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 concentrations370 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.