no E. lecontei-specific population
models were found in the literature, a
development index based age-structured population
model is sufficient to capture many of the
important population dynamics. Such an
age-structured population model is defined in
the following way. Simulatio
n Results The system of differential equations
was solved numerically using a Runge Kutta
method. Simulations were run to interrogate the
impact of stocking adult weevils at the start of
a growth season. For each scenario, the left
graph represents M. spicatum biomass, the right
represents the E. lecontei population.
Conclusions Here, we sought to
capture the largest components of this system
while maintaining some degree of mathematical
simplicity. As a result, certain simplifying
assumptions were made during the development of
the mathematical model. Understanding these
assumptions and their implications is important
to interpreting model results and insights. Of
particular note is the influence of M. spicatum
on the E. lecontei population. This interaction
is as follows. Meristems are considered
occupied if a larva is present and available
otherwise. This implies that a meristem becomes
immediately available following a larval
occupation without the need for a period of
repair or replacement. In addition, availability
is not a function of current egg population.
Thus, the initial egg population and subsequent
larval population can be super saturated under
certain conditions, as in the 300 adult scenario
for example. As survivability rates in the model
are unaffected by overpopulation, accuracy should
be suspect under high initial populations.
Scenario 3 Initial adult weevil population of 100
Scenario 4 Initial adult weevil population of 300
Table 1. Description of model parameters and
dependent variables.
Field Work Field work was completed in May, 2009
to assess the distribution of M. spicatum in Six
Mile Lake, Michigan, and to evaluate meristem
density. Data concerning the number of meristems
per plant were recorded and used to estimate
relevant model parameters.
W(t) M. spicatum biomass
C(t) M. spicatum stored energy (e.g. nonstructural carbohydrates)
T(t) Temperature at time t
L(t) E. lecontei larval population at time t
µ0 M. spicatum base growth rate
k1 Irradiance half saturation constant
?g Growth related temperature base (constant)
Tb Base temperature at which other constants are measured
Kwt Light attenuation constant due to water clarity
km Light attenuation constant due to blockage by biomass
d Water depth
h M. spicatum stand height
I0 Surface irradiance level
µ1 Stored energy usage rate
usubscript Step function (value 1 or 0) taking a value of 1 at the appropriate time for term subscript
?r Biomass decline related temperature base (constant)
?0, ?1 Base rate of biomass loss and energy usage
?1, ?2, ?3, ?4 Various rate constants for relating terms
e1, e2 Constants with small value
Dc E. lecontei development class egg, larva, pupa, adult
NDc Number of individuals in development class Dc
RDc Rate of development of E. lecontei individuals
mDc, vDc Death and immigration/emigration rates
aDc Base development rate for development class Dc
Tw Minimum temperature for E. lecontei development
µ3 Base egg laying rate for E. lecontei adults
k2 Egg laying rate temperature half saturation constant
?w Meristem count exponential base (constant)
Ps M. spicatum stand density
Parameter Estimation of ?max(W)/ ?parameter
µ0 base growth rate for M. Spicatum 8.34103
Ps M. spicatum stand density -1.32100
aE base egg laying rate -2.32101
aegg base egg development rate 9.97101
alarva base larval development rate -3.13102
apupa base pupa development rate -1.49102
ln(?w ) log of meristem count exponential base -5.60102
SENSITIVITY ANALYSIS
Figure 1. Known current distribution of
Myriophyllum spicatum (Eurasian watermilfoil)
Figure 2. Euhrychiopsis lecontei (milfoil weevil)

Selected parameters were interrogated over a
range of /- 10 for an initial population of 100
adults. ln(?w) was varied as opposed to ?w
directly, as ?w is the base of an exponential
term. The results were then used to approximate
the ratio of change in maximal biomass to change
in parameter value.
Figure 3. The developmental stages of E. lecontei
(egg, larva, pupa, adult)
Table 2. Estimated effect of parameter variation.
Figure 5. Meristem of M. spicatum
Figure 6.
,
Literature cited Grace, J., and R. Wetzel. 1978.
The production biology of Eurasian watermilfoil
(Myriophyllum spicatum L.) a review. Journal
of Aquatic Plant Management 16111. Herb, W.,
and H. Stefan. 2006. Seasonal growth of submersed
macrophytes in lakes The e?ects of biomass
density and light competition. Ecological
Modelling . 193560-574 Mazzei, K., R. Newman, A.
Loos, and D. Ragsdale. 1999. Development rates of
the native milfoil weevil, Euhrychiopsis
lecontei, and damage to Eurasian watermilfoil at
constant temperatures. Biological Control
16139143. Sheldon, S., and L. OBryan. 1996.
Life history of the weevil Euhrychiopsis
lecontei, a potential biological control agent of
Eurasian watermilfoil. Entomological News
1071622. Titus, J., R. Goldstein, M.
Adams, J. Mankin, R. ONeill, P. Weiler, H.
Shugart, and R. Booth. 1975. A production model
for Myriophyllum spicatum L. Ecology 5611291138.
Scenario 1 Initial adult weevil population of 0
1 Meristem
7 Meristems
15 Meristems
3 Meristems
Acknowledgments We thank Dr. Peter Niewiarowski,
Dr. Young math department, and our colleagues in
the Integrated Biosciences Program. Funding for
this project was provided by the University of
Akron Department of Biology.
Scenario 2 Initial adult weevil population of 50
Length 3 P0
Length P0
Length 2 P0
Length 4 P0

For further information Please contact
jkm29_at_uakron.edu, hwg3_at_uakron.edu, or
ldr11_at_uakron.edu