1. 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. Simulation 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. 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 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 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 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 Development MERISTEM MODEL DEVELOPMENT The number of meristems on a plant can be modeled in the following way. Consider a plant of initial length P=P 0 and increment plant length using a branching pattern. From this illustration it is clear that the number of meristems is equal to 2 P -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)) = f 0  ^[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 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 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. 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. Kyle Miller, Heath Garris and Lara Roketenetz Integrated 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 μ 0 – base growth rate for M. Spicatum8.34×10 3 P s – M. spicatum stand density-1.32×10 0 α E – base egg laying rate-2.32×10 1 α egg – base egg development rate9.97×10 1 α larva – base larval development rate-3.13×10 2 α pupa – base pupa development rate-1.49×10 2 ln(θ w ) – log of meristem count exponential base -5.60×10 2 SENSITIVITY ANALYSIS Selected parameters were interrogated over a range of +/- 10% for an initial population of 100 adults. ln(θ w ) was varied as opposed to θ w directly, as θ 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. Length P 0 Length 2 P 0 Length 3 P 0 1 Meristem 3 Meristems 7 Meristems Length 4 P 0 15 Meristems Table 1. Description of model parameters and dependent variables. Table 2. Estimated effect of parameter variation., W(t)W(t)M. spicatum biomass C(t)C(t)M. spicatum stored energy (e.g. nonstructural carbohydrates) T(t)T(t)Temperature at time t L(t)L(t)E. lecontei larval population at time t μ0μ0 M. spicatum base growth rate k1k1 Irradiance half saturation constant θgθg Growth related temperature base (constant) TbTb Base temperature at which other constants are measured K wt Light attenuation constant due to water clarity kmkm Light attenuation constant due to blockage by biomass dWater depth hM. spicatum stand height I0I0 Surface irradiance level μ1μ1 Stored energy usage rate u subscript Step function (value 1 or 0) taking a value of 1 at the appropriate time for term subscript θrθr Biomass decline related temperature base (constant) λ 0, λ 1 Base rate of biomass loss and energy usage γ 1, γ 2, γ 3, γ 4 Various rate constants for relating terms ε 1, ε 2 Constants with small value DcE. lecontei development class {egg, larva, pupa, adult} N Dc Number of individuals in development class Dc R Dc Rate of development of E. lecontei individuals m Dc, v Dc Death and immigration/emigration rates α Dc Base development rate for development class Dc TwTw Minimum temperature for E. lecontei development μ3μ3 Base egg laying rate for E. lecontei adults k2k2 Egg laying rate temperature half saturation constant θwθw Meristem count exponential base (constant) PsPs M. spicatum stand density 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. Literature cited Grace, 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 e ff ects 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. 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. For further information Please contact jkm29@uakron.edu, hwg3@uakron.edu, or ldr11@uakron.edujkm29@uakron.eduhwg3@uakron.edu, 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.