27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.


e. Sketch a representative solution curve in the xy-plane and indicate the direction in which the solution evolves.


x′(t) = −3x + 6xy, y′(t) = y − 4xy

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample

d. According to Newton’s Law of Cooling, the temperature of a hot object will reach the ambient temperature after a finite amount of time.

U.S. population projections According to the U.S. Census Bureau, the nation’s population (to the nearest million) was 296 million in 2005 and 321 million in 2015. The Bureau also projects a 2050 population of 398 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Here is one approach:

d. Estimations of this kind must be made and interpreted carefully. Suppose the projected population for 2050 is 410 million rather than 398 million. What is the value of the carrying capacity in this case?

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

d. In general, how does halving the time step affect the error at t=0.2 and t=0.4?

y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²