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Chapter 5 Conservation of Mass: Chemistry, Biology, and Thermodynamics

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Chapter 5 Conservation of Mass: Chemistry, Biology, and Thermodynamics Chapter 5 Conservation of Mass: Chemistry, Biology, and Thermodynamics 5.1 An Environmental Pollutant Consider a region in the natural environment where a water borne pollutant enters and leaves by stream flow, rainfall, and evaporation. A plant species absorbs this pollutant and returns a portion of it to the environmentafter death. A herbivore species absorbs the pollutant by eating the plants and drinking the water, and it return a portion of the pollutant to the environ- ment in excrement and after death. We are given (by ecologists) the initial concentrations of the pollutant in the ambient environment, the plants, and the animals together with the rates at which the pollutant is transported by the water flow, the plants and the animals. A basic problem is to determine the concentrations of the pollutant in the water, plants, and animals as a function of time and the parameters in the system. Figure 5.1 depicts the transport pathways for our pollutant, wherethe state variable x1 denotes the pollutant concentration in the environment, x2 its concentration in the plants, and x3 its concentration in the herbivores. These three state variables all have dimensions of mass per volume (in some consistent units of measurement). The pollutant enters the environment at the rate A (with dimensions of mass per volume per time) and leaves at the rate ex1 (where e has dimensions of inverse time). Likewise the remaining 37 38 Conservation of Mass A b Plants Animals Environment c x1 x2 f x3 a e d Figure 5.1: Schematic representation of the transport of an environmental pollutant. rate constants a, b, c, d,andf all have dimensions of inverse time. The scenario just described is typical. We will derive a model from the law of conservation of mass: the rate of change of the amount of a substance (which is not created or destroyed) in some volume is its rate in (through the boundary) minus its rate out; or, to reiterate, time rate of change of amount of substance = rate in rate out. − This law is easy to apply by taking the units of measurement into account. Often we are given the volume V of a container, compartment, or region measured in some units of volume, the concentration x of some substance in the compartment measured in amount (mass) per volume, and rates k of inflow and ! of outflow for some medium carrying the substance measured in units of volume per time. The rate of change dx/dt in this case, where t denotes time measured in some consistent unit, has units of amount per volume per time. The rate of outflow ! times the concentration x has units of amount per time. Thus, we must multiply the time-derivative by the total volume to achieve consistent units in the differential equation d(Vx) = rate in !x, dt − where the inflow rate must have units of amount per time. In case V is constant, we may divide both sides of the equation by V . The quantity !/V Conservation of Mass 39 has units of inverse time. In this case, it is called the rate constant for the outflow of the substance. For the environmental pollutant, the rate constants are supposed to be given; the volumes of the compartments (the environment, herbivores, and plants) are in the background but not mentioned. Using conservation of mass, the transport equations are x˙ = A + ax + bx (c + d + e)x , 1 2 3 − 1 x˙ = cx (a + f)x , 2 1 − 2 x˙ = dx + fx bx . (5.1) 3 1 2 − 3 This type of model, which is ubiquitous in mathematical biology, is called a compartment model. The parameters in the model (5.1) would be measured by the field work of ecologists. When the parameters are determined, the model canbeused to make quantitative predictions of the future concentrations of the pollutant by solving the system of ODEs (5.1). Some of the rates might be difficult to measure directly (for example, the rate constant f that determines the transport of the pollutant from plants to the herbivore). These rates might be estimated using a two step procedure: The pollutant concentrations are measured in the field over some suitable period of time and the parameters are chosen in the model to match these measurements. Exactly how to choose the parameters based on available data is the parameter estimation problem; it is one of the most important and difficult issues in mathematical modeling. Our model may also be used to make qualitative predictions. For example, we might make a mathematical deduction from the form of the equations that determines the long-term substance concentrations for some range of parameter values. We might also use our model to predict the outcome of some intervention in the environment. For example, suppose thatsome portion of the plants are harvested by humans and removed from the region under study. The effect of this change can be predicted by solving our model equations with appropriate assumptions. The ability to make predictions without conducting new field studies is one of the most important motivations for developing a mathematical model. Another reason to develop a model is to test hypotheses about the underlying physical process. For example, a model that leads to predictions which do not agree with data obtainedfrom field observations must be based on at least one false hypothesis. 40 Conservation of Mass The linear system (5.1) can be solved explicitly. But, since the eigenvalues of the system matrix are roots of a cubic polynomial, explicit formulas for the time evolution of the state variables x1, x2,andx3 are complicated. This is to be expected. Differential equations with simple explicit solutions arerare. Most differential equations do not have explicit solutions and most explicit solutions are too complicated to be useful. For this reason, model validation and predictions are usually obtained by a combination of qualitative methods, analytic approximations, and numerical approximations. The most valuable information is usually obtained by qualitative analysis. Physically meaningful concentrations are all nonnegative. Hence, our model should have the property that nonnegative initial concentrations re- main nonnegative as time evolves. Geometrically, the positive octant of the three-dimensional state space for the variables u, v,andw should be posi- tively invariant (that is, a solution starting in this set should stay in theset for all positive time or at least as long as the solution exists). To checkthe invariance, it suffices to show that the vector field given by the right-hand side of the system of differential equations is tangent to the boundary of the positive octant or points into the positive octant along its boundary, which consists of the nonnegative parts of the coordinate planes. In other words, positive invariance follows becausex ˙ 0wheneverx =0,˙x 0whenever 3 ≥ 3 2 ≥ x2 =0,and˙x1 0wheneverx1 =0. What happens≥ to the concentrations of the pollutant after a long time? Our expectation is that the system will evolve to a steady state (that is, a zero of the vector field that defines the ODE). To determine the steady state, we simply solve the system of algebraic equations obtained by setting the time derivatives of the system equal to zero. A computation shows that if be(a + f) = 0, then the solution of the system # A + ax + bx (c + d + e)x =0, 2 3 − 1 cx (a + f)x =0, 1 − 2 dx + fx bx = 0 (5.2) 1 2 − 3 is A Ac A(ad +(c + d)f) x = ,x= ,x= . 1 e 2 e(a + f) 3 be(a + f) If be(a + f) > 0, then there is a steady state in the positive first octant. If b =0,e =0,or(a + f) = 0, then there is no steady state. This is reasonable on physical grounds. If the pollutant is not returned to the environment Conservation of Mass 41 by the herbivores, then we would expect the pollutant concentrations in the herbivores to increase. If the pollutant does not leave the environment, then the concentration of the pollutant in our closed system will increase.Ifthe plants do not transfer the pollutant to the environment or the herbivores, we would expect their pollutant concentration to increase. Suppose that there is a steady state. Do the concentrations of the pollu- tant approach their steady state values as time passes? This basic question is answered for our linear system by determining the stability of the corre- sponding rest point in our dynamical system. The stability of a steadystate can usually be determined from the signs of the real parts of the eigenvalues of the system matrix of the linearization of the system at the steadystate (see Appendix A.16). In particular, if all real parts of these eigenvalues are negative, then the steady state is asymptotically stable. The system matrix at our steady state is c d ea b − − − c (a + f)0 . (5.3) − dfb − To show the asymptotic stability of the rest point, it suffices to provethat all the real parts of the eigenvalues are negative. This follows by a direct computation using the Routh-Hurwitz criterion (see Appendix A.16 and Ex- ercise 5.1). Of course, this also agrees with physical intuition. Since our system is linear, we can make a strong statement concerning stability: if e = 0, then all initial concentrations evolve to the same steady state concen- trations.# Our qualitative analysis makes numerical computation unnecessary for answering some questions about our system, at least in those caseswherewe can reasonably assume the system is in steady state. For example, if the rate at which the pollutant leaves the environment is decreased by 50%, then we can conclude that the pollutant concentration in the plants will be doubled.
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