《数学建模》美赛优秀论文：01C Identifying Potential Zebra Mussel Colonization

Identifying Potential Zebra Mussel Colonization 385 Identifying Potential Zebra Mussel Colonization David e stier Marc Alan Leisenring Matthew glen Kenned Humboldt State University Arcata. ca 95221 Advisor: Eileen m. cashman Summary Both environmental and anthropogenic factors influence the spread of zebra mussels to new areas. Variations in water quality can affect both proliferation and mortality, which greatly influence colonization rate. High levels of calcium and alkalinity in fresh waters tend to increase juvenile zebra mussel population Dreissena also requires specific ranges of pH, temperature, and potassium con- centration for propagation. Consumption by predators and spread by humans also influence colonization and population dynamics We develop a lumped-parameter stochastic model using data from a lake with known water quality, using optimal water quality parameter ranges for zebra mussel survival. The model predicts the susceptibility to colonization of a lake with known water quality. We find a significant probability for seasonal colonization in Lake B but gligible probability for Lake C The use of de-icing agents in the vicinity of Lake B may increase the proba bility of colonization, due to elevated calcium concentrations in the lake Literature review ry The zebra mussel originated in the Caspian and Black Sea regions. By the early 19th century, a well-developed population was established throughout The UMAP Journal 22 (4)(2001)385-397. Copyright 2001 by COMAP, Inc. Allrights reserved Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial dvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP

Identifying Potential Zebra Mussel Colonization 385 Identifying Potential Zebra Mussel Colonization David E. Stier Marc Alan Leisenring Matthew Glen Kennedy Humboldt State University Arcata, CA 95221 Advisor: Eileen M. Cashman Summary Both environmental and anthropogenic factors influence the spread of zebra mussels to new areas. Variations in water quality can affect both proliferation and mortality, which greatly influence colonization rate. High levels of calcium and alkalinity in fresh waters tend to increase juvenile zebra mussel population. Dreissena also requires specific ranges of pH, temperature, and potassium con￾centration for propagation. Consumption by predators and spread by humans also influence colonization and population dynamics. We develop a lumped-parameter stochastic model using data from a lake with known water quality, using optimal water quality parameter ranges for zebra mussel survival. The model predicts the susceptibility to colonization of a lake with known water quality. We find a significant probability for seasonal colonization in Lake B but negligible probability for Lake C. The use of de-icing agents in the vicinity of Lake B may increase the proba￾bility of colonization, due to elevated calcium concentrations in the lake. Literature Review History The zebra mussel originated in the Caspian and Black Sea regions. By the early 19th century, a well-developed population was established throughout The UMAP Journal 22 (4) (2001) 385–397.
c Copyright 2001 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP

386 The UMAP Journal 22. 4(2001) the major drainages of Europe in connection with extensive canal building lUSGS 2001. Researchers surmise that the zebra mussel first arrived in north America in the mid-1980s in a ballast tank of a commercial vessel the first recorded population appeared in Lake St Clair, Canada [herbert et al. 1989 By 1990, the zebra mussel habitat encompassed the great lakes and soon after entered the Mississippi River drainage via the Illinois River. Today, zebra mussels exist in at least 21 states [ USGs 2001] Factors Influencing Propagation Physical Mechanism of Propagation Anthropogenic activities are considered the most influential factor in spread ing zebra mussels [Mackie and Schloessler 1996]. Zebra mussels attach them selves to firm surfaces including boat hulls, nets, buoys, and floating debris Balcom and rohmer 1994; Ram and mcmahon 1996. a zebra mussel dis lodged in transport can start a new population Natural dispersion mechanisms include birds, water currents, insects, and other animals [mackie and schloesser 1996; Hincks and Mackie 1997 . When carried by currents, microscopic zebra mussel larvae, called veligers, can quickly disperse themselves [Mackie and Schloesser 1996]. The mussels can travel large distances in the two-to three-week free-swimming veliger stage [Rice 1995] The species has demonstrated resilience to long-overland trips. Zebra mus sels survive longest under cool, moist conditions, similar to the environment in a boat hull Payne 1992] Habitat Zebra mussel habitat includes freshwater lakes and reservoirs as well as cooling ponds, quarries, and irrigation ponds of golf courses. However, the species can survive where salinity does not exceed 8 to 12 parts per thousand (ppt)[Mackie and schloesser 1996] sea abra mussels prefer hard substrates[Heath 1993] but can survive on soft sediment [Stoeckel et al. 1997 ]. Current velocities up to 2 m/s provide opti mal settlement conditions, while speeds ranging from 0.5 m/s to 1.5 m/s best support growth [Rice 1995] Water Quality H Zebra mussels have colonized areas with pH values ranging from 7.0 to 9.0. A pH of 7.5 promotes optimum growth [ Rice 1995 Potassium The optimal range of potassium in the environment is 0.5-1.5 mg/L, with survival at 2-3 mg/L [Dietz et al. 1996 Calcium and Alkalinity Calcium and alkalinity are the strongest infuences on zebra mussel growth and reproduction [Heath 1993]. Zebra mussels require

386 The UMAP Journal 22.4 (2001) the major drainages of Europe in connection with extensive canal building [USGS 2001]. Researchers surmise that the zebra mussel first arrived in North America in the mid-1980s in a ballast tank of a commercial vessel; the first recorded population appeared in Lake St. Clair, Canada [Herbert et al. 1989]. By 1990, the zebra mussel habitat encompassed the Great Lakes and soon after entered the Mississippi River drainage via the Illinois River. Today, zebra mussels exist in at least 21 states [USGS 2001]. Factors Influencing Propagation Physical Mechanism of Propagation Anthropogenic activities are considered the most influential factor in spread￾ing zebra mussels [Mackie and Schloessler 1996]. Zebra mussels attach them￾selves to firm surfaces including boat hulls, nets, buoys, and floating debris [Balcom and Rohmer 1994; Ram and McMahon 1996]. A zebra mussel dis￾lodged in transport can start a new population. Natural dispersion mechanisms include birds, water currents, insects, and other animals [Mackie and Schloesser 1996; Hincks and Mackie 1997]. When carried by currents, microscopic zebra mussel larvae, called veligers, can quickly disperse themselves [Mackie and Schloesser 1996]. The mussels can travel large distances in the two- to three-week free-swimming veliger stage [Rice 1995]. The species has demonstrated resilience to long-overland trips. Zebra mus￾sels survive longest under cool, moist conditions, similar to the environment in a boat hull [Payne 1992]. Habitat Zebra mussel habitat includes freshwater lakes and reservoirs, as well as cooling ponds, quarries, and irrigation ponds of golf courses. However, the species can survive where salinity does not exceed 8 to 12 parts per thousand (ppt) [Mackie and Schloesser 1996]. Zebra mussels prefer hard substrates [Heath 1993] but can survive on soft sediment [Stoeckel et al. 1997]. Current velocities up to 2 m/s provide opti￾mal settlement conditions, while speeds ranging from 0.5 m/s to 1.5 m/s best support growth [Rice 1995]. Water Quality pH Zebra mussels have colonized areas with pH values ranging from 7.0 to 9.0. A pH of 7.5 promotes optimum growth [Rice 1995]. Potassium The optimal range of potassium in the environment is 0.5–1.5 mg/L, with survival at 2–3 mg/L [Dietz et al. 1996]. Calcium and Alkalinity Calcium and alkalinity are the strongest influences on zebra mussel growth and reproduction [Heath 1993]. Zebra mussels require

Identifying Potential Zebra Mussel Colonization 387 a Ca* concentration of 12 mg/l and CaCO3 concentration of 50 mg/I[Heath 1993]. Ramcharan et al. [1992] found that European lakes with pH below 7.3 and Ca+2 concentration below 28. 3 mg/l lacked zebra mussels, but in North America there are numerous examples ofinvasion at far lower calcium concentrations Dissolved Oxygen Heath [1993] indicates a minimum oxygen threshold of 25% oxygen saturation, or 2 mg/I at 25C. Dense overgrowths of zebra mussels may deplete dissolved oxygen enough to cause large die-offs of Dreissena and other aquatic species [Ramcharan et al. 1992 Nutrients and Phytoplankton A water body s chlorophyll-a concentration is a factorin growth variability of the zebra mussel [Mackie and Schloesser 1996 Zebra mussels compete with herbivorous zooplankton and fish for phyto- plankton [Ramcharan et al. 1992]. Zebra mussels collect their food through ciliary filter feeding processes [McMahon 1996 that filtering increases water clarity, and light penetration fosters growth in the lake's benthic population [Maclsaac 1996], which can increase the nuisance aquatic weed biomass Salinity Research suggests optimal salinity for adults is 1 ppt at high temper atures(18-20 C)and 2-4 ppt in lower temperatures(3-12 C)Kilgour et al 1994; Mackie and Schloesser 1996]. Rice [1995] suggests 1 ppt as optimal for growth and short-term tolerance of 12 ppt; but zebra mussels have high adaptive ability to nonideal conditions in salinity and other water quality parameters. Temperature For reproduction, the zebra mussel requires prolonged periods above 12C and maximum temperatures ranging from 18 to 23C [Heath 1993: McMahon 1996]. It cant survive in temperatures greater than 32 C, the lower temperature survival threshold is 0oC [Heath 1993 Predators Crustacean zooplankton and larval fish consume the larval stages of the mussel [Mackie and Schloesser 1996]. Adult Dreissena provide food for crayfish, fish, and waterfowl [Mackie and Schloessler 1996]. Fish ob- served consuming zebra mussels include yellow perch, white perch, wall- eye,white bass, lake whitefish, lake sturgeon, and the round goby [Maclsaac 1996: French 1993. Potential consumers include the freshwater drum, re dear sunfish, pumpkinseed, copper and river redhorse, and common carp Round gobies consume 50-100 zebra mussels per day, depending on the size of the mollusk [Ghedottiet al. 1995. Diving waterfowl consume significant amounts of zebra mussels in proper conditions. Hamilton et al. [1994] found the ducks devoured 57% of the autumn mussel biomass in lake erie: but due to icing over of the lake and consequent lack of winter predation, continued juvenile growth diminished the effects of the consumption

Identifying Potential Zebra Mussel Colonization 387 a Ca+2 concentration of 12 mg/l and CaCO3 concentration of 50 mg/l [Heath 1993]. Ramcharan et al. [1992] found that European lakes with pH below 7.3 and Ca+2 concentration below 28.3 mg/l lacked zebra mussels, but in North America there are numerous examples of invasion at far lower calcium concentrations. Dissolved Oxygen Heath [1993] indicates a minimum oxygen threshold of 25% oxygen saturation, or 2 mg/l at 25◦C. Dense overgrowths of zebra mussels may deplete dissolved oxygen enough to cause large die-offs of Dreissena and other aquatic species [Ramcharan et al. 1992]. Nutrients and Phytoplankton A water body’s chlorophyll-a concentration is a factor in growth variability of the zebra mussel [Mackie and Schloesser 1996]. Zebra mussels compete with herbivorous zooplankton and fish for phyto￾plankton [Ramcharan et al. 1992]. Zebra mussels collect their food through ciliary filter feeding processes [McMahon 1996]; that filtering increases water clarity, and light penetration fosters growth in the lake’s benthic population [MacIsaac 1996], which can increase the nuisance aquatic weed biomass. Salinity Research suggests optimal salinity for adults is 1 ppt at high temper￾atures (18–20◦C) and 2–4 ppt in lower temperatures (3–12◦C) [Kilgour et al. 1994; Mackie and Schloesser 1996]. Rice [1995] suggests 1 ppt as optimal for growth and short-term tolerance of 12 ppt; but zebra mussels have high adaptive ability to nonideal conditions in salinity and other water quality parameters. Temperature For reproduction, the zebra mussel requires prolonged periods above 12◦C and maximum temperatures ranging from 18 to 23◦C [Heath 1993; McMahon 1996]. It can’t survive in temperatures greater than 32◦C; the lower temperature survival threshold is 0◦C [Heath 1993]. Predators Crustacean zooplankton and larval fish consume the larval stages of the mussel [Mackie and Schloesser 1996]. Adult Dreissena provide food for crayfish, fish, and waterfowl [Mackie and Schloessler 1996]. Fish ob￾served consuming zebra mussels include yellow perch, white perch, wall￾eye, white bass, lake whitefish, lake sturgeon, and the round goby [MacIsaac 1996; French 1993]. Potential consumers include the freshwater drum, re￾dear sunfish, pumpkinseed, copper and river redhorse, and common carp. Round gobies consume 50–100 zebra mussels per day, depending on the size of the mollusk [Ghedotti et al. 1995]. Diving waterfowl consume significant amounts of zebra mussels in proper conditions. Hamilton et al. [1994] found the ducks devoured 57% of the autumn mussel biomass in Lake Erie; but due to icing over of the lake and consequent lack of winter predation, continued juvenile growth diminished the effects of the consumption

388 The UMAP Journal 22. 4(2001) Modeling Zebra mussels Zebra mussel populations demonstrate high sensitivity to small changes in water quality parameters. In some lakes, the long-term population size remains fairly constant, while populations in other lakes fluctuate greatly from year to yea Modeling History Some of the more common types of models developed include multivariate, bioenergetic, and probabilistic . Multivariate models have been used to determine the environmental factors that most influence the ability of Dreissena to establish viable populations IRamcharan et al 19921 Bioenergetic models focused on modeling individual zebra mussel growth as a function of certain environmental factors [ Schneider 1992 Probabilistic models used discrete probabilities associated with environmen- tal variables known to contribute to the successful colonization of freshwater bodies to evaluate the susceptibility of certain lakes to zebra mussel colo nization [Miller and ignacio 1994 Model Development Model Choice and approach We develop an analytical model that is transient, lumped-parameter, and stochastic We obtained from the literature ranges of water quality the parameters that are necessary for survival. Using a time step of one year, we determine the probability of survival based on those and determine the population. We use the data on Lake a to calibrate and verify the model's ability to predict colonization Data Considerations The data files provided contain water quality and population data for Lake a Shared by most files were calcium concentration(mg/L), chlorophyll concen tration(ug/L), potassium concentration(mg/L), temperature(C), and pH, all of which the literature shows are important factors We use the average juvenile population for a given year for comparison with the model results, regardless of the amount of data available for that year. Therefore, for each time step, we need an annual average and standard deviation for each parameter and each population. We assume that the average value is the average for the year

388 The UMAP Journal 22.4 (2001) Modeling Zebra Mussels Zebra mussel populations demonstrate high sensitivity to small changes in water quality parameters. In some lakes, the long-term population size remains fairly constant, while populations in other lakes fluctuate greatly from year to year. Modeling History Some of the more common types of models developed include multivariate, bioenergetic, and probabilistic: • Multivariate models have been used to determine the environmental factors that most influence the ability of Dreissena to establish viable populations [Ramcharan et al. 1992]. • Bioenergetic models focused on modeling individual zebra mussel growth as a function of certain environmental factors [Schneider 1992]. • Probabilistic models used discrete probabilities associated with environmen￾tal variables known to contribute to the successful colonization of freshwater bodies to evaluate the susceptibility of certain lakes to zebra mussel colo￾nization [Miller and Ignacio 1994]. Model Development Model Choice and Approach We develop an analytical model that is transient, lumped-parameter, and stochastic. We obtained from the literature ranges of water quality the parameters that are necessary for survival. Using a time step of one year, we determine the probability of survival based on those and determine the population. We use the data on Lake A to calibrate and verify the model’s ability to predict colonization. Data Considerations The datafiles provided contain water quality and population data for Lake A. Shared by most files were calcium concentration (mg/L), chlorophyll concen￾tration (µg/L), potassium concentration (mg/L), temperature (◦C), and pH, all of which the literature shows are important factors. We use the average juvenile population for a given year for comparison with the model results, regardless of the amount of data available for that year. Therefore, for each time step, we need an annual average and standard deviation for each parameter and each population. We assume that the average value is the average for the year

Identifying Potential Zebra Mussel Colonization 389 Review of literature Calcium, alkalinity, phytoplankton, potassium, water temperature, and ph are important for survival. Because of the dependence between alkalinity and calcium concentration, we use only calcium. We use chlorophyll-a in place of phytoplankton to represent available food. We summarize in Table 1 the ranges of water quality parameters required for survival Optimal water quality conditions for survival of each age class. Constituents Age Group Ca(mg/L) Chl-a(ug/L) K(mg/L) Temp LL UL LL UL LL UL LL UL LL UL Birth 50+ 1234 0038 0051.2778.51221 0051.2778.51221 00051.37387528 300051.5529.3031 10 0051.5529.3031 Methodology The model uses assumptions about probabilities of survival at specific age classes Age Classes We divide zebra mussels into four distinct age classes: class 1(0-l years) class 2(1-2 years), class 3(2-3 years), and class 4(3-4 years). At the end of each time step( one year), the population of each age class moves into the next age class, except that class 4 dies. Values for each water quality parameter are specified at each time step Survival Probabilities The ranges of values for each parameter are divided into smaller rang and assigned survival probabilities. A normal distribution is used to create a probability distribution for each parameter. For each age class, we take the mean of the optimal range found in the literature. Newborns and age class 1 use the same ranges and probabilities: classes 3 and 4 also use their own same ranges and probabilities; age class 2 has its own ranges and probabilities. A normal distribution is fit to the average; we assume that the limits of the optimal ranges in the literature represent one standard deviation from the mean

Identifying Potential Zebra Mussel Colonization 389 Review of Literature Calcium, alkalinity, phytoplankton, potassium, water temperature, and pH are important for survival. Because of the dependence between alkalinity and calcium concentration, we use only calcium. We use chlorophyll-a in place of phytoplankton to represent available food. We summarize in Table 1 the ranges of water quality parameters required for survival. Table 1. Optimal water quality conditions for survival of each age class. Constituents Age Group Ca (mg/L) Chl-a (µg/L) K (mg/L) pH Temp LL UL LL UL LL UL LL UL LL UL Birth 20 50+ 0 15 0.05 1.2 7.7 8.5 12 21 1 20 50+ 0 15 0.05 1.2 7.7 8.5 12 21 2 15 50+ 3 20 0.05 1.3 7.3 8.7 5 28 3 10 50+ 8 30 0.05 1.5 5.2 9.3 0 31 4 10 50+ 8 30 0.05 1.5 5.2 9.3 0 31 Methodology The model uses assumptions about probabilities of survival at specific age classes. Age Classes We divide zebra mussels into four distinct age classes: class 1 (0–1 years), class 2 (1–2 years), class 3 (2–3 years), and class 4 (3–4 years). At the end of each time step (= one year), the population of each age class moves into the next age class, except that class 4 dies. Values for each water quality parameter are specified at each time step. Survival Probabilities The ranges of values for each parameter are divided into smaller ranges and assigned survival probabilities. A normal distribution is used to create a probability distribution for each parameter. For each age class, we take the mean of the optimal range found in the literature. Newborns and age class 1 use the same ranges and probabilities; classes 3 and 4 also use their own same ranges and probabilities; age class 2 has its own ranges and probabilities. A normal distribution is fit to the average; we assume that the limits of the optimal ranges in the literature represent one standard deviation from the mean

390 The UMAP Journal 22. 4(2001) Constraints and assumptions For each age group the probabilities of survival at each time step for each of the water quality parameters are assumed to be mutually independent. Thus the probability of survival of each age class is the product of the probabilitic of its survival at each water quality value Additional constraints are also included Age classes 2, 3, and 4 are able to reproduce in water above 12C The survival of eggs and larvae to age class 1 depends on their probability of migration out of the system and the probability of survival at the current water quality conditions. The probability of migration is calculated as a function of calcium concentration [Hincks and Mackie 19971 Since the number of eggs per adult female varies in the literature (4000- 100,000), we use its value as a parameter for calibration An initial number of juveniles (age class 1), specified by the user, is intro- duced at the first time step, and no additional veligers or juveniles enter the system from outside sources The model allows the user to decide which parameters to consider in the probability calculations depending on the availability of data The model was programmed in Fortran 90 with a Lahey ile Suse linux operating system. Calibration The model was calibrated using the data in the files LakeAChem1xls and LakeAPopulation1. xls. The water quality data are provided as the median, maximum, minimum, and 25th and 75th percentiles of data for 1992 to 1999 We assume that the median equals the mean and that the average difference between the mean and the 25th and 75th percentiles is the standard deviation We use a random number generator to create two sets of random numbers between 0 and 1, for n years. The value of each water quality parameter for each of the years is given by X2=R+Par;×Pan1×ox, where Xi is the value of the parameter at time step i X is the parameter mean, Ox is the parameter standard deviation, Pranl i is the random number at time step i, and

390 The UMAP Journal 22.4 (2001) Constraints and Assumptions For each age group, the probabilities of survival at each time step for each of the water quality parameters are assumed to be mutually independent. Thus, the probability of survival of each age class is the product of the probabilities of its survival at each water quality value. Additional constraints are also included: • Age classes 2, 3, and 4 are able to reproduce in water above 12◦C. • The survival of eggs and larvae to age class 1 depends on their probability of migration out of the system and the probability of survival at the current water quality conditions. The probability of migration is calculated as a function of calcium concentration [Hincks and Mackie 1997]. • Since the number of eggs per adult female varies in the literature (4000– 100,000), we use its value as a parameter for calibration. • An initial number of juveniles (age class 1), specified by the user, is intro￾duced at the first time step, and no additional veligers or juveniles enter the system from outside sources. • The model allows the user to decide which parameters to consider in the probability calculations depending on the availability of data. The model was programmed in Fortran 90 with a Lahey compiler under a Suse Linux operating system. Calibration The model was calibrated using the data in the files LakeAChem1.xls and LakeAPopulation1.xls. The water quality data are provided as the median, maximum, minimum, and 25th and 75th percentiles of data for 1992 to 1999. We assume that the median equals the mean and that the average difference between the mean and the 25th and 75th percentiles is the standard deviation. We use a random number generator to create two sets of random numbers between 0 and 1, for n years. The value of each water quality parameter for each of the years is given by Xi = X¯ + Pvar i × Pran1 i × σX, where • Xi is the value of the parameter at time step i, • X¯ is the parameter mean, • σX is the parameter standard deviation, • Pran1 i is the random number at time step i, and

Identifying Potential Zebra Mussel Colonization 391 1, if Pranli<0.5 +1, if Prank≥0.5 Using this method, we created a file of n years of generated data for each parameter for each of 10 sites at Lake A. We calibrated the model for its abil ity to predict susceptibility of a location to colonization by varying the initial population of juveniles and adjusting the number of eggs per adult female At these sites, trends in the model results replicate trends in the populations At a site susceptible to colonization, a higher initial population of juveniles yields faster establishment and propagation; at a site not susceptible to infes- tation, the population does not establish any structure and dies off. However, increasing the number of eggs per female produces colonization at some sites that were not possible at lower levels of egg production; at these sites, water quality is near a juvenile survival threshold. [EDITOR'S NOTE: Space does not permit reproducing the authors graphs illustrating these conclusions. The model is qualitatively accurate. It predicts zebra mussel colonization where and under circumstances when colonization actually occurs, and pre- dicts no colonization when observed juveniles are low or nonexistent. The ability of a population to proliferate is apparent in the development of a pop- ulation age class structure over time if an age structure is not established the location does not experience successful colonization Verification The model predicts whether or not colonization will occur, but the and magnitude of the colonization are not accurately approximated. Also, spee the water quality levels were artificially generated from descriptive statistics the performance of the model with actual data is unknown. With data on the annual accumulation of zebra mussels and the distribution of water quality con- stituents, as provided in the files LakeAChem2xls and LakeAPopulation2xls the model can be tested, adjusted and verified Figures 1 and 2 compare 5 of the 10 sites for the two data sets at Lake A similar trends appear at each site. Running the model with the second set of data indicates that populations proliferate where they have been observed in high numbers. Though the model predictions for juveniles are an order of mag nitude greater than the observed values, the model correctly predicts whether populations survive: we attribute the difference to incomplete calibration Model sensitivities The dominant model sensitivities in predicting the magnitude of prolifer ation are to the number of water quality constituents incorporated and to the initial juvenile population. When more probabilities are considered in the cal- culation, overall probability is lowered. Since the model was calibrated using all parameters, using fewer parameters results in a more conservative estimate

Identifying Potential Zebra Mussel Colonization 391 • Pvar i = −1, if Pran1 i < 0.5 +1, if Pran1 i ≥ 0.5. Using this method, we created a file of n years of generated data for each parameter for each of 10 sites at Lake A. We calibrated the model for its abil￾ity to predict susceptibility of a location to colonization by varying the initial population of juveniles and adjusting the number of eggs per adult female. At these sites, trends in the model results replicate trends in the populations. At a site susceptible to colonization, a higher initial population of juveniles yields faster establishment and propagation; at a site not susceptible to infes￾tation, the population does not establish any structure and dies off. However, increasing the number of eggs per female produces colonization at some sites that were not possible at lower levels of egg production; at these sites, water quality is near a juvenile survival threshold. [EDITOR’S NOTE: Space does not permit reproducing the authors’ graphs illustrating these conclusions.] The model is qualitatively accurate. It predicts zebra mussel colonization where and under circumstances when colonization actually occurs, and pre￾dicts no colonization when observed juveniles are low or nonexistent. The ability of a population to proliferate is apparent in the development of a pop￾ulation age class structure over time; if an age structure is not established, the location does not experience successful colonization. Verification The model predicts whether or not colonization will occur, but the speed and magnitude of the colonization are not accurately approximated. Also, since the water quality levels were artificially generated from descriptive statistics, the performance of the model with actual data is unknown. With data on the annual accumulation of zebra mussels and the distribution of water quality con￾stituents, as provided in the files LakeAChem2.xls and LakeAPopulation2.xls, the model can be tested, adjusted, and verified. Figures 1 and 2 compare 5 of the 10 sites for the two data sets at Lake A; similar trends appear at each site. Running the model with the second set of data indicates that populations proliferate where they have been observed in high numbers. Though the model predictions for juveniles are an order of mag￾nitude greater than the observed values, the model correctly predicts whether populations survive; we attribute the difference to incomplete calibration. Model Sensitivities The dominant model sensitivities in predicting the magnitude of prolifer￾ation are to the number of water quality constituents incorporated and to the initial juvenile population. When more probabilities are considered in the cal￾culation, overall probability is lowered. Since the model was calibrated using all parameters, using fewer parameters results in a more conservative estimate,

392 The UMAP Journal 22.4( 2001) 30000 25000 -Location 2 20000 △ Location3 15000 餐 Location5 -Location 6 Location 7 10000 Location 8 -Location 9 5000 Location 10 199319941995199619971998199920002001 Year Figure 1. Annual average accumulation rates using the lst population data for Lake a 30000 25000 手2000 → Location1 Location 2 15000 Location 5 Location 6 10000 Location 10 5000 0 199219941996199820002002 Year Figure 2. Annual average accumulation using the 2nd population data set for Lake A

0 5000 10000 15000 20000 25000 30000 1993 1994 1995 1996 1997 1998 1999 2000 2001 Year Location 1 Location 2 Location 3 Location 4 Location 5 Location 6 Location 7 Location 8 Location 9 Location 10 Annual Average Accumulation (juveniles/m2/day) 0 5000 10000 15000 20000 25000 30000 1992 1994 1996 1998 2000 2002 Year Location 1 Location 2 Location 5 Location 6 Location 10 Annual Average Accumulation (juveniles/m2/day) 392 The UMAP Journal 22.4 (2001) Figure 1. Annual average accumulation rates using the 1st population data for Lake A. Figure 2. Annual average accumulation using the 2nd population data set for Lake A

Identifying Potential Zebra Mussel Colonization 393 that is, the model over-predicts. The dominant factor in the rate of proliferation is the number of eggs or veligers that are allowed to survive Model limitations The model becomes more conservative as the number of variables consid ered decreases. It predicts either the occurrence of a large outbreak or that a population never establishes The model assumes that the survival probabilities for each parameter range are independent, but in actuality some parameters have strong dependencies, such as between pH and calcium concentration [Hincks and Mackie 1997] Application Lake b Lake b is at the threshold for zebra mussel survival for the only variables on which we have data pH, calcium concentration, and chlorophyll concentration With so few water quality indicators, we expect a conservative estimate (i an overestimate of survivability and colonization potential). We ran the model population introduced to Lake b will not proliferate survive to age class 2.A with an initial juvenile population of 1,000; only 10 Lake c Lake C has a very low average pH and a low annual average calcium concen- tration; it is not suitable for colonization. The probability of survival predicted by the model is zero Impacts of De-icing Near Lake B win any de-icing agents used to remove snow and ice from roads during the nter contain calcium salts, specifically calcium chloride(Cacl2) Repeated application of calcium chloride to roads may accumulate cal cium in Lake B. A small increase in its available calcium level of 11.5 mg/L could allow colonization The model indicates that a calcium concentration of 21.5 mg/L would allow zebra mussel colonization, but continuing low values for pH and and chlorophyll concentration force the colony to die out eventually Other de-icing agents, such as sodium chloride(NaCD), increase sodium concentrations in freshwater bodies, which can inhibit propagation of zebra mussels; however, zebra mussels can adapt to higher levels of salinity

Identifying Potential Zebra Mussel Colonization 393 that is, the model over-predicts. The dominant factor in the rate of proliferation is the number of eggs or veligers that are allowed to survive. Model Limitations The model becomes more conservative as the number of variables consid￾ered decreases. It predicts either the occurrence of a large outbreak or that a population never establishes. The model assumes that the survival probabilities for each parameter range are independent, but in actuality some parameters have strong dependencies, such as between pH and calcium concentration [Hincks and Mackie 1997]. Application Lake B Lake B is at the threshold for zebra mussel survival for the only variables on which we have data: pH, calcium concentration, and chlorophyll concentration. With so few water quality indicators, we expect a conservative estimate (i.e., an overestimate of survivability and colonization potential). We ran the model with an initial juvenile population of 1,000; only 10 survive to age class 2. A population introduced to Lake B will not proliferate. Lake C Lake C has a very low average pH and a low annual average calcium concen￾tration; it is not suitable for colonization. The probability of survival predicted by the model is zero. Impacts of De-icing Near Lake B Many de-icing agents used to remove snow and ice from roads during the winter contain calcium salts, specifically calcium chloride (CaCl2). Repeated application of calcium chloride to roads may accumulate cal￾cium in Lake B. A small increase in its available calcium level of 11.5 mg/L could allow colonization. The model indicates that a calcium concentration of 21.5 mg/L would allow zebra mussel colonization, but continuing low values for pH and and chlorophyll concentration force the colony to die out eventually. Other de-icing agents, such as sodium chloride (NaCl), increase sodium concentrations in freshwater bodies, which can inhibit propagation of zebra mussels; however, zebra mussels can adapt to higher levels of salinity

394 The UMAP Journal 22. 4(2001) References Balcom N. C. and e.m. rohmer. 1994. Zebra mussel awareness and boat use patterns among boaters using three"high risk"Connecticut lakes. Univer sity of Connecticut, Oberlin College, Connecticut Sea Grant College Pro- gram Bowman, Michelle F, andR.C. Bailey. 1998. Upper pH limit of the zebra mussel (Dreissena polymorpha). Canadian Journal of Zoology 76(11): 2119-2123 Dietz, Thomas H, Shawn J. Wilcox, Roger A. Byrne, John W. Lynn, and Harold Silverman. 1996. Osmotic and ionic regulation of North American zebra mussels. American Zoologist 36(3): 364-372 French, J.R. P, Ill. 1993. How well can fishes prey on zebra mussels in Eastern North America. Fisheries 18(6): 13-19. Ghedotti, Michael J, Joseph C. Smihula, and gerald R. Smith. 1995. Zebra mussel predation by round gobies in the laboratory. Journal of Great Lake Research21(4):665-669 Hamilton, D.J.,D. Ankney, and R.C. Bailey. 1994. Predation of zebra mussels by diving ducks. Ecology 75: 521-531 Heath, R.T. 1993. Zebra mussel migration to inland lakes and reservoirs: A guide for lake managers. Kent State University, Ohio Sea Grant College Herbert, P.D.N., B W. Muncater, and G.L. Mackie. 1989. Ecological and genetic studies on Dreissena polymorpha(Pallas): A new mollusk in the Great Lakes Canadian Journal of fisheries and Aquatic Sciences 46( 9): 1587-1591 Hincks, Sheris, and Gerald L Mackie. 1997. Effects of pH, calcium, alkalinity hardness, and chlorophyll on the survival, growth, reproductive success of zebra mussel (Dreissena polymorpha)in Ontario Lakes. Canadian Journal of Fisheries and Aquatic Sciences 54: 2049-2057 Kilgour, B W, G.L. Mackie, A. Baker, and R. Keppel. 1994. Effects of salinity on the condition and survival of zebra mussels Estuaries 17: 385-393 Mackie, G L, and D.W. Schloesser. 1996. Comparative biology of zebra mussels in North America and Europe. American Zoologist 36 (3): 300-310 Maclsaac, Hugh J. 1996. Potential abiotic and biotic impacts of zebra mussels on the inland waters of North America. American Zoologist 36 ( 3): 287-299 McMahon, Robert F. 1996. The physiological ecology of the zebra mussel, Dreissena polymorpha, in North America and Europe. American Zoologist 36 (3):339-363

394 The UMAP Journal 22.4 (2001) References Balcom, N.C., and E.M. Rohmer. 1994. Zebra mussel awareness and boat use patterns among boaters using three “high risk” Connecticut lakes. Univer￾sity of Connecticut, Oberlin College, Connecticut Sea Grant College Pro￾gram. Bowman, Michelle F., and R.C. Bailey. 1998. Upper pH limit of the zebra mussel (Dreissena polymorpha). Canadian Journal of Zoology 76 (11): 2119–2123. Dietz, Thomas H., Shawn J. Wilcox, Roger A. Byrne, John W. Lynn, and Harold Silverman. 1996. Osmotic and ionic regulation of North American zebra mussels. American Zoologist 36 (3): 364–372. French, J.R.P., III. 1993. How well can fishes prey on zebra mussels in Eastern North America. Fisheries 18 (6):13–19. Ghedotti, Michael J., Joseph C. Smihula, and Gerald R. Smith. 1995. Zebra mussel predation by round gobies in the laboratory. Journal of Great Lakes Research 21 (4): 665–669. Hamilton, D.J., D. Ankney, and R.C. Bailey. 1994. Predation of zebra mussels by diving ducks. Ecology 75: 521–531. Heath, R.T. 1993. Zebra mussel migration to inland lakes and reservoirs: A guide for lake managers. Kent State University, Ohio Sea Grant College Program. Herbert, P.D.N., B.W. Muncater, and G.L. Mackie. 1989. Ecological and genetic studies on Dreissena polymorpha (Pallas): A new mollusk in the Great Lakes. Canadian Journal of Fisheries and Aquatic Sciences 46 (9): 1587–1591. Hincks, Sheri S., and Gerald L. Mackie. 1997. Effects of pH, calcium, alkalinity, hardness, and chlorophyll on the survival, growth, reproductive success of zebra mussel (Dreissena polymorpha) in Ontario Lakes. Canadian Journal of Fisheries and Aquatic Sciences 54: 2049–2057. Kilgour, B.W., G.L. Mackie, A. Baker, and R. Keppel. 1994. Effects of salinity on the condition and survival of zebra mussels. Estuaries 17: 385–393. Mackie, G.L., and D.W. Schloesser. 1996. Comparative biology of zebra mussels in North America and Europe. American Zoologist 36 (3): 300–310. MacIsaac, Hugh J. 1996. Potential abiotic and biotic impacts of zebra mussels on the inland waters of North America. American Zoologist 36 (3): 287–299. McMahon, Robert F. 1996. The physiological ecology of the zebra mussel, Dreissena polymorpha, in North America and Europe. American Zoologist 36 (3): 339–363