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Unformatted text preview: DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS 9 9.4 Models for Population Growth In this section, we will: Investigate differential equations used to model population growth. DIFFERENTIAL EQUATIONS One of the models for population growth we considered in Section 9.1 was based on the assumption that the population grows at a rate proportional to the size of the population: NATURAL GROWTH dP kP dt = Is that a reasonable assumption? NATURAL GROWTH Suppose we have a population (of bacteria, for instance) with size P = 1000. At a certain time, it is growing at a rate of P = 300 bacteria per hour. NATURAL GROWTH Now, lets take another 1,000 bacteria of the same type and put them with the first population. Each half of the new population was growing at a rate of 300 bacteria per hour. NATURAL GROWTH We would expect the total population of 2,000 to increase at a rate of 600 bacteria per hour initiallyprovided theres enough room and nutrition. So, if we double the size, we double the growth rate. NATURAL GROWTH In general, it seems reasonable that the growth rate should be proportional to the size. NATURAL GROWTH In general, if P ( t ) is the value of a quantity y at time t and, if the rate of change of P with respect to t is proportional to its size P ( t ) at any time, then where k is a constant. This is sometimes called the law of natural growth. LAW OF NATURAL GROWTH dP kP dt = Equation 1 If k is positive, the population increases. If k is negative, it decreases. LAW OF NATURAL GROWTH Equation 1 is a separable differential equation. Hence, we can solve it by the methods of Section 9.3: where A (= e C or 0) is an arbitrary constant. LAW OF NATURAL GROWTH ln kt C C kt kt dP k dt P P kt C P e e e P Ae + = = + = = = To see the significance of the constant A , we observe that: P (0) = Ae k = A Thus, A is the initial value of the function. LAW OF NATURAL GROWTH The solution of the initialvalue problem is: LAW OF NATURAL GROWTH (0) dP kP P P dt = = Equation 2 ( ) kt P t Pe = Another way of writing Equation 1 is: This says that the relative growth rate (the growth rate divided by the population size) is constant. Then, Equation 2 says that a population with constant relative growth rate must grow exponentially. LAW OF NATURAL GROWTH 1 dP k P dt = We can account for emigration (or harvesting) from a population by modifying Equation 1as follows. LAW OF NATURAL GROWTH If the rate of emigration is a constant m , then the rate of change of the population is modeled by the differential equation See Exercise 13 for the solution and consequences of Equation 3. LAW OF NATURAL GROWTH dP kP m dt = Equation 3 As we discussed in Section 9.1, a population often increases exponentially in its early stages, but levels off eventually and approaches its carrying capacity because of limited resources....
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This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.
 Fall '11
 AlanS.Grave
 Differential Equations, Equations

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