Predator - Euhrychiopsis lecontei Growth ModelWhile 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.

Simulation ResultsThe 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.

IntroductionECOLOGICAL BACKGROUNDInvasive 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 floating mats that can reduce native macrophyte populations, as well as associated invertebrate and fish 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-specific predators for biological control is a relatively novel concept.

MATHEMATICAL MODELING BACKROUNDOwing 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 O’Bryan, 1996; Mazzei et al., 1999). Using this information it was possible to construct an age-structured population model based on development index.

The M. spicatum–E. 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 it’s 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 DevelopmentMERISTEM MODEL DEVELOPMENTThe number of meristems on a plant can be modeled in the following way. Consider a plant of initial length P=P0 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) > 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 ModelThe 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 Beer’s 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.

ConclusionsHere, 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.

Kyle Miller, Heath Garris and Lara RoketenetzIntegrated Biosciences, University of Akron

Modeling the interactions between an exotic invasive aquatic macrophyte (Myriophyllum spicatum L.) and a native

biocontrol agent (Euhrychiopsis lecontei Dietz).

Figure 3. The developmental stages of E. lecontei (egg, larva, pupa, adult)

Figure 1. Known current distribution of Myriophyllum spicatum (Eurasian watermilfoil)

Figure 5. Meristem of M. spicatum

Figure 2. Euhrychiopsis lecontei (milfoil weevil) Parameter

Estimation of Δmax(W)/

Δparameter μ0 – base growth rate for M. Spicatum 8.34×103

Ps – M. spicatum stand density -1.32×100

αE – base egg laying rate -2.32×101

αegg – base egg development rate 9.97×101

αlarva – base larval development rate -3.13×102

αpupa – base pupa development rate -1.49×102

ln(θw ) – log of meristem count

exponential base-5.60×102

SENSITIVITY ANALYSISSelected parameters were interrogatedover a range of +/- 10% for an initialpopulation of 100 adults. ln(θw) was variedas opposed to θw directly, as θw 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.

Length P0 Length 2 P0 Length 3 P0

1 Meristem 3 Meristems 7 Meristems

Length 4 P0

15 Meristems

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Table 1. Description of model parameters and dependent variables.

Table 2. Estimated effect of parameter variation.

,

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

θg 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

θr Biomass decline related temperature base (constant)

λ0, λ1 Base rate of biomass loss and energy usage

γ1, γ2, γ3, γ4 Various rate constants for relating terms

ε1, ε2 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

αDc 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

θw Meristem count exponential base (constant)

Ps M. spicatum stand density

Field WorkField 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.

Literature citedGrace, J., and R. Wetzel. 1978. The production biology of Eurasian watermilfoil (Myriophyllum spicatum L.): a review. Journal of Aquatic Plant Management 16:1–11.

Herb, W., and H. Stefan. 2006. Seasonal growth of submersed macrophytes in lakes: The effects of biomass density and light competition. Ecological Modelling . 193:560-574

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 16:139–143.

Sheldon, S., and L. O’Bryan. 1996. Life history of the weevil Euhrychiopsis lecontei, a potential biological control agent of Eurasian watermilfoil. Entomological News 107:16–22.

Titus, J., R. Goldstein, M. Adams, J. Mankin, R. O’Neill, P. Weiler, H. Shugart, and R. Booth. 1975. A production model for Myriophyllum spicatum L. Ecology 56:1129–1138.

AcknowledgmentsWe 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.

For further informationPlease contact jkm29@uakron.edu, hwg3@uakron.edu, or ldr11@uakron.edu

Scenario 1: Initial adult weevil population of 0

Scenario 2: Initial adult weevil population of 50

Scenario 3: Initial adult weevil population of 100

Scenario 4: Initial adult weevil population of 300

Figure 6.