Midterm 2. MAT17C, Spring 2014, Walcott

Transcription

1 Mierm 2 MAT7C, Spring 24, Walcott Note: There are SEVEN questions. Points for each question are shown in a table below and also in parentheses just after each question (and part). Be sure to budget our time accordingl. Student ID Name Section TA/time/number Please write our name at the top of each page (in case the staple fails). Please ensure ou have pages (including this page, the equation page and scrap paper at the end). Please bo our answers. Scrap paper is provided at the end of the eam; if ou need more, just ask. Calculators, books, etc. are not allowed; ou ma onl have a pen or pencil/eraser. Partial credit is available for all questions, but onl if ou show our work and it is legible. Read ever question carefull and completel Ma 2, 24; 2: pm 3: pm / /8 /8 / TOTAL /2 / /25 /25 /

2 Potentiall Useful Equations For these equations, unless otherwise stated, assume there is a function of two variables f(, ). The vector is [ = the notation f() = f(, ), the vector f() is f() = [ f (, ) f 2 (, ) f() f( ) + J ( ) where the Jacobian matri is J f f 2 f f 2 The solution to the sstem of ODEs d = M where the matri M has eigenvectors s and s 2 with associated eigenvalues λ and λ 2 is = as e λt + bs 2 e λ2t the constants a and b are determined b the initial condition () = as + bs 2. If the real parts of ever eigenvalue of the matri M are less than zero, then all trajectories head toward =. This point is then said to be stable. Eigenvalues real, positive Unstable Node. Eigenvalues real, negative Stable Node. Eigenvalues real, one positive, one negative Saddle (unstable). Eigenvalues imaginar, with negative real part Stable Spiral. Eigenvalues imaginar, with positive real part Unstable Spiral. Eigenvalues imaginar, with no real part Center (trajectories form closed paths) Linearizing about a fied point,, we get or, if u =, d = J ( ) du = J u 2

3 . ( pts.) Consider the linear ODE [ suppose = d = M and M has the following eigenvalue/vector pairs: λ =, s = [ λ 2 =, s 2 = On the aes provided below, sketch a trajector (t) starting at () = indicated as an and labeled I.C.). [ [ (this initial condition is I. C. 2. (8 pts.) Linearize the following non-linear ODEs about =, =. d d = e = 2e To receive full credit, please write our answer in matri-vector form. 3

4 3. (8 pts.) Suppose ou ve linearized a non-linear ODE about =, = and found d = [ 2 3 where = Classif the fied point =, = as one of the following: a. Unstable Node; b. Stable Node; c. Saddle; d. Stable Spiral; e. Unstable Spiral. [ () 4. ( pts.) Suppose that = [ and that the linear ODE d = M Given the following flow field, please do these three tasks:. Sketch the directions of the eigenvectors; 2. Label each eigenvector with the sign of the associated eigenvalue (positive or negative); and 3. Classif the fied point =, = as one of the following: a. Unstable Node; b. Stable Node; c. Saddle; d. Stable Spiral; e. Unstable Spiral

5 5. (2 pts.) Suppose ou have two coupled non-linear ODEs d d = f (, ) = f 2 (, ) (2) There are si separate plots below. Each one in the top row shows null clines. Each one in the bottom row shows a trajector, labeled a), b) and c), from left to right. Each trajector plot at the bottom corresponds to ONE of the null cline plots at the top. In the bo at the top left corner of each null cline plot, write the letter of the corresponding trajector plot. NOTES: For our reference, the flow is indicated at the same point in all plots. The null clines are labeled d/ = or d/ = as appropriate. The scale on all plots is the same. d = d = d = d = d = d = a) b) c) 5

6 6. (25 pts.) Speciation part II. Let s revisit speciation. Here, animals either have antlers (A) or the don t (). Antlered animals onl reproduce with other antlered animals; animals without antlers onl reproduce with animals lacking antlers (see picture, below). Both parents with antlers (A) A with -or- with A Both parents without antlers () No reproduction Successful Reproduction Successful Reproduction The number of animals with antlers N A and without antlers N change according to the following ODEs. ( dn A NA = N A N ) + N A N A + N K dn ( N = N N + N A N A + N K The terms on the right hand side are ) the chance of an antlered (top) or non-antlered (bottom) animal encountering a similar animal 2) a limit on growth due to scarcit of resources (the equations reduce to a resource-limited growth model if N = (top) or N A = (bottom)). a. (4 pts.) I ve used a computer to generate a series of trajectories (below, right). Based on this plot, please indicate the likel stabilit of the four fied points:. N A =, N =. (circle one) Stable Unstable 2. N A = K, N =. (circle one) Stable Unstable 3. N A =, N = K. (circle one) Stable Unstable 4. N A = K/4, N = K/4. (circle one) Stable Unstable ) Number without antlers (N ) K K/4 K/4 K Number with antlers (N ) A 6

7 b. ( pts.) On the plot at the bottom of the previous page, in the bo at the right, draw a phase portrait near the fied point N A = K/4, N = K/4. This bo represents a blow-up of the bo in the computer-generated plot at the left. To help ou, here is the Jacobian matri at that point: [ 3/4 J = 4 3/4 In order to receive partial credit, please show all calculations ou use in our phase portrait. In these equations, antlered animals have a fitness advantage the have twice as man offspring as animals lacking antlers. ( dn A 2NA = N A N ) + N A N A + N K dn ( N = N N + N A N A + N K c. (7 pts.) Below, I ve drawn the null-clines and flow direction at one point. At each point a g, please indicate the correct flow direction (to get full credit, our arrows should point in the correct quadrant). ) Number without antlers (N ) K K/2 dn = b a g f c d e dn K/4 2K Number with antlers (N ) A A = 7

8 Number without antlers (N ) K K/2 dn = I. C. dn K/4 2K Number with antlers (N A) A = d. (4 pts.) In the plot above, I ve re-drawn the null-clines. Based on our answer to c., sketch a trajector from the given initial condition (marked as an and labeled I.C.). 8

9 7. (25 pts.) Sliding on a W In the following problem, a hocke puck slides on an ic surface shaped like a rounded W (pictured in the top drawing) Force -2-2 Gravit applies a force on the puck, there is some (small) friction between the puck and the ice. We can determine its horizontal position () and velocit (v) from the following equations: d dv = v = ( )( + ).v The function ( )( + ) is the force eperienced b the puck due to gravit, and it is pictured above (bottom drawing). a. (2 pts.) There is a fied point at =, v =. Determine its stabilit and sketch a phase portrait nearb. 9

10 b. (8 pts.) Determine the stabilit and sketch a phase portrait nearb the fied point =, v =. The Jacobian associated with this fied point is: [ J = 2. It has the following eigenvalue/vector pairs: [ λ =.95, s =.95 [ λ 2 =.5, s 2 =.5 c. (5 pts.) Based on our answers to a. and b., circle the letter (a f) associated with the phase portrait most consistent with our local phase portraits. a. v b. v c. v d. v e. v f. v

11 Scrap Paper