{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

hw6sol - Homework 6 Solutions a = aL and = bK Eq(3 is...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Homework 6, Solutions a) α = aL and β = bK . Eq. (3) is obtained from (1) by dividing both sides by K . Eq. (4) is obtained from (2) by dividing both sides by L . b) ( x , y ) is a fixed point of (3), (4) if and only if x = 0 and y = 0 , (1) or 1 x α y = 0 and y = 0 , (2) or x = 0 and 1 y β x = 0 , (3) or 1 x α y = 0 and 1 y β x = 0 . (4) (1), (2), and (3) correspond to the three fixed points ( 0 , 0 ) , ( 1 , 0 ) , and ( 0 , 1 ) . The Jacobi matrix in ( 0 , 0 ) is J = bracketleftbigg r 0 0 s bracketrightbigg . This matrix has two positive eigenvalues, therefor the origin is an unstable node. c) To find a fixed point ( x , y ) in which the two species coexist, we have to solve the linear system (4). In matrix-vector form: bracketleftbigg 1 α β 1 bracketrightbiggbracketleftbigg x y bracketrightbigg = bracketleftbigg 1 1 bracketrightbigg . (5) The system is non-singular if αβ negationslash = 1. In that case, Cramer’s Rule tells us that x = 1 α 1 αβ and y = 1 β 1 αβ .
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern