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Chapter 10

# Chapter 10 - 5E-10(pp 622-631 9:18 AM Page 622 CHAPTER 10...

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Differential Equations By analyzing pairs of differ- ential equations we gain insight into population cycles of predators and prey, such as the Canada lynx and snowshoe hare. C H A P T E R 1 0 0 R W 1000 150 100 50 2000 3000 0 t R W 120 80 40 t™ 2000 1000 R W 3000 5E-10(pp 622-631) 1/18/06 9:18 AM Page 622

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Perhaps the most important of all the applications of cal- culus is to differential equations. When physical scientists or social scientists use calculus, more often than not it is to analyze a differential equation that has arisen in the pro- cess of modeling some phenomenon that they are studying. Although it is often impossible to find an explicit formula for the solution of a dif- ferential equation, we will see that graphical and numerical approaches provide the needed information. |||| 10.1 Modeling with Differential Equations In describing the process of modeling in Section 1.2, we talked about formulating a math- ematical model of a real-world problem either through intuitive reasoning about the phe- nomenon or from a physical law based on evidence from experiments. The mathematical model often takes the form of a differential equation, that is, an equation that contains an unknown function and some of its derivatives. This is not surprising because in a real- world problem we often notice that changes occur and we want to predict future behavior on the basis of how current values change. Let’s begin by examining several examples of how differential equations arise when we model physical phenomena. Models of Population Growth One model for the growth of a population is based on the assumption that the population grows at a rate proportional to the size of the population. That is a reasonable assumption for a population of bacteria or animals under ideal conditions (unlimited environment, ade- quate nutrition, absence of predators, immunity from disease). Let’s identify and name the variables in this model: The rate of growth of the population is the derivative . So our assumption that the rate of growth of the population is proportional to the population size is written as the equation where k is the proportionality constant. Equation 1 is our first model for population growth; it is a differential equation because it contains an unknown function P and its derivative . Having formulated a model, let’s look at its consequences. If we rule out a population of 0, then for all t . So, if , then Equation 1 shows that for all t . This means that the population is always increasing. In fact, as increases, Equation 1 shows that becomes larger. In other words, the growth rate increases as the popula- tion increases. dP dt P t P t 0 k 0 P t 0 dP dt dP dt kP 1 dP dt P the number of individuals in the population the dependent variable t time the independent variable 623 |||| Now is a good time to read (or reread) the discussion of mathematical modeling on page 25.
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Chapter 10 - 5E-10(pp 622-631 9:18 AM Page 622 CHAPTER 10...

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