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The following systems are models of the populations of pairs of species that either compete for resources (an increase in one species decreases the growth rate of the other) or cooperate (an increase in one species increases the growth rate of the other). For each system, identify the variables (independent and dependent) and the parameters (carrying capacity, measures of interaction between species. etc.) Do the species compete or cooperate? (Assume all parameters are positive.)
(a)
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Differential Equations
- Dieting A persons weight depends both on the daily rate of energy intake, say C calories per day, and on the daily rate of the energy consumption, typically between 15 and 20 calories per pound per day. Using an average value of 17.5 calories per pound per day, a person weighing w pounds uses 17.5wcalories per day. If C=17.5w, then weight remains constant, and weight gain or loss occurs according to whether C is greater or less than 17.5w. Source: The College Mathematics Journal. To determine how fast a change in weight will occur, the most plausible assumption is that dwdt is proportional to the net excess or deficit C17.5w in the number of calories per day. a. Assume C is constant and write a differential equation to express this relationship. Use k to represent the constant of proportionality. What does C being constant imply? b. The units of dwdt are pounds per day, and the units of C17.5w are calories per day. What units must k have? c. Use the fact that 3500 calories is equivalent to 1 lb to rewrite the differential equation in part a. d. Solve the differential equation. e. Let w0 represent the initial weight and use it to express the coefficient of e0.005t in terms of w0 and C.arrow_forwardRedo Exercise 5, assuming that the house blend contains 300 grams of Colombian beans, 50 grams of Kenyan beans, and 150 grams of French roast beans and the gourmet blend contains 100 grams of Colombian beans, 350 grams of Kenyan beans, and 50 grams of French roast beans. This time the merchant has on hand 30 kilograms of Colombian beans, 15 kilograms of Kenyan beans, and 15 kilograms of French roast beans. Suppose one bag of the house blend produces a profit of $0.50, one bag of the special blend produces a profit of $1.50, and one bag of the gourmet blend produces a profit of $2.00. How many bags of each type should the merchant prepare if he wants to use up all of the beans and maximize his profit? What is the maximum profit?arrow_forward23. Consider a simple economy with just two industries: farming and manufacturing. Farming consumes 1/2 of the food and 1/3 of the manufactured goods. Manufacturing consumes 1/2 of the food and 2/3 of the manufactured goods. Assuming the economy is closed and in equilibrium, find the relative outputs of the farming and manufacturing industries.arrow_forward
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