= 2. If a 2 is halved to a = 1, what is the effect on the amplitude and period of both x(t) and y(t)? Does the effect on the solutions curves make sense? = = 3. If b 1.2 is doubled to b 2.4, what is the effect on the amplitude and period of both x(t) and y(t)? What does b represent in the Lotka-Volterra Models? Does the effect on the solutions curves make sense? 4. If b = 1.2 is halved to b = 0.6, what is the effect on the amplitude and period of both x(t) and y(t)? Does the effect on the solutions curves make sense? 5. If c 1 is doubled to c = 2, what is the effect on the amplitude and period of both x(t) and y(t)? What does c represent in the Lotka-Volterra Models? Does the effect on the solutions curves make sense? 6. If cis halved to c = 0.5, what is the effect on the amplitude and period of both x(t) and y(t)? Does the effect on the solutions curves make sense? 7. If d=0.9 is doubled to d = 1.8, what is the effect on the amplitude and period of both x(t) and y(t)? What does d represent in the Lotka-Volterra Models? Does the effect on the solutions curves make sense? 8. If d = 0.9 is halved to d = 0.45, what is the effect on the amplitude and period of both x(t) and y(t)? Does the effect on the solutions curves make sense? 9. What is the effect on the solution curves of setting b = 0? 10. What is the effect on the solution curves of setting d = 0? 11. What effect does changing the initial values have on the solution curves? 12. Upload a graph for the case when all of the default values in slopes app are doubled and the initial value is x(0) = 5, y(0) = 2 and describe how the graphs differ from the graph of the default values. 1. PART I: Investigating Key Concepts Using Graphs. Although the Lotka-Volterra system can be solved analytically, the solutions are beyond the scope of our class, so we will will explore key concepts such as Cyclic Behavior, Parameter Sensitivity and Equilibrium Points and other features of the Lotka-Volterra predator-prey model graphically by using the Slopes App by Tim Lucas. You can use any other app or program to graph the solutions of the predator-prey system, but the Slopes App is really convenient. Cyclic Behavior: The model predicts that predator and prey populations can exhibit cyclic or oscillatory behavior over time. When the prey population is high, it provides abundant food for predators, causing the predator population to increase. As the predator population increases, it exerts greater predation pressure on the prey, causing the prey population to decline. This, in turn, leads to a decrease in the predator population due to reduced food availability. The cycle then repeats. Parameter Sensitivity: The behavior of the model is highly sensitive to changes in the values of its parameters, such as the intrinsic growth rates, predation rates, and death rates. Small changes in these parameters can lead to significant differences in the dynamics of the predator-prey system. To investigate how the solutions depend on the parameters we will use this system: dx =2x-1.2xy dt 33 dy == ―y + 0.9xy dt For each of the constants a, b, c, d, you are to individually look at what the effect of changing that constant has on the amplitude and period of the solutions. Use the Slopes App to investigate the effect changing the parameters on the solutions curves. Upload your answers as a pdf file to GS. For the following questions (except questions 11 and 12), keep the initial conditions at x(0) = 1, y(0) = 1. 1. If a = 2 is doubled to a = 4, what is the effect on the amplitude and period of both x(t) and y(t)? What does a represent in the Lotka-Volterra Models? Does the effect on the solutions curves make sense?