A Multiple regression Model 367 A Multiple regression model to Predict Zebra mussel population Growth Michael p. schubmehl Marcy A. la violette Deborah a chun Harvey Mudd College Claremont. ca 91711 Advisor: Michael E. Moody Summary Zebra mussels(Dreissena polymorpha) are an invasive mollusk accidentally troduced to the united States by transatlantic ships during the mid-1980s Because the mussels have few natural predators and adapt quickly to new envi- ronments, they have spread quickly from the great lakes into many connected waterways. Although the mussel is hardy, sometimes little or no growth is observed in lakes to which it has been introduced extensive research indicates that the chemical concentrations in these bodies of water may be unsuitable for the mussels To quantify the relationship between chemical contents and mussel popu lation growth, we first use the logistic equation, dy y to model Dreissena population as a function of time. After modeling growth rates under a variety of conditions, we used multiple regression to determine which chemicals affect this growth rate. An extensive literature search sup- ported our findings that population growth is linearly dependent on two pri- mary factors: calcium concentration and pH. After further refining our model using the second set of data from Lake A, we obtained the regression equation maximum growth rate=2338[Ca2++39202 pH-334089 The UMAP Journal22(4)(2001)367-383. 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
A Multiple Regression Model 367 A Multiple Regression Model to Predict Zebra Mussel Population Growth Michael P. Schubmehl Marcy A. LaViollette Deborah A. Chun Harvey Mudd College Claremont, CA 91711 Advisor: Michael E. Moody Summary Zebra mussels (Dreissena polymorpha) are an invasive mollusk accidentally introduced to the United States by transatlantic ships during the mid-1980s. Because the mussels have few natural predators and adapt quickly to new environments, they have spread quickly from the Great Lakes into many connected waterways. Although the mussel is hardy, sometimes little or no growth is observed in lakes to which it has been introduced; extensive research indicates that the chemical concentrations in these bodies of water may be unsuitable for the mussels. To quantify the relationship between chemical contents and mussel population growth, we first use the logistic equation, dy dt = ry 1 − y K , to model Dreissena population as a function of time. After modeling growth rates under a variety of conditions, we used multiple regression to determine which chemicals affect this growth rate. An extensive literature search supported our findings that population growth is linearly dependent on two primary factors: calcium concentration and pH. After further refining our model using the second set of data from Lake A, we obtained the regression equation maximum growth rate = 2338 [Ca2+] + 39202 pH − 334089, The UMAP Journal 22 (4) (2001) 367–383. 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.
368 The UMAP Journal 22. 4(2001) where the maximum growth rate is in juveniles settling per day, and [Ca2t is in mg/L. Using this model, we predict that lakes B and C cannot support Dreissena population. Because the levels of calcium in lake b are close to those required to support a Dreissena population, however, we advise the community near Lake B to use de-icing agents that do not contain calcium Environmental Factors Affecting Dreissena a large body of research links environmental factors such as temperature pH, calcium ion concentration, and alkalinity to the success or failure of zebra mussel populations. The two factors repeatedly most closely associated with survival are calcium concentration and pH. In a survey of 278 lakes, for exam- le, Ramcharan et al. [1992] found no populated lakes with ph below 7.3 or Ca content below 28.3 mg/L. Recent studies have lowered the minimum Ca con- centration to 15 mg/L for adults and 12 mg/L for larvae [McMahon 1996]. The upper bound for pH is somewhere near 9.4 [McMahon 1996]. The optimum conditions for growth are a pH of 8.4 and 34 mg/L of Ca[McMahon 1996 Other requirements for survival include alkalinity, which must be kept above 50 mg/L [Balog et al. 1995, and dissolved oxygen, which must be above 0.82 ppm(approximately 10% of saturation) Johnson and McMahon 1996 Dreissena also cannot survive in magnesium-deficient water; they require a minimum concentration of 0.03 mM for a low-density population [Dietz and Byrne 1994]. Sulfate(SO4) is also required in small amounts for survival [Dietz and b Zebra mussels can survive in an amazingly wide range of temperatures, but Van der Velde et al. [1996] determined that exposure to 34 C is lethal within 114 minutes and that any temperature above 25 C inhibits movement and feeding Some individuals can tolerate short-term sub-freezing air temperatures [Pauk stis et al. 1996 Although not used by the mussels themselves, phosphorus and nitrogen are essential for freshwater phytoplankton survival, and phytoplankton are the main source of food for Dreissena. Densities of mussel populations are neg- atively related to both phosphates and nitrates; but iron, chlorine, and sodium have no relationship to the existence or density of populations [Ramcharan et al. 1992]. Chlorophyll content measures the density of phytoplankton and thus decreases drastically after the establishment of a zebra mussel colony [Miller and Haynes 19971 Surprisingly, food availability is not an important factor once a zebra musse is established. In one study, Dreissena were able to survive without food for 524 days with only a 60% mortality rate [Chase and McMahon 1995]. Once a population has acclimatized, limited reproduction can occur in brackish water below 7.0 ppt salinity [Fong et al. 1995, with little mortality even up to 10 ppt Kennedy et al. 1996]. Potassium can be tolerated only in low concentrations up to 0.3-0.5 mM. Ammonia(NH3)is lethal in doses as low as 2 mg/L[Baker et
368 The UMAP Journal 22.4 (2001) where the maximum growth rate is in juveniles settling per day, and [Ca2+] is in mg/L. Using this model, we predict that lakes B and C cannot support Dreissena population. Because the levels of calcium in Lake B are close to those required to support a Dreissena population, however, we advise the community near Lake B to use de-icing agents that do not contain calcium. Environmental Factors Affecting Dreissena A large body of research links environmental factors such as temperature, pH, calcium ion concentration, and alkalinity to the success or failure of zebra mussel populations. The two factors repeatedly most closely associated with survival are calcium concentration and pH. In a survey of 278 lakes, for example, Ramcharan et al. [1992] found no populated lakes with pH below 7.3 or Ca content below 28.3 mg/L. Recent studies have lowered the minimum Ca concentration to 15 mg/L for adults and 12 mg/L for larvae [McMahon 1996]. The upper bound for pH is somewhere near 9.4 [McMahon 1996]. The optimum conditions for growth are a pH of 8.4 and 34 mg/L of Ca [McMahon 1996]. Other requirements for survival include alkalinity, which must be kept above 50 mg/L [Balog et al. 1995], and dissolved oxygen, which must be above 0.82 ppm (approximately 10% of saturation) [Johnson and McMahon 1996]. Dreissena also cannot survive in magnesium-deficient water; they require a minimum concentration of 0.03 mM for a low-density population [Dietz and Byrne 1994]. Sulfate (SO4) is also required in small amounts for survival [Dietz and Byrne 1999]. Zebra mussels can survive in an amazingly wide range of temperatures, but Van der Velde et al. [1996] determined that exposure to 34◦C is lethal within 114 minutes and that any temperature above 25◦C inhibits movement and feeding. Some individuals can tolerate short-term sub-freezing air temperatures [Paukstis et al. 1996]. Although not used by the mussels themselves, phosphorus and nitrogen are essential for freshwater phytoplankton survival, and phytoplankton are the main source of food for Dreissena. Densities of mussel populations are negatively related to both phosphates and nitrates; but iron, chlorine, and sodium have no relationship to the existence or density of populations [Ramcharan et al. 1992]. Chlorophyll content measures the density of phytoplankton and thus decreases drastically after the establishment of a zebra mussel colony [Miller and Haynes 1997]. Surprisingly, food availability is not an important factor once a zebra mussel is established. In one study, Dreissena were able to survive without food for 524 days with only a 60% mortality rate [Chase and McMahon 1995]. Once a population has acclimatized, limited reproduction can occur in brackish water below 7.0 ppt salinity [Fong et al. 1995], with little mortality even up to 10 ppt [Kennedy et al. 1996]. Potassium can be tolerated only in low concentrations up to 0.3–0.5 mM. Ammonia (NH3) is lethal in doses as low as 2 mg/L [Baker et
A Multiple regression Model 369 al. 1994. An extensive literature search revealed no correlation between NHg and zebra mussel populations Constructing the Model We need to quantify Dreissena population growth, then examine how this growth is affected by the environment. We use the logistic equation, a standard modeling device in ecology [Gotelli 1998]. We choose a continuous approach because of the huge number of individuals involved, and the logistic equation in particular because its simplicity allows us to make as few assumptions as possible Standard techniques for examining the influence of variables like calcium ion concentrations, pH, and temperature on Dreissena populations include mul tiple regression and discriminant analysis [Ramcharan et al. 1992]. We want to predict actual population growth rates and not just state whether or not population could exist in certain conditions, so we use multiple regression te relate population growth to chemical concentrations Assumptions Population growth rate is proportional to total population We assume that the growth rate of an areas population is proportional to the rate at which juveniles settle on plates there. This rate is, in turn, proportional to the total number of larvae present in the water, which is proportional to the total population. Thus, the population growth rate is proportional to the population level e Carrying capacity is constant Larvae can be thought of as a resource necessary for juveniles to exist breeding season, only a certain number of larvae are produced, so the popu lation can increase only to a certain point. Thus, there is effectively a carrying capacity at work. We assume that this carrying capacity does not depend explicitly on time once the breeding season begins Migration, genetic structure, and age structure do not affect the popula tion Although Dreissena populations spread quickly from one region to another individuals can move only at a slow crawl. Thus, migration of existing population into or out of a region is negligible. Also, there is no evidence for the existence of individuals whose ages or genes dramatically affect their influence on the population, so we neglect age and genetic variation Predation is negligible
A Multiple Regression Model 369 al. 1994]. An extensive literature search revealed no correlation between NH4 and zebra mussel populations. Constructing the Model We need to quantify Dreissena population growth, then examine how this growth is affected by the environment. We use the logistic equation, a standard modeling device in ecology [Gotelli 1998]. We choose a continuous approach because of the huge number of individuals involved, and the logistic equation in particular because its simplicity allows us to make as few assumptions as possible. Standard techniques for examining the influence of variables like calcium ion concentrations, pH, and temperature on Dreissena populations include multiple regression and discriminant analysis [Ramcharan et al. 1992]. We want to predict actual population growth rates and not just state whether or not a population could exist in certain conditions, so we use multiple regression to relate population growth to chemical concentrations. Assumptions • Population growth rate is proportional to total population. We assume that the growth rate of an area’s population is proportional to the rate at which juveniles settle on plates there. This rate is, in turn, proportional to the total number of larvae present in the water, which is proportional to the total population. Thus, the population growth rate is proportional to the population level. • Carrying capacity is constant. Larvae can be thought of as a resource necessary for juveniles to exist. Each breeding season, only a certain number of larvae are produced, so the population can increase only to a certain point. Thus, there is effectively a carrying capacity at work. We assume that this carrying capacity does not depend explicitly on time once the breeding season begins. • Migration, genetic structure, and age structure do not affect the population. Although Dreissena populations spread quickly from one region to another, individuals can move only at a slow crawl. Thus, migration of existing population into or out of a region is negligible. Also, there is no evidence for the existence of individuals whose ages or genes dramatically affect their influence on the population, so we neglect age and genetic variation. • Predation is negligible
370 The UMAP Journal 22. 4(2001) We assume that Dreissena are so numerous that any species that prey on them-and there are few-do not have a substantial impact. Sites within a lake can be treated as distinct lakes Although all of the data came from a single lake, we model each site as a separate lake. That is, we assume that the introduction of mussels from another part of the lake is equivalent to their introduction into a fresh lake and we model the population at the new site independently Population Growth Model: The Logistic Equation We model a Dreissena population with the logistic equation dy where r is the intrinsic growth rate of the population and K is the carrying capacity. For simplicity, we let a=r and b=r/K, so that dy With the initial condition y(0)=yo, the equation has closed-form solutions shown in Figure 1. Because the data from Lake A measure the population growth rate, what we really want to fit to the data is the derivative of this y'(t aeatyo(a +b( whose graph is shown in Figure 2. We can convert the parameters a, b, and yo into the position, height, and full width at half maximum(FWHM) of this peak, making it easy to fit to data Because the first data set did not include information about changes in chemical concentration over time, we average the population growth rates over all years after the introduction of Dreissena and fit the model curve to this"average year"at each site. The position and width of the peak are fairl constant from site to site, as we expect, since the breeding season usually peaks around mid-to late August and lasts for about three months. The peak heights, however, are radically different at different sites, ranging from about 38,000 juveniles per day at site 2(Figure 3) to just 1 juvenile per day at site 10. This variation can be explained only by the environmental conditions there, so we determine how these growth rates varied with chemical concentrations
370 The UMAP Journal 22.4 (2001) We assume that Dreissena are so numerous that any species that prey on them—and there are few—do not have a substantial impact. • Sites within a lake can be treated as distinct lakes. Although all of the data came from a single lake, we model each site as a separate lake. That is, we assume that the introduction of mussels from another part of the lake is equivalent to their introduction into a fresh lake, and we model the population at the new site independently. Population Growth Model: The Logistic Equation We model a Dreissena population with the logistic equation dy dt = ry 1 − y K , where r is the intrinsic growth rate of the population and K is the carrying capacity. For simplicity, we let a = r and b = r/K, so that dy dt = ay − by2. With the initial condition y(0) = y0, the equation has closed-form solutions y(t) = aeaty0 a − by0 + beaty0 , shown in Figure 1. Because the data from Lake A measure the population growth rate, what we really want to fit to the data is the derivative of this function, y (t) = a2eaty0(a − by0) (a + b(−1 + eat)y0)2 , whose graph is shown in Figure 2. We can convert the parameters a, b, and y0 into the position, height, and full width at half maximum (FWHM) of this peak, making it easy to fit to data. Because the first data set did not include information about changes in chemical concentration over time, we average the population growth rates over all years after the introduction of Dreissena and fit the model curve to this “average year” at each site. The position and width of the peak are fairly constant from site to site, as we expect, since the breeding season usually peaks around mid- to late August and lasts for about three months. The peak heights, however, are radically different at different sites, ranging from about 38,000 juveniles per day at site 2 (Figure 3) to just 1 juvenile per day at site 10. This variation can be explained only by the environmental conditions there, so we determine how these growth rates varied with chemical concentrations.
A Multiple regression Model 371 Time Figure 1. Solution to a generic logistic equation, y'= ay- by with population plotted as a function of time Time Figure 2. The derivative of the solution to a generic the logistic equation, showing the time rate of change of population. The peak corresponds to Dreissena breeding season in our model
A Multiple Regression Model 371 Population Time Figure 1. Solution to a generic logistic equation, y = ay − by2, with population plotted as a function of time. Growth Rate Time Figure 2. The derivative of the solution to a generic the logistic equation, showing the time rate of change of population. The peak corresponds to Dreissena breeding season in our model