1st EditionJoseph S. Levine, Kenneth R. Miller1,773 explanations

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6th EditionDavid L Nelson, Michael M. Cox616 explanations

BIOLOGYUse the compartment model defined in Subsection 8.2.2 to investigate how the size of a lake influences nutrient dynamics in the lake after a perturbation. Mary Lake and Elizabeth Lake are two fictitious lakes in the North Woods that are used as experimental lakes to study nutrient dynamics. Mary Lake has a volume of 6.8 × 103 m3, and Elizabeth Lake has twice that volume, or 13.6 × 103 m3. Both lakes have the same inflow/outflow rate q = 170 liter s−1. Because both lakes share the same drainage area, the concentration CI of the incoming solute is the same for both lakes, namely, CI = 0.7mg liter−1. Assume that at the beginning of the experiment both lakes are in equilibrium; that is, the concentration of the solution in both lakes is 0.7mg liter−1. Your experiment consists of increasing the concentration of the solution by 10% in each lake at time 0 and then watching how the concentration of the solution in each lake changes with time. Assume the single-compartment model to make predictions about how the concentration of the solution will evolve. (Note that 1m3 of water corresponds to 1000 liters of water.) (a) Find the initial concentration C0 of the solution in each lake at time 0 (i.e., immediately after the 10% increase in concentration of the solution). (b) Use Equation (8.58) to determine how the concentration of the solution changes over time in each lake. Graph your results. (c) Which lake returns to equilibrium faster? Compute the return time to equilibrium, TR, for each lake, and explain how it is related to the eigenvalues corresponding to the equilibrium concentration CI for each lake. (REFERENCE 8.22) This example is adapted from DeAngelis (1992). Compartment models are frequently used to model the flow of matter (nutrients, energy, and so forth). The simplest such model consists of one compartment—for instance, a fixed volume V of water (such as a tank or lake) containing a solute (such as phosphorus). Assume that water enters the compartment at a constant rate q and leaves the compartment at the same rate. (See Figure 8.17.) (Having the same input and output rate keeps the volume of the pool constant.) We will investigate the effects of different input concentrations of the solution on the concentration of the solution in the pool.