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ma266-proj3-f10 - to eat other destructive insects If she...

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Computer Project 3. Predator-Prey Equations Goal: Investigate the qualitative behavior of a nonlinear system of differential equations. Tools needed: pplane8 Description: A farmer has ladybugs and aphids in her fields. The helpful ladybugs (preda- tor) eat the destructive aphids (prey) who devour her crops. Let ( x ( t ) = aphid population (in millions) at time t, y ( t ) = ladybug population (in millions) at time t. The farmer knows that the growth rates of the aphid and ladybug populations are given respectively by dx dt = x (1 - y ) , dy dt = y ( x - 1) . Questions: Assume there are initially 800 , 000 aphids and 400 , 000 ladybugs in all that follows below. 1. Use pplane8 to plot the trajectory through (0 . 8 , 0 . 4). As t increases, describe what happens to each population. Is the aphid population ever smaller than 300 , 000? Are the aphids ever eradicated? Does the ladybug population ever exceed 2 million? 2. A fellow farmer suggests that she use pesticide to kill the aphids. She is reluctant be- cause it also kills the helpful ladybugs and she prefers to have some ladybugs remaining
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Unformatted text preview: to eat other destructive insects. If she were to use a pesticide, the growth rates would then become dx dt = x (1-y )-sx, dy dt = y ( x-1)-sy. ( * ) where s ≥ 0 is a measure of the “strength” of the pesticide – the larger the s , the stronger the pesticide. Currently there are only two commercially available strengths: s = 0 . 5 and s = 0 . 75. Plot the trajectories for the new system of equations ( * ) with these values of s . Will the aphids ever be totally eliminated? 3. If she knows her crops will survive if the aphid population never exceeds 2 . 6 million, which strength (if any) would you recommend she use: s = 0 . 0 (no pesticide), s = 0 . 5, s = 0 . 75? 4. By special permission, she could get a pesticide with the maximum strength of s = 1 . 5. Plot this trajectory. What happens to the ladybugs and aphids if she uses this pesticide?...
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