Local Stability of Competition Model At the equilibrium Y ce Y ke u v 25864

# Local stability of competition model at the

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Local Stability of Competition Model: At the equilibrium, ( Y ce , Y ke ) = ( 0 , 0 ) ˙ u ˙ v = 0 . 25864 0 0 0 . 057443 u v , which has eigenvalues λ 1 = 0 . 25864 and λ 2 = 0 . 057443, so this equilibrium is an Unstable Node At the equilibrium, ( Y ce , Y ke ) = ( 12 . 742 , 0 ) ˙ u ˙ v = - 0 . 25864 0 . 72649 0 - 0 . 0031847 u v , which has eigenvalues λ 1 = - 0 . 25864 and λ 2 = - 0 . 0031847, so this equilibrium is a Stable Node Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (65/68) 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 Local Stability of Competition Model Local Stability of Competition Model: At the equilibrium, ( Y ce , Y ke ) = ( 0 , 5 . 8802 ) ˙ u ˙ v = - 0 . 076620 0 0 . 027979 - 0 . 057443 u v , which has eigenvalues λ 1 = - 0 . 07662 and λ 2 = - 0 . 057443, so this equilibrium is a Stable Node At the equilibrium, ( Y ce , Y ke ) = ( 4 . 4407 , 2 . 9554 ) ˙ u ˙ v = - 0 . 090137 0 . 25319 0 . 014062 - 0 . 021428 u v , which has eigenvalues λ 1 = - 0 . 1246 and λ 2 = 0 . 01307, so this equilibrium is a Saddle Node Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (66/68) 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 Competition Model Phase Portrait: Plot shows nullclines and solution trajectory Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (67/68) 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 Competition Model Time Series: Plot shows the solution trajectories 0 100 200 300 400 500 600 0 2 4 6 8 10 12 S. cerevisiae S. kephir t (hr) Yeast (Vol) Competition Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (68/68)
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