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# ch07_04 - 7-35 SECTION 7.4 Systems of First-Order...

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7-35 SECTION 7.4 . . Systems of First-Order Differential Equations 599 the integral x n + 1 x n f ( x , y ( x )) dx . One such estimate is a Riemann sum using left-endpoint evaluation, given by f ( x n , y ( x n )) x . Show that with this estimate you get Eu- ler’s method. There are numerous ways of getting better es- timates of the integral. One is to use the Trapezoidal Rule, x n + 1 x n f ( x , y ( x )) dx f ( x n , y ( x n )) + f ( x n + 1 , y ( x n + 1 )) 2 x . The drawback with this estimate is that you know y ( x n ) but you do not know y ( x n + 1 ). Briefly explain why this state- ment is correct. The way out is to use Euler’s method; you do not know y ( x n + 1 ) but you can approximate it by y ( x n + 1 ) y ( x n ) + h f ( x n , y ( x n )). Put all of this together to get the Improved Euler’s method: y n + 1 = y n + h 2 [ f ( x n , y n ) + f ( x n + h , y n + h f ( x n , y n ))] . Use the Improved Euler’s method for the IVP y = y , y (0) = 1 with h = 0 . 1 to compute y 1 , y 2 and y 3 . Compare to the exact values and the Euler’s method approximations given in exam- ple 3.4. 2. As in exercise 1, derive a numerical approximation method based on (a) the Midpoint rule and (b) Simpson’s rule. Compare your results to those obtained in example 3.4 and exercise 1. 3. In sections 7.1 and 7.2, you explored some differential equa- tion models of population growth. An obvious flaw in those models was the consideration of a single species in isolation. In this exercise, you will investigate a predator-prey model. In this case, there are two species, X and Y , with populations x ( t ) and y ( t ), respectively. The general form of the model is x ( t ) = ax ( t ) bx ( t ) y ( t ) y ( t ) = bx ( t ) y ( t ) cy ( t ) for positive constants a, b and c . First, look carefully at the equations. The term bx ( t ) y ( t ) is included to represent the ef- fects of encounters between the species. This effect is negative on species X and positive on species Y . If b = 0, the species don’t interact at all. In this case, show that species Y dies out (with death rate c ) and species X thrives (with growth rate a ). Given all of this, explain why X must be the prey and Y the predator. Next, you should find the equilibrium point for co- existence. That is, find positive values ¯ x and ¯ y such that both x ( t ) = 0 and y ( t ) = 0. For this problem, think of X as an in- sect that damages farmers’ crops and Y as a natural predator (e.g., a bat). A farmer might decide to use a pesticide to reduce the damage caused by the X ’s. Briefly explain why the effects of the pesticide might be to decrease the value of a and increase the value of c . Now, determine how these changes affect the equilibrium values. Show that the pest population X actually increases and the predator population Y decreases. Explain, in terms of the interaction between predator and prey, how this could happen. The moral is that the long-range effects of pesti- cides can be the exact opposite of the short-range (and desired) effects.

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ch07_04 - 7-35 SECTION 7.4 Systems of First-Order...

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