model considered before, but the prey obey logistic growth in the absence of predators. In

Transcription

1 5.2. First Orer Systems of Differential Equations. Phase Portraits an Linearity. Section Objective(s): Moifie Preator-Prey Moel. Graphical Representations of Solutions. Phase Portraits. Vector Fiels an Direction Fiels. Matrix Notation. Linear Systems Moifie Preator-Prey Moel. Example 1: Consier a preator-prey moel, where all the assumptions are like in the moel consiere before, but the prey obey logistic growth in the absence of preators. In this case, the equations moeling the rabbit an fox populations as functions of t can be given by ( R = αr 1 R ) βrf N F = γf + δrf Here all the parameters are as in Section?? an in aition, N enotes the carrying capacity of the rabbit population. Fin the equilibrium solutions for a particular example of this system, where the parameters have been chosen as follows. ( R = R 1 R ) RF 4 F = 4F + 2RF. 1

2 Graphical Representations of Solutions. Phase Portraits. Figure 1. R(t) an F (t) as functions of time. Figure 2. Phase portrait of the preator-prey system. (1) Sketch the solution curve in the phase portrait corresponing to the equilibrium solutions an the one corresponing to initial conition R(0) = 3.5, F (0) = 3. (2) Sketch the graphs of the solutions R(t) an F (t), corresponing to these initial conitions, as functions of t.

3 Vector Fiels an Direction Fiels. Let s come back to the mass-spring system (harmonic oscillator system) with mass m an spring constant k. Written as a secon orer ODE, it takes the form y + y = 1. Written as a system of 2 first orer ODES, it takes the form = v v = k m y. Choose k/m = 1. How o we sketch the the phase portrait of the system? Define the vector fiel F(y, v) =(v, y). Then, if we efine the vector r(t) =(y(t), v(t)), we can write the above system of ODEs as r = F. Let us plot the vector fiel F(y, v) at various points in the yv plane: F(1, 0) = F(0, 1) = F( 1, 0) = F(0, 1) = F( 2, 2) = F(2, 0) = F(0, 2) =

4 4 Given an autonomous system with of two ifferential equations, = f(x, y) = g(x, y), let r(t) = (x(t), y(t)) an let the vector fiel F(r) = F(x, y) = (f(x, y), g(x, y)) Then, we can write the system in vector form as follows. r = = f(x, y) g(x, y) = F((r)). Definition 1. (Direction Fiel) Given an autonomous system = f(x, y) = g(x, y), the the vector fiel corresponing to the system is a collection of vectors F(x, y) = (f(x, y), g(x, y)) at points (x, y) in the xy plane. This may cause ifficulty in interpreting the picture when the vectors are too long. If each of the vectors in the vector fiel is scale to the same (short) length, the resulting picture is calle a irection fiel of the system. Figure 3. Vector fiel. Figure 4. Direction fiel.

5 Matrix Notation. Example 2: Consier the system = x y = x + y. Note that it can be written in the form x y = x y Example 3: Write the unampe harmonic oscillator system using matrix notation. my + ky = 0. y v = 0 1 k/m 0 y v Example 4: Write the ampe harmonic oscillator system using matrix notation. my + by + ky = 0. y v = 0 1 k/m b y v

6 Linear Systems. Definition 2. (Linear System of ODEs) A two-imensional linear system with constant coefficients is a system of the form = ax + by = cx +, where a, b, c, an are constants. Using matrix notation, the system can be written in the form x y = a c b x y This form allows us to generalize the efinition of linear systems of ODEs to any number n of epenent variables y 1, y 2,..., y n. A constant coefficient linear system with n epenent variables is a system of the form 1 = a 11y 1 + a 12 y a 1n y n 2 = a 21y 1 + a 22 y a 2n y n... n = a n1 y 1 + a n2 y a nn y n In matrix form this can be written as Y = AY, where Y = y 1 y 2. an A = a 11 a a 1n a 21 a a 2n.... y n a n1 a n2... a nn

7 7 Theorem 1. (The Linearity Principle/ Principle of Superposition) If Y = AY is a linear system of ifferential equations, then it satisfies the Linearity Principle, i.e., (1) If Y is a solution an k R is an arbitrary constant, then ky is also a solution. (2) If Y 1 an Y 2 are two solutions, then Y 1 + Y 2 is also a solution. Example 5: Consier the system Y = Y. (a) Notice that this system is a partially ecouple system. Use this to solve it base in techniques for scalar ODEs. (b) Using the above, you can show that two solutions of the system are, for example, Y 1 = e2t 0 an Y 2 = e 4t 2e 4t (c) Using the Linearity Principle, what other solutions are there?

8 8 Example 6: Which of the following systems are linear systems of ODEs? = y 1 = x 1 = x2 1 = y = x + 2y = y = 2x = y = x = 2y = 1 y = 1 + x = x2 1 = y = x 2y = y Match each of the following irection fiels to one of the above systems. Figure 5. Fig. A. Figure 6. Fig. B. Figure 7. Fig. C. Figure 8. Fig. D.