interspecies competition from the second species Joseph M Mahaffy h

Interspecies competition from the second species

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interspecies competition from the second species Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (60/68)
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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 Parameters Competition Model: Assume a competition model of the form dY c dt = a 1 Y c - a 2 Y 2 c - a 3 Y c Y k dY k dt = b 1 Y k - b 2 Y 2 k - b 3 Y k Y c The monoculture experiments give the values: a 1 = 0 . 25864 a 2 = 0 . 020298 b 1 = 0 . 057443 b 2 = 0 . 0097689 The competition experiments give the best interspecies competition parameters a 3 = 0 . 057015 b 3 = 0 . 0047581 These experiments also fit the best initial conditions: Y c (0) = 0 . 41095 Y k (0) = 0 . 62579 More details for fitting a 3 , b 3 , Y c (0) , and Y k (0) are available from Math 636 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (61/68) 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 Fit Competition Model: dY c dt = 0 . 25864 Y c - 0 . 020298 Y 2 c - 0 . 057015 Y c Y k , Y c (0) = 0 . 41095 dY k dt = 0 . 057443 Y k - 0 . 0097689 Y 2 k - 0 . 0047581 Y k Y c , Y k (0) = 0 . 62579 0 5 10 15 20 25 30 35 40 45 50 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 — (62/68) 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 Equilibria for Competition Model Equilibria for Competition Model: Let the equilibria for S. cerevisiae and S. kephir be Y ce and Y ke , respectively Y ce (0 . 25864 - 0 . 020298 Y ce - 0 . 057015 Y ke ) = 0 Y ke (0 . 057443 - 0 . 0097689 Y ke - 0 . 0047581 Y ce ) = 0 Must solve the above equations simultaneously, giving 4 equilibria Extinction equilibrium , ( Y ce , Y ke ) = ( 0 , 0 ) Carrying capacity equilibria , ( Y ce , Y ke ) = ( 12 . 742 , 0 ) and ( Y ce , Y ke ) = ( 0 , 5 . 8802 ) Coexistence equilibrium , ( Y ce , Y ke ) = ( 4 . 4407 , 2 . 9554 ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (63/68) 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 Linearization of Competition Model Linearization of Competition Model: With equilibria Y ce and Y ke , let u = Y c - Y ce and v = Y k - Y ke ˙ u ˙ v = f 1 ( Y ce , Y ke ) u f 1 ( Y ce , Y ke ) v f 2 ( Y ce , Y ke ) u f 2 ( Y ce , Y ke ) v u v so the linear system is ˙ u ˙ v = a 1 - 2 a 2 Y ce - a 3 Y ke a 3 Y ce b 3 Y ke b 1 - 2 b 2 Y ke - b 3 Y ce u v , where a 1 = 0 . 25864 a 2 = 0 . 020298 a 3 = 0 . 057015 b 1 = 0 . 057443 b 2 = 0 . 0097689 b 3 = 0 . 0047581 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (64/68)
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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
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