Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of

Linear applications of systems of 1 st order des

This preview shows page 10 - 11 out of 17 pages.

Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Two Species Competition Model 1 Two Species Competition Model : Let X ( t ) be the density of one species of yeast and Y ( t ) be the density of another species of yeast. Assume each species follows the logistic growth model for interactions within the species. Model has a Malthusian growth term . Model has a term for intraspecies competition . The differential equation for each species has a loss term for interspecies competition . Assume interspecies competition is represented by the product of the two species. If two species compete for a single resource, then 1. Competitive Exclusion - one species out competes the other and becomes the only survivor 2. Coexistence - both species coexist around a stable equilibrium Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (37/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Two Species Competition Model 2 Two Species Competition Model : The system of ordinary differential equations (ODEs) for X ( t ) and Y ( t ) : dX dt = a 1 X - a 2 X 2 - a 3 X Y = f 1 ( X, Y ) dY dt = b 1 Y - b 2 Y 2 - b 3 Y X = f 2 ( X , Y ) First terms with a 1 and b 1 represent the exponential or Malthusian growth at low densities The terms a 2 and b 2 represent intraspecies competition from crowding by the same species The terms a 3 and b 3 represent interspecies competition from the second species Unlike the logistic growth model , this system of ODEs does not have an analytic solution, so we must turn to other analyses. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (38/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Competition Model – Analysis 1 Competition Model : Analysis always begins finding equilibria , which requires: dX dt = 0 and dY dt = 0 , in the model system of ODEs. Thus, a 1 X e - a 2 X 2 e - a 3 X e Y e = 0 , b 1 Y e - b 2 Y 2 e - b 3 X e Y e = 0 . Factoring gives: X e ( a 1 - a 2 X e - a 3 Y e ) = 0 , Y e ( b 1 - b 2 Y e - b 3 X e ) = 0 . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (39/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Competition Model – Analysis 2 The equilibria of the competition model satisfy: X e ( a 1 - a 2 X e - a 3 Y e ) = 0 , Y e ( b 1 - b 2 Y e - b 3 X e ) = 0 . This system of equations must be solved simultaneously. The first equation gives: X e = 0 or a 1 - a 2 X e - a 3 Y e = 0.
Image of page 10

Subscribe to view the full document.

Image of page 11
  • Fall '08
  • staff

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

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes