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Module Two Guided Activity Page 1 of 31 July 2022 Intro to Mathematical Modeling Analysis Activity This Guided Activity consists of three parts. Part 1 (Instruction) walks you through an example. Part 2 (Independent Practice) is where you will take what you have learned in the example, apply it to a similar problem, and analyze your findings. Part 3 (Quiz) is taken in Brightspace where you will enter your answers from Part 2. You need to complete Part 2 of the assignment before submitting your work for credit via a Brightspace Guided Activity quiz. Please note that no credit will be given for the assignment outside of the quiz. The quiz must be completed by the due date. You will have three attempts on each quiz. You have Guided Activities in modules 1, 2, 3, 4, 5, 6, and 8. Each assignment builds upon what you have done before, so it is imperative that you complete each assignment. MAT 375 Module Two Guided Activity: (Continuous) Dynamical Systems Our textbook has a series of Exercises and Examples dealing with a model of whales. The textbook starts off with relatively simpler models and then adds features. This Guided Activity is part of Exercise 5 in Section 4.4 of your textbook. Exercise 5 is based on Example 4.2 which is in turn based on Exercise 1 in Section 2.4. Module Two Guided Activity Page 2 of 31 July 2022 Section 2.4 Exercise 1. Ecologists use the following model to represent the growth process of two competing species, x and y 1 1 1 1 dx x r x x y dt K = − − 2 2 2 1 dy y r y x y dt K = − − The variables x and y represent the number in each population; the parameters ri represent the intrinsic growth rates of each species; Ki represents the maximum sustainable population in the absence of competition; and αi represents the effects of competition. Studies of the blue whale and fin whale populations have determined the following parameter values (t in years) Blue Fin r 0.05 0,08 K 150,000 400,000 α 10^(–8) 10^(–8) (The Exercise continues by asking several questions about population levels.) Instructor’s Note: Instead of x and y, we will use the variable B for blue whales and F for fin whales. It’s easier to remember! Module Two Guided Activity Page 3 of 31 July 2022 Example 4.2. The blue whale and fin whale are two similar species that inhabit the same areas. Hence, they are thought to compete. The intrinsic growth rate of each species is estimated at 5% per year for the blue whale and 8% per year for the fin whale. The environmental carrying capacity (the maximum number of whales that the environment can support) is estimated at 150,000 blues and 400,000 fins. The extent to which the whales compete is unknown. In the last 100 years intense harvesting has reduced the whale population to around 5,000 blues and 70,000 fins. Will the blue whale become extinct? Module Two Guided Activity Page 4 of 31 July 2022 Section 4.4 Exercise 5. In the whale problem of Example 4.2 we used a logistic model of population growth, where the growth rate of population P in the absence of interspecies competition is 1 P g(P) r K –P = In this problem we will be using a more complex model 1 P P g(P) r P P c – c – K = + in which the parameter c represents a minimum viable population level below which the growth rate is negative. Assume that … the minimum viable population level is 3,000 for blue whales and 15,000 for fin whales. a) Can the two species of whales coexist? Use the five-step method, and model as a dynamical system in steady state. b) Sketch the vector field for this model. Classify each equilibrium point as stable or unstable. c) Assuming that there are currently 5,000 blue whales and 70,000 fin whales, what does this model predict about the future of the two populations. d) Suppose that we have underestimated the minimum viable population for the blue whale, and that it is actually closer to 10,000. Now what happens to the two species? Module Two Guided Activity Page 5 of 31 July 2022 Instructor’s Note: This can be summarized as Blue Fin r 0.05 0,08 K 150,000 400,000 α 10^(–8) 10^(–8) c 3,000 15,000 Part 1, Instruction, provides • A CoCalc / SageMath based solution for Exercise 5 parts a, b, and c in Section 4.4. Part 2, Independent Practice, consists of • A CoCalc / SageMath based solution for Exercise 6 parts a, b, and c in Section 4.4. Part 1 (Instruction) In this module, we are going to investigate dynamical systems. It is important to build on what we have done before, so we are going to work through an example problem together, emphasizing the key concepts, and then you will build upon this on your own. In Module One, we analyzed discrete time dynamical systems. They are called discrete because time is measured in discrete increments (month 1, month 2, month 3, etc.). In Module Two, we analyze a continuous dynamical system (also called a real dynamical system) where time is measured as a real number. Solutions to these continuous time models often use differential equations instead of spreadsheet recursions. Section 4.2 in the text discusses continuous dynamical systems while section 4.3 discusses discrete dynamical systems. In the readings, Example 4.2 was an elementary population dynamics problem. Read the problem and example carefully before moving forward with this activity. We are going to use the same scenario, but extend it together, and then you will extend in a different way on your own. Scenario In the whale problem of Example 4.2, we used a logistic model of population growth where the growth rate of population, P, in the absence of interspecies competition is Module Two Guided Activity Page 6 of 31 July 2022 ( ) 1 – = P g P r P K This logistic model builds in a maximum population K. Applied to blue whales (B) and fin whales (F), we get two equations in our logistic model = 1 B B B – B g r B K = 1 F F F – F g r F K To that, we added an interspecies competition factor resulting in a logistic model where the two variables impact each other. Using interspecies competition factors cB and cF with cB = cF = α B F B’ = gB –cB F’ = gF – cF we get equations for the growth rates B’ and F’ of = − 1 B B B K –B' r B B F = − 1 F F F F K –' r F B F In this Guided Activity, we will start with a more complex model for each species, a model that not just includes a maximum population K but also a minimum population c. The following growth function includes a maximum value K for the population and a minimum value c. Module Two Guided Activity Page 7 of 31 July 2022 ( ) 1 – – = + P P g P r P P c K c We can add the interspecies competition factor to this model, getting the two growth equations = − + 1 B B B BB BB' r B B F B – c – K c = − + 1 F F F FF FF' r F B F F – c – K c In this Guided Activity, we will be using a more complex model ( ) 1 – – = + P P g P r P P c K c Here the parameter c represents a minimum viable population level below which the growth rate is negative. Assume that the minimum viable population levels are 3,000 for blue whales and 15,000 for fin whales. Assume that α =10-8 for the interspecies competition factor. a) Can the two species of whales coexist? Use the five-step method and model this scenario as a dynamical system in steady state to determine whether there are any possible coexistence equilibrium points. b) Sketch the vector field for this model. Classify each equilibrium point as stable or unstable. Conclude whether the two species can or cannot coexist. c) Assuming there are currently 5,000 blue whales and 70,000 fin whales, what does this model predict about the future of these populations? Solution to Question a: Step 1: Determine what question is to be answered. Module Two Guided Activity Page 8 of 31 July 2022 In this problem, we have a few things that the problem is asking